A review of supersymmetry and supergravity

A REVIEW OF SUPERSYMMETRY AND SUPERGRAVITY

J. G. TAYLOR

Department of Mathematics, King's College, London, U.K.

C O N T E N T S

1. INTRODUCTION 1.1. Preamble 1.2. Towards the unification of all forces 1.3. The search for a new symmetry

2. SUPERSYMMETRIC MODELS 2.1. The Wess-Zumino model 2.2. Supersymmetric gauge theories 2.3. Supersymmetry breaking 2.4. Towards a realistic SUSY gauge model

*2.5. Extended supersymmetric gauge theories 3. SUPERGRAVITY

3.1. The construction ofsupergravity 3.2. Extended supergravities 3.3. Supergravity in higher dimensions 3.4. N = 8 supergravity and further symmetries 3.5. Spontaneous compactifieations 3.6. Superunitication

4. GLOBAL SUPERSPACE 4.1. The super-Poincar6 algebra 4.2. SupertieMs 4.3. The Wess-Zumino model in supcrspace 4.4. Supergraph rules

5. GAUGED SUPERSPACE 5.1. Supersymmetric gauge theories 5.2. N = 1 SYM in superspace

*5.3. N = 2 SYM in snperspace 5.4. The N = 3 barrier in gauged superspace 5.5. Quantum properties of supcrgauge theories

*5.6. SYM in light cone supcrspace 6. LOCAL SUPERSPACE

6.1. Super-differential geometry 6.2. Discovering auxiliary fields 6.3. Constrained superspace 6.4. Supcrspace supergravity actions 6.5. Supergravity ultra-violet divergences 6.6. The N = 3 barrier in local superspace

*6.7. Central charges 7. CONCLUSIONS AND FUTURE DIRECTIONS ACKNOWLEDGEMENTS REFERENCES

1 1 2 8

11 11 14 15 20 27 30 3o 32 33 35 38 40 43 43 47 52 53 55 55 56 58 60 64 67 71 71 74 79 85 86 88 91 95 96 96

1. I N T R O D U C T I O N

1.1. P r e a m b l e

S u p e r s y m m e t r y is a p r o p o s e d s y m m e t r y o f n a t u r e i n w h i c h b o s o n s a n d f e r m i o n s a r e

i n d i s t i n g u i s h a b l e . I n t h e f o r m i t is k n o w n a n d u s e d t o d a y i t w a s f i rs t i n t r o d u c e d b y W e s s

2 J.G. Taylor

and Zumino (1974a), after earlier discussions of a similar symmetry in dual resonance models formulated as two-dimensional field theories (Neveu and Schwarz 1971; Ramond 1971; Aharanov, Casher and Susskind 1971; Gervais and Sakita 1971) though there had already been an independent expression of this idea in a little-read paper by Golfand and Likhtman (1971) and a little later by Volkov and Akulov (1973) from the somewhat different viewpoint (of a non-linear realization in terms of a single spinor field). Initial interest in this symmetry grew rapidly both because it apparently allowed an intertwining of space-time symmetries and internal symmetries (Haag, Lopuszanski and Sohnius 1975) previously forbidden (O'Raifeartaigh 1965; Coleman and Mandula 1967) and also because it endowed non-trivial quantum field theories respecting it with less virulent ultra-violet divergences (Zumino 1974). The subsequent construction of supergravity, in which supersymmetry was now elevated to the status of a local gauge theory (Freedman, Ferrara and van Nieuwenhuizen 1976; Deser and Zumino 1976) increased this interest, especially because it hinted of the possibility of unifying gravity and matter. The combination of internal and space-time symmetries by means of extended supersymmetry (Salam and Strathdee 1974a) and the subsequent construction of extended supersymmetric gauge theories and supergravities has brought about an explosive interest in the subject. This has arisen both from the point of view of possible applications to high-energy physics and more recently early cosmology and nuclear physics, and also at the theoretical level of constructing a sensible quantum theory of gravity.

In this review we will try to describe the general ideas behind these developments and how successfully the hopes described above have been realised. We will also try to show where the main challenges presently lie in finally achieving the ultimate goal of all physics, that of constructing a completely unified theory of all of the forces of nature.

The subject of supersymmetry and supergravity has now a large literature. A review of the subject written in 1976 (Fayet and Ferrara 1977) cited 140 references, with a note added 9 months later including 50 more papers; a more recent review of supergravity (van Nieuwenhuizen 1981) has a list of 608 references. If we were to quote all the papers written on the subject to date (April 1983) this list would be at least doubled (and cover about 30 pages). It is therefore necessary to be selective at this juncture, which is how we have proceeded. Some important contributions may then have been unjustly neglected, but we can plead that this review is written for those non-experts interested in finding out what supersymmetry and supergravity are achieving without being exhaustive. In other words we regard this review as a large-scale map of the subject area, showing the principal features of the geography and the main roads, etc., without describing all of the secondary roads or the footpaths. There are sections in the review which are undoubtedly more technical than the rest (and are marked with an asterisk) but they are not used in further developments, so can be neglected for those only interested in the very largest scale map. Further advance into the subject is recommended by way of various School reports (Ferrara and Taylor 1982, Ferrara, Taylor and van Nieuwenhuizen 1983) and some of the numerous excellent review articles (Salam and Strathdee 1975, 1978; Wess 1981; Deser, S. 1980; Ferrara and van Nieuwenhuizen 1980) and earlier books (Hawking and Rocek 1981; van Nieuwenhuizen and Freedman 1980).

1.2. Towards the unification of all forces

The ultimate goal of physics is to give a description of the nature of matter using the fewest possible underlying concepts. The process of "whittling down" these concepts has

A Review of Supersymmetry and Supergravity 3

become, in the last two decades, that of unifying the forces of nature. The final aim is to describe nature in terms, hopefully, of one "super-force", with the particles we observe around us, or create under special circumstances, being various manifestations of the quanta associated with this superforce. The first step along this path was in fact taken over a hundred years ago by James Clerk Maxwell in his well-known unification of electricity and magnetism and continued by the successful discovery of the intermediate W-vector boson (Rubbia et al. 1983) on the basis of the Glashow-Salam-Weinberg theory.

Further unification of the resulting electroweak (EW) force with the strong force of quantum chromodynamics (QCD) into the so-called grand-unified gauge theory (GUTS) has been suggested on the basis of various larger groups starting with SU(5) (Georgi and Glashow 1974) but still awaits experimental validation by means of the observation of proton decay.

The remaining force of nature, that of gravity, is still to be satisfactorily unified with these other forces, even at the theoretical level. Many approaches have been taken towards this goal, numerous models suggested in which this unification occurs, but there are still a host of difficulties to be overcome in this quest for complete unification. These problems stem from two apparently different sources which appear to be ultimately related. They arise in attempting to answer the following two questions:

(1) How can we unify gravity with quantum mechanics? (2) How can we unify gravity, which can be regarded as the curvature of space-time,

with the other forces of nature which cannot so be regarded ?

Question (1) is difficult to answer for the reason that the standard approach to quantising a field theory, that of quantum field theory, gives poorly defined answers for any physical prediction when applied to the standard classical model of gravity, that of Einstein. The reason for this is the presence of the dimensional constant G or the associated dimensional length Ep = (hG/ca) */2 ~ 1.6 x 10- 33 cm (the Planck length), the Planck time tp = (hG/cS) 1/2 .,~ 0.5 x 10 -43 sec, and the Planck energy

Ep = (hcS/G) x/2 .~ 1.2 x 1019 B e V ~ 2 x 10- 5 gr,

when Planck's constant h and the velocity of light c are included, as they must be in a quantum theory of gravity. Any process involving gravity will be described by a quantum amplitude A(-- - ) , depending on the energies and momenta of the various participating objects, which we expect to be expressible as power series in G as

A(---) = ~. G " A , ( - - - ) . (1.2.1) n

On dimensional grounds we expect to expand A( - - - ) in powers of a dimensionless constant, which we take as ( E ) / E v , where ( E ) is an average energy involved in the process. Thus we may re-express A( - - - ) as

A ( - - - ) = ~ [ ( E ) / E v ] " B , ( - - - ) . (1.2.2)

We see that this expression may become poorly defined at energies

( E ) ~ E v (1.2.3)

when the perturbation expansion may fail to converge. Whilst this may be avoided by attempting to work outside perturbation theory we may have serious difficulties if the high-energy behaviour predicted by (1.2.2), that of increasing in ( E ) faster than any

PPNP-A~

J. G. Taylor

FIG. 1.

polynomial, is used in a rescattering effect, such as shown in Fig. 1. This depicts the process of graviton-graviton re-scattering, and integration over the energy-momentum transferred from 1 ~ 1' and 1' -o 3, taken as (E) in the scattering amplitudes for 1 + 2 -* 1' + 2' -o 3 + 4, respectively, will lead to a divergent result. It may be that the use of perturbation theory (1.2.2) to determine the high-energy behaviour of the scattering processes is completely misleading, but that feature alone differentiates gravity from the other forces. For these latter the direction of unification and predictions of the existence of charm and the W-boson were based on the criterion of renormalisability of the perturbative expansions of all scattering amplitudes in the relevant physical processes. Renormalisability here allows for certain ultra-violet divergences which are, however, removable (renormalisable) by means of the redefinition of suitable particle masses and coupling constants. The results for gravity indicated above show that there are an infinite number of ultra-violet divergences in the perturbation expansion of quantum gravitational processes; it seems unlikely to be able to remove an infinite number of infinite parameters using only a finite number of physical observables.

It might be claimed that the difficulty described above is avoided by a more careful and complete analysis of the situation. Thus we would follow the usual path of quantising the gravitational field potentials guv(#,v = 0, 1,2,3) about the flat Minkowski background quv = diag (1, - 1 , - 1 , - 1 ) as

gu~, = flu v + xh~,~. (1.2.4)

We then consider the Einstein-Hilbert Lagrangian density

Lr~n = x - 2 ~ / ~ " R (1.2.5)

(where g = det gu~, with g~,vO va = 6 I, R = RU'o~,, is the curvature scalar defined from the Riemann tensor Ru~ with

R # v P P a a p a = Fur,v- Fu,,v- Fu,F~ p + F~vFup

and the Christoffel symbol F~p being defined in terms of the metric gu, by

_ 1 , . , p a t , . , -'1- "

where the comma denotes differentiation; see any standard text on general relativity such as Weinberg (1972) or Misner, Thorne and Wheeler (1973) for a full development) as a power series expansion in x with quadratic term L o giving the usual kinetic energy for the spin-2 graviton field hu,. The interaction term at each order always contains, however, two derivatives, as can be seen from the Goldberg expression for Lnn (Goldberg 1958) in terms

of~ ~ = x / ~ O ~v and its inverse Ou~ as

Lr~ri = 8-~ {20wO~O,~-OP'~ O~,cO~, -46~ft]#~,}O'~'¢, pO~-'a~- (1.2.6)

The usual assessment of the degree of divergence of an L-loop perturbation graph with


L internal loops then gives the degree of overall divergence of the momentum space integrals as

d = 2(L+ 1) (1.2.7)

which increases without limit as L increases. The more careful analysis has not helped. There are various ways out of this dilemma. One is to modify the Einstein-Hilbert action

so that it becomes renormalizable at the quantum level but does not disturb the impressive agreement with experiment so far obtained by use of LEn alone at the classical level. This may be done by adding to LEa terms quadratic in the curvature tensor Ruva,, constructed from Our (Utiyama and De Witt 1962). These may be expressed as the linear combination

L' = aR 2 + bRuvRUL (1.2.8)

The resulting propagators now behave for large momentum like 1/k 4 for large k instead of 1/k 2 for LEt I, since the kinetic energy term in L' has four powers of derivatives. The resulting quantum theory may be shown to be renormalizable in perturbation theory (see the detailed analysis of this given by Stelle 1977) but has the drawback that there must be ghost particles present with negative probability. These particles cannot be decoupled (Nouri- Moghadam and Taylor 1976), nor can they be avoided by a suitable redefinition of the state-space of the theory without loss of causality (Dirac 1942; Lee and Wick 1969; Taylor 1970).

Another alternative is to suppose that the infinite number of regularisation parameters, corresponding to momentum-space cut-offs on the infinite set of divergent integrals arising in the perturbation expansion of quantum gravity, do not prevent non-trivial predictions being made in the theory. This has, in particular, been suggested, under the name "asymptotic safety", by Weinberg (1976), starting with the theory in (2 + e) dimensions with e small and positive. Such a programme has allowed renormalisability of quantum gravity to be proven near two dimensions (Gastmans, Kallosh and Truffin 1978). Since the Einstein-Hilbert action is a total derivative in two dimensions, and in three dimensions there are no physical modes of the graviton field this result is not easy to extend to four dimensions.

Yet a further approach is to suppose that the gravitational force is not a fundamental one but is more correctly an effective interaction arising from an underlying force which has more satisfactory quantum properties (Sakharov 1968). An interesting attempt along these lines has been made by Adler (1980) (see also Zee 1981), who proposed an underlying scale- and gauge-invariant theory. The Einstein-Hilbert action (1.2.5) is then supposed to arise by dynamical symmetry breaking due to the presence of a scalar field (though this latter is not essential if fermion condensates arise spontaneously). All of these theories require the presence of the quadratic terms R 2 and Ru,,R~", so suffer from ghost particles already discussed in association with (1.2.8). If we wish to construct a theory of quantum gravity which can be used, for example, to discuss the nature of the early universe before the Planck time tp was reached, then ghosts or causality violations which become apparent at such energies will be highly undesirable.

We might also hope that non-perturbative techniques might avoid the problems of non-renormalizability of gravity in perturbation theory. Thus the non-polynomial- Lagrangian approach (Salam 1971) was developed so as to provide a natural regularization of quantum gravity leading to a finite theory. However whilst the resulting theory can be proven unitary (Taylor 1972) the ambiguity of regularisation appears to be too great to give a satisfactory theory.

6 J.G. Taylor

Strong-coupling methods might also be used, such as that by perturbing the Einstein- Hilbert action not about a fiat background as given in (1.2.4) but taking the limit x ~ (Liang 1972), with x 2 = 16rrG. In this limit all spatial derivatives disappear from the dynamics, so that nearby spatial points are uncoupled and light cones collapse to vertical lines. Metrics giving representations of this case have been analysed (Pilati 1982, Isham 1983), but there will be expected to be severe problems over the ultra-violet divergences arising on perturbation expansion around this limit (but see Klauder 1981).

Another non-perturbative approach is to use the functional integral formulation of quantum gravity and attempt to sum over metrics (to some loop order) so as to obtain a damping factor in that manner (Wheeler 1957, Hawking 1978). However the Euclidean action for gravity is not positive definite because a conformal transformation can make the action arbitrarily negative. One may add the quadratic terms (1.2.8) to achieve positivity but then one again has the ghost problem mentioned earlier. The possibility of propagating torsion, which involves the quadratic terms (1.2.8), though now with a different interpretation, has also been discussed (Hehl, von der Heyde and Kerlick 1976; for quantum aspects see Neville 1982 and references) but this does not seem to help the renormalizability problems if ghosts are explicitly excluded (Sezgin and van Nieuwenhuizen 1980).

At this point it might be thought that these difficulties indicate the need to leave gravity unquantized. This approach is supported if one considers the difficulty of satisfactorily determining the background space-time on which any quantization is to be performed (see Taylor 1979a). However this retreat from attempting to quantize gravity would destroy any hope in unifying gravity with the other forces unless these latter were also considered in a non-quantized version. Problems over the nature of the measurement process in quantum mechanics have led some to try to follow this path, which was originally taken by Einstein himself. However the recent success of tests of quantum mechanics associated with locality and Bell's inequalities (Aspect, Dalibard and Roger 1982) and the almost impossible task of having to give as convincing an explanation of modem successes of quantum field theory (whose predictions agree with experiment to better than 1 in 106) but in classical terms, indicate that avenue to be unprofitable.

We can only return, then, to probe the ultra-violet divergences of quantum gravity in the presence of matter in more detail to see if there is any choice of the matter interactions which would cause these divergences to cancel. We need only require that this cancellation occurs of physical processes, or in other words for S-matrix elements; off-shell the Green's functions could be allowed to be very poorly defined. At one loop the divergences are known to cancel for pure gravity ('t Hooft and Veltmann 1974). However, this situation is lost in the presence of matter either as electromagnetic or fermi fields (Deser and van Nieuwenhuizen 1974) or as scalar fields (Nouri-Moghadam and Taylor 1975).

There is, however, a simple argument that leads to ultra-violet divergence amelioration. The vacuum fluctuations of the electromagnetic field are known to lead to the experimentally observed Casimir effect (Sabisky and Anderson 1973). We would expect, therefore, that finiteness of these vacuum fluctuations would be a valuable result; even a reduction of the total divergence of this expression could give an important indication of where to analyse in more detail. For free fields the contribution from bosons to the vacuum energy is 1(~2-+-m~)1/2 for each momentum/~ and degree of freedom, (where m8 is the associated boson mass), with the opposite sign for fermions. The free vacuum energy is therefore

½ ~ (2s + 1) ( - 1)~2" + ,) rda~(~p2 + m~)'/2 (1 o2w9) 8 d


where all modes of spin s are assumed, for simplicity, to have the same mass ms. The quartic divergence in (1.2.9) cancels provided

(2s + 1 ) ( - 1) 2s+1 = 0 (1.2.10) s

whilst the quadratic and logarithmic divergences also cancel provided

Y' (2s+ 1 ) ( - 1)2S+am2 = 0 (1.2.11) $

~ (2s+ 1 ) ( - 2s+ t_* 1) ,n~ = 0. (1.2.12) S

The requirement (1.2.10) is that of equality of the number of bosonic and fermionic degrees of freedom:

NF = NB. (1.2.13)

The condition (1.2.11) is required in any theory for which the vacuum energy is only logarithmically divergent, and is an important sum rule arising in many supersymmetric and supergravity models. We will discuss this and (1.2.12) at later points in the review.

We have learnt from the above that a bose-fermi "equality" is required for ultra-violet divergence amelioration of the vacuum energy for free fields. This feature leads us naturally to consider elevating this relationship between bosonic and fermionic modes to the level of a symmetry principle, that of supersymmetry mentioned earlier. This may not be the only way of ensuring ultra-violet divergence cancellation in interacting quantum field theories (Allcock 1980) but such a route has allowed many interesting interacting cases to be studied, leading to the complete cancellation of vacuum divergences (Zumino 1975) and to the discovery of completely finite interacting quantum field theories (Mandelstam 1982; Howe, Stelle and Townsend 1982). Ultra-violet divergence cancellation also occurs in supergravity, as we will discuss later.

A completely different possibility arises when we turn to question 2, that of unification of two apparently qualitatively different classes of forces. Gravity, we have learnt from Einstein, is described as curvature in our four-dimensional world, leaving no room for the others to be so considered. Nor can they be; the principle of equivalence is true only for gravity. Unification, at a qualitative level, can only proceed if we introduce some extra, possibly fictitious, dimensions. The other forces, besides gravity, are now to be regarded as curvature in these new directions. Such an approach was begun over fifty years ago (Kaluza 1921; Klein 1926) with the identification of electromagnetism as curvature in one extra dimension. Such an approach has been developed considerably since then, (see for example, Duff 1982, 1983). In particular the use of up to seven extra dimensions has allowed a very elegant formulation to be given of the largest possible unification of gravity with other forces so that all forces are now curvatures in some dimension. This result, which we will discuss at some length later, leads to the questions (i) are these extra dimensions actually accessible to us, possibly at a suitably high energy? (ii) do the extra forces, other than gravity, correspond to the ones we observe, or at least give predictions in agreement with experiment ? (iii) do the extra dimensions help to alleviate ultra-violet divergences ?

We will consider the answers to these questions that are presently available in later parts of this review.

8 J.G. Taylor

1.3. The search for a new symmetry

We wish to introduce a new symmetry, that of supersymmetry (or SUSY, for short) between bosons and fermions. Such a symmetry is clearly absent in nature, since there is absolutely no evidence for the existence of SUSY partners of the electron, muon, neutron or proton. We would wish, therefore, to combine SUSY in a non-trivial fashion with space-time symmetry so that the eigenvalues of P~ are not the same for superpartners. (Pz being the translation generators of the Poincar6 group ~ , which also contains rotations and Lorentz boosts, all generated by J~,v-)

We have already mentioned that there are strong criteria on the manner in which internal symmetries may be combined with the symmetry group of space-time, the Poincar~ group. These theorems divide into two sorts: those which are purely group theoretic (O'Raifertaigh 1965, Jost 1966) and those which are in the framework of S-matrix theory (Coleman and Mandula 1967). Under the assumption, used in the S-matrix approach, that the generators of the symmetry group G commute with the S-matrix, act additively on incoming many-particle states and connect particles with the same mass, we can summarise the results of these theorems as essentially: any Lie group G combining the Poincar6 group with that of an internal symmetry group g is of the form of a direct product ~ ® g (so that the generators of ~' and g commute). In particular, all particles in any irreducible representation of G will therefore have the same mass. Since no particle has the same mass as any other this latter result shows there can be no such group G which is a perfect symmetry of nature.

In order to implement the idea of a symmetry between bosom and fermions we must introduce a generator, which we can call S, which achieves the transformation of a boson into a fermion and vice-versa:

S= : boson ~ fermion. (1.3.1)

We have attached a spinor label e to S in (1.3.1) since we know that fermions are described by spinors, whereas bosons are represented by scalars or vectors. We could have included a vector label on S=, but this would have created fermions of spin ] from scalars, so would not correspond to any known fermions.

Since S= itself has the character of a fermion we expect it to satisfy anticommutation relations with itself of some form or other. By the attachment of the spinor label a to S we expect S= to behave as a spin ½ field under the Lorentz group, so we expect the commutation relation

EJz,, S j _ = ½(~z,S)= (1.3.2)

where we have defined

1 %, = ~ D;,, ~,]-

in terms of the four Dirac matrices ~,~ wi th the usual algebra [7~, 7v] + = 2~v. Since we do not wish S~ to bring about a translation we also expect i t to commute with P~,, so that

EP~,, sO_ = 0. (1.3.3)

If there is an internal symmetry group g we would also expect S= to commute with the generators of g. We are thus left with the problem of discovering the form of the anticommutator [S=,Sp]+. To do that we will consider how the action of S= is to be represented more explicitly on bosonic and fermionic fields.


If we are given a fermionic field ¢~(x) then it may be transformed into a scalar field A(x) by means of an infinitesimal spinor parameter e= by the "rotation"

6A(x) = -g~(x) (1.3.4)

(where we take ~ to be the usual Dirac conjugate spinor e+~°). If fiA and ~ have the canonical length dimensions - 1 and _3 (since in an action S d4x[(a~ A)2+ iff¢O] is dimensionless) then e has dimension ½. We can then vary A into by the rotation

6d/= i(¢e)A (1.3.5)

where the factor ¢ = duyu is required on dimensional grounds and the factor of i to preserve reality if A and ~, are real (in the Majorana representation) and the ~, then purely imaginary. We may regard the rotations (1.3.4) and (1.3.5) as produced by the generator Sa as

6T = i[gS, T]_ (1.3.6)

for any field T, and we assume that the e's anticommute with all other spinors. We can then deduce, by performing the SUSY transformation twice on A, using (1.3.4) and (1.3.5) but with different infinitesimals el and/~2, that

[ - 6 2 , 6 1 ] _ A = --62(e1~/)-~-61(~2~/) : ( - - e l j ~ 2 + ~ 2 ~ l ) A = 2(g2/~,1)A (1.3.7)

where p~, = idv. We may also use (1.3.6) to write

[62, 6x]A = - l-g2 S, [-~1 S, A] _] _ + [gl S, [g2 S, A] _] _ (1.3.8)

and the use of the Jacobi identity allows us to rewrite the r.h.s, of (1.3.8) as - [[~2S, et S] _ A] _, so that since e I and e 2 anti-commute

IS=, S#] + = - 2(pC)~# (1.3.9)

where C is the charge-conjugation matrix with C-I?~,C = -?~. In other words the generators S= are effectively the square root of the translation generators P~,. We can reduce the number of degrees of freedom in S~ by requiring it to be a Majorana spinor, so that

= s T c = S+70, and (1.3.9) can also be written as

[S=, SP] + = 2 ~ . (1.3.10)

The anti-commutation relations (1.3.2), (1.3.3) and (1,3.9) [or (1.3.10)], together with the usual commutation relations for the generators P~,, J,~ of the Lorentz group, form those for the generators of the supersymmetry algebra of Wess and Zumino (1974a), often called the super-Poincar6 algebra and denoted by 6 a t-

It is straightforward to extend this algebra to include an internal symmetry group S U(N) by adding an index i, onto S=, with 1 ~< i ~< N. This is most convenient to do in the chiral representation, with chiral projectors ½(l+i?s), where Y5 = )~o)~1Y2)~3, so that ~_+ = [½(1 +i75)~k']~ are the chiral projections of any spinor. Then we take i S~+ to belong to the fundamental representation N of SU(N) in the index i and its conjugate S=_~ to belong to the conjugate representation N. We modify the only non-zero part of (1.3.9) to be

i [ S : + , Sp_ j ] + = - 2 ( f C ) : + # _ 6~.. (1.3.11 )

This modified algebra is the N-extended supersymmetry algebra, which we denote by ~'~N; the case N = 1 clearly reduces to the case (1.3.9).

10 J.G. Taylor

Simple supersymmetry combines particles with spins differing by ½ into what are called supermultiplets. The simplest example is the pair (A, ~) of (1.3.4), (1.3.5) with spin (0,½). We can expect, however, that any pair (j,j +½) can be combined by SUSY, and we will later consider in some detail the cases (½, 1), combining spin -½ matter and a gauge vector field A, (supersymmetric Yang-Mills or SYM theory), and (~, 2), combining the spin 2 graviton field h,v and a new spin-~ field ~,~, the gravitino (supergravity or SGR). Extended supersymmetry has more than one SUSY generator, so is to be expected to combine particles spanning a range of spins. In fact if we regard the SUSY generator S~ as a creation or annihilation operator of a spin -½ the presence of N of these in N-SUSY should allow multiplets with particle spin varying over N/2 adjacent half-integer values. Simultaneously we expect non-trivial behaviour under the internal SU(N) symmetry group mentioned above. We have therefore combined particles with differing spins and SU(N) transformations into the same multiplet, in apparent contradiction to the No-Go theorems mentioned at the beginning of this section.

The reason why such a contradiction might be avoided is that the usual commutation relations for the generators of the Poincar6 group have been augmented by both commutation relations [in the cases (1.3.2) and (1.3.3)] and anti-commutators [for (1.3.9)]. The set of generators of ~1 is therefore not a Lie algebra but what is termed a graded Lie algebra or a Lie super-algebra. (Berezin and Katz 1970; Corwin, Ne'eman and Sternberg 1975; Kac 1975; Pais and Rittenberg 1975). Since graded Lie algebras were not considered in the proofs of the No-Go theorems discussed earlier it might be hoped that by means of supersymmetry (or more general graded Lie symmetries) particles of different spins and internal symmetries can be combined together into irreducible representations of the supersymmetry algebra, so as to have differing masses in such representations. This last criterion is crucial, as it was in the non-graded case, in order that a symmetry can be applied exactly to the known elementary particles.

This hope of avoiding the No-Go theorems by extending the Lie algebra symmetry to that of a graded-algebra was dashed by Haag, Lopuszanski and Sohnius (1975), who showed under similar assumptions to the non-graded case, that the maximal symmetry G of the S-matrix is the direct product of an internal symmetry group g with supersymmetry (with associated SUSY algebra ~N). This latter is actually allowed to be more general than that given by (1.3.2), (1.3.3) and (1.3.9) in that the generators i S~t+, Sot_ j can more generally transform according to conjugate representations of a compact Lie group (in terms of the labels i, j). If { T t } are the set of generators of this group then this latter generalization is

[S~+, Tt]- = (a~)iiS~+ (1.3.12)

and its complex conjugate, where a e is the hermitian matrix of the representation. There is also a further relaxation of (1.3.9) allowed by the introduction of what are called central charges and denoted by ziL These are defined by

i j [S~+, S~+] + = 2C~+#+ Z ij. (1.3.13)

Since the 1.h.s. of (1.3.13) is symmetric in the interchange ( ~ + , i ) ~ (fl+,j) whilst C~+#+ is an antisymmetric matrix, the central charge matrix Z ij must be antisymmetric. The epithet central is applied to Z ° since they are required to commute with all generators: those of YN and of #. In the case of zero mass particles the compact Lie group ff can be restricted even further to U(N), which is the situation we will consider in most detail from now on.

Since from the above result the mass operator P~ must commute with the internal


symmetry generators Te the mass of all states in an irreducible representation of g will be the same. There is, in fact, another case, incorrectly excluded by Haag, Lopuszanski and Sohnius, which is the Konopelchenko algebra. However this case still has the same mass for all particles of an irreducible representation (Flato and Neiderle 1982; Konopelchenko 1974). Thus in spite of achieving the remarkable new combination of particle states with differing spins and internal symmetries in a given irreducible representation of a symmetry group it will be necessary to break this symmetry either explicitly or spontaneously before we can hope to relate it to properties of the known elementary particles. In other words the Coleman-Mandula No-Go theorem (1967) has not been avoided, nor is it expected to be so by the addition of further symmetries which do not change the purely bosonic part of the theory, to which the original No-Go theorem still applies. In fact the addition of these further supersymmetry generators produces an even greater degeneracy of particle states of given mass. This means that we will have to work even harder to break this degeneracy and obtain agreement with the particle masses observed in nature. Before the possibility of SUSY breaking can be analysed in detail we will first have to construct theories which do have exact supersymmetry but which may then be broken in a controlled fashion.

2. SUPERSYMMETRIC MODELS

2.1. The Wess-Zumino model

In this section we propose first to construct field theory models which are supersymmetric and which may be applicable to nature. We will then discuss how this supersymmetry may be broken and what experimental limits can be put on the existence of super-partners. We will choose here the simplest SUSY possible, that of 6~ 1, without central charges or any internal symmetry group. This case can be developed purely in terms of component fields and without the use of superspace techniques (which we will certainly discuss later). Since these latter require a certain amount of investment in learning a new formalism we felt it better to delay its introduction till later in the review. Those readers who only wish to see supersymmetry in action at the simplest level can therefore read this and the next section (which deals with supergravity also in components) and then turn to the conclusions. Since there are too many components to handle satisfactorily when extended SUSY is being used it becomes essential that superspace tools be available since they allow multiplets of component fields to be manipulated simultaneously. Furthermore it is only through quantum field techniques in superspace that the fermi-bose ultra-violet divergence cancellation be achieved to all orders in perturbation theory. Those readers wishing to enter the field of supersymmetry or supergravity or understand the most recent developments are also advised to read through the next two sections but persevere through the later ones.

The simplest model of global supersymmetry is that based on the SUSY multiplet with spins (0,½). In order to have supersymmetry realized there must be equal numbers of bose and fermi degrees of freedom. Since the spinor ~k~ has 2 real degrees of freedom on-shell (4 real degrees reduced to 2 by the Dirac equation) there must also be two scalar degrees of freedom. By parity we expect these to be a scalar A and a pseudo-scalar B, so that we must extend (1.3.4) and (1.3.5) to the SUSY transformations

5 A = - - g ~

6 B = - -~s~k

3~k = #(A +?,B)~. (2.1.1)

12 J .G. Taylor

The free Lagrangian invariant under (2.1.1) is

1 2 2 i L o = ~ [(OuA ) + (OuB) ] +~ ~¢~0. (2.1.2)

If we now take e to be space-time dependent [but no derivatives of e in (2.1.1)] then the variation of Lo can only be proportional to Out:

(2.1.3) ~SL o = (OuU)Ju~

where Ju~, is the super current (Iliopoulos and Zumino 1974)

(2.1.4)

The super-current J~,~ is the usual Noether current of the symmetry being considered, that is supersymmetry, as is to be expected by analogy to the Noether-current associated with translation invariance (being the energy-momentum tensor Tuv ) and that associated with Lorentz invariance (being the rotation tensor Juv~)" Since the generator of SUSY, S~, now has a spinor index we expect the associated Noether current to be a vector-spinor, as it is from (2.1.4). Finally we note that

OuJ~ = 0 (2.1.5)

if the free-field equations of motion [--]A = ITB = ¢~0 = 0 are used, as again expected for a Noether current of a global symmetry. The super-charge Q~ = S d3xJo~ is therefore a constant in time.

If we calculate the commutator of two supersymmetries (2.1.1) we find that for A and B we obtain the result (1.3.7) but a similar result arises for ~k only if we assume the field equation ¢~O = 0. This is to be expected since there are only 2 bose degrees of freedom off-shell whilst there are 4 fermi degrees of freedom (for a Majorana spinor ~b). We must add to the set of fields (A, B, ~k) a further scalar F and a pseudo-scalar G. These fields must vanish on-shell, in order to preserve equality of bose and fermi degrees of freedom, but off-shell allow for closure of the full SUSY algebra. The fields F and G are called "auxiliary" fields, this epithet applying to the way they may be eliminated algebraically by their equations of motion. Such auxiliary fields are necessary in all extended supersymmetries; their description in these cases has proved one of the outstanding difficulties in the development of the subject, as we will indicate shortly.

In this particular case the only terms which we can add to L o of (2.1.2) which will be quadratic in F or in G, and have the trivial equation of motion F = G = 0 are proportional to F 2, G 2 and FG. If we diagonalize them we may take the additional term as ½(F2+ G 2) so that the total free Lagrangian is now

L o = 1 [(o,a)2 + (duB) 2 + F 2 + G2] + 2 ff¢~b. (2.1.6)

Since F and G must therefore both have (length) dimension - 2 their SUSY transformations must be proportional to g¢~O and g¢~5~b. The proportionality co-efficients and the manner in which F and G enter 6~k is now fixed by requiring SUSY-invariance of L0 and closure of the SUSY algebra of (1.3.9), and we have

6A = -gd/ 6B = -g)'5$ 3F = --ig~$ 6G = --ig(~5~k 6¢t = [i¢(a +y5 B) + ( F - y~G)]e. (2.1.7)


We can now determine possible mass and interaction terms which may be added to (2.1.6) and which are invariant under (2.1.7). The mass term can be seen, on dimensional and parity-conserving grounds, as involving i~k, A F and BG. Its detailed form, in order to have invariance under (2.1.7), is easily seen to be

L,,, = m(½~$ + A F - - BG). (2.1.8)

Elimination of the auxiliary fields F and G from (L o +Lm) by means of their equations of motion

F = - m A , G = - m B (2.1.9)

gives the usual mass terms -½m2(A 2 + B2); we could have taken these terms and the spinor term in (2.1.8) without any auxiliary fields, but would have the disadvantage off-shell of non-closure of the algebra.

A similar situation arises for the interaction terms. If only a dimensionless coupling constant is allowed, on grounds of renormalizability, then we may construct invariant parity-conserving cubic terms of dimension four and which are at most linear in the auxiliary fields F and G (so they can still be eliminated algebraically) as proportional to FA 2, F B 2, GAB, ff~,A, ff75 ~kB. A straightforward calculation shows the invariant interaction term to be

Lin t = g(FA 2 - F B 2 - 2GAB + ~d/A - ff75~bB). (2.1.10)

The interaction terms resulting from the elimination of F and G from (Lo + L , + Lin t) are

- ½ [ m A + g(A 2 - B2)'] 2 - ½ [ m B + 2AB] 2

so that the total Lag, angian in the massive interacting case, without auxiliary fields, is

i 2 [(a~ A)2 _ m2A 2 ] + ~ [-(t~/~B)2 _ m2B 2-] + 2 iff~¢ + gEffeA - ff75 ~kB

- m A ( A 2 + B 2 ) ] - g 2 ( A 2 +B2) 2. (2.1.11)

We note that the elimination of the auxiliary fields has generated quartic terms as well as new cubic ones. These have only one independent coupling constant, however, as required by supersymmetry. Moreover we could have written (2.1.11) down directly and checked its invariance under (2.1.1); however the structure of (2.1.10) is clearly simpler, with only one overall constant of proportionality. This feature is always associated with the explicit appearance of the auxiliary fields, for then the SUSY transformations (2.1.7) are linear on the fields. To arrive at (2.1.11) we eliminated these fields from (L o +Lm+Lint) using their equations of motion

F = - m A - g ( A 2 - B 2 ) , G = m B + 2 g A B . (2.1.12)

This procedure clearly introduces non-linearity depending on the details of the dynamics, for example by the presence of the coupling constant g in (2.1.12).

We could also have added other cubic or quartic or higher terms to Lint of (2.1.10). If these involved the auxiliary fields F and G non-trivially they would necessarily have required the introduction of couplings with a positive length dimension. By standard arguments along the lines of those discussed in Section 1.2 this would have led to non- renormalizable quantum field theories in perturbation theory, so would be quite unacceptable. The only other terms allowed with a dimensionless coupling constant would be proportional to a fourth order polynomial in A and B. This can easily be seen to be

14 J.G. Taylor

non-invariant under (2.1.7) for any choice of coefficients in that polynomial, so can be discarded.

The quantum field theory of the model described by (L o + L , + Lint) of (2.1.6), (2.1.8) and (2.1.10), or equivalently as (2.1.11), can be quantized in a manner preserving supersymmetry, and can be shown to be less ultra-violet divergent than one would have expected on simple power-counting arguments (Wess and Zumino 1974a; Iliopoulos and Zumino 1974). Only a single logarithmic divergence is present, leading to a common wave function renormalization of the fields A, B and ~O, and with mass and charge renormalization also being determined by this wave function renormalization constant Z as

m, = Zm, g, = Z3/2g. (2.1.13)

We will not discuss the quantum features further here, but delay a more complete analysis till superspace techniques are available, since these are considerably more powerful than component techniques.

2.2. Supersymmetrie gauge theories

We might expect that the multiplet (A, B, F, G, ~k) discussed in the previous section would be a suitable prototype to describe matter in a supersymmetric fashion. In order to relate supersymmetry to the modern developments of gauge theories we must also construct supersymmetric analogues of non-Abelian gauge theories of the Yang-Mills type. We will do so in this section; we will proceed by analogy with the somewhat extended discussion of the previous section, but not give details which are similar to that case.

The multiplet to be used to describe the SYM case is expected to have spins (½, 1). Since off-shell there are 4 fermi degrees of freedom per gauge mode and only 3 bosonic ones (given the gauge invariance of the vector field) we need a real auxiliary scalar field to give equality of the bose and fermi degrees of freedom. The resulting set of fields V u, 2 and D have transformation rules under global SUSY as

~V~ = g?~2 64 = i(¢3ye)+ i(?se)D 6D = -g~5~2

and the associated invariant action is

LYoM _~[F~(V)] 2 i 1 2 = + +

(2.2.1)

(2.2.2)

where F~v (V) = ~[~ Vv] (where we define antisymmetrization and symmetrization without a factor of ½:T[~3 = T , , ~ - T ~ , T~uv)= T~v+Tv~). We note that there is an additional gauge transformation involved in (2.2.1), since V~, acquires an additional scalar contribution in ~7~2. The multiplet (2.2.1) with Lagrangian (2.2.2) describes an Abelian gauge theory or the kinetic terms of a non-Abelian gauge theory if V~, 2 and D are taken in the adjoint representation of the gauge group G and an additional trace over the gauge group indices is taken in (2.2.2).

We may couple the above theory to charged matter supersymmetrically if the latter is described by a pair of multiplets described in the last section combined to make complex component fields. Thus we take (A, B, F, G, ~) to be complex, with real and imaginary parts transforming as (2.1.7); we take these parts A 1, A2 etc. to be defined as

1 1 A = ~ ( A I +ia2), B = --~_(B 1 +iB2). (2.2.3)

,/2 ,/e

A Review of Supersymmetry and Supergravity

. -" v.

" - (A,B) q' ",,t k

Vp.x t , / (A ,B )

V - - "', (A B)

(A.B) , , .// .,.., (A,B)

(A,B) - " ' , (A,B)

FIG. 2.

15

The resulting Lagrangian for the matter and gauge field is then the kinetic term for the matter field as in (2.1.6) with additional cubic and quartic interactions, plus a mass term if so desired, together with the free Yang-Mills Lagrangian (2.2.2) (Wess and Zumino 1974a)

L = r ~matter) 4-/(matter) -4- L TM + L (int) (2.2.4) ~ K ' e ~ ~ m a s s - -

Z (int) = g [ - D ( a 1 n 2 - - a 2 n 1 ) - - V u J I~ - - i]({ ( a 1 + Y 5 n l )~k 2 - - (A 2 + ~ 5 8 2 )~1 }] _½02 2 2 2 V/~(A, +A2+BZ~ +B 2) (2.2.5)

where J~ is the usual charged current of the scalar multiplet

Ju = A,~'uA 2 + BlUrB 2 - i~17~2 (2.2.6)

where A~uB = AO~,B-(OuA).B. The expression (2.2.4) is invariant under SUSY transformations combined with a suitable (non-linear) gauge transformation, whose details are too complicated to discuss in detail here. Elimination of the auxiliary scalars Fi, Gi and D give the expected kinetic term for the scalars Ai, pseudoscalars B~, spinors qJ~ and 2 and vector V~, with gauge coupling o V f l ~, together with a Yukawa coupling between 2, qJ~ and (A~ + ysBj) with strength 9, and quartic couplings with strength 9 given by the last term in (2.2.5) and the term (A1B 2 - A z B 1)2.

A similar situation arises when the gauge group is extended from U(1) to any Lie group G (Ferrara and Zumino 1974; Salam and Strathdee 1974b). We may assume G can be divided into a product of simple factors, and for each of these there will be an associated coupling constant 9. Without auxiliary fields there will be a gauge vector multiplet of vectors and spinors, all in the adjoint representation, and a set of matter multiplets in various representations, each composed of a scalar, a pseudoscalar and a Majorana spinor. With auxiliary scalars the SUSY-invariant Lagrangian will have the same form as (2.2.4)-(2.2.6), though now with more general commutator brackets instead of terms like D(A1B 2-A2B1). There will, in any case, still be the same general feature of the sum of gauge-covariant kinetic matter terms plus Yukawa terms coupling the gauge spinors to the matter scalars and spinors and I-D + (scalar) 2] couplings. On elimination of the auxiliary gauge pseudoscalar D a quartic scalar self-interaction is generated.

We may summarise by giving the vertices for the various interactions (after elimination of auxiliary fields) in Fig. 2, with straight single and double lines denoting the matter and gauge spinors, and dashed and wavy lines the scalars/pseudoscalars and the gauge vectors: All 3-line vertices have strength #, all 4-line vertices strength g2.

2.3. Supersymmetry breaking

In the previous section we described how supersymmetric gauge theories interacting supersymmetrically with matter fields could be constructed. The physical fields or particles

16 J.G. Taylor

for the gauge sector were the vector fields V, and Majorana (real) spinors 2, both in the adjoint representation. The physical matter fields were complex scalars and pseudoscalars and Weyl spinors (since we require complex representations); the interaction vertices were as shown in Fig. 2.

Since there are no SUSY partners of the observed particles we must discover ways of breaking the supersymmetry. If we return to the anticommutation relation for the SUSY generators S~, in the form (1.3.10), multiply both sides by ~o and set P~ = (H, 0) (the rest frame) we obtain

= + ] +. (2 .3 .1) 8~

Thus H, being the sum of non-negative operators of the form I-S, S ÷ ] ÷ is always positive or zero; in any state l a)

(a ln la> >t O. (2.3.2)

If there exists a state 10> in which

then by (2.3.1) we have

S~ 10) = 0 (2.3.3)

(01 HI 0) = 0. (2.3.4)

The condition (2.3.3) states that 10) is a supersymmetry-invariant state, whilst (2.3.4) allows us to consider it as a vacuum state.

There may be more than one state left invariant under the action of S~, but in any case the theory has at least one SUSY-invariant ground state, and so there is no breaking of supersymmetry. The problem of distinguishing between several ground states is by now standard, and we will not discuss it further here.

Supersymmetry may be broken by various mechanisms:

(i) explicitly, by addition of terms to the Lagrangian which are not SUSY invariant. For example we may add mass terms solely for the scalars and pseudoscalars in (2.1.8), so splitting the mass of the fermions and bosons. Since the basic commutation relation (1.3.9) can no longer be valid the argument we have just outlined is not applicable. Such an approach may be followed in this situation (see, for example, Dimopoulos and Georgi 1981; Sakai 1982), though it is somewhat unappealing. However, in the case of extended SUSY (N = 2 and 4 super Yang-Mills theories), to be discussed later, the addition of terms which explicitly break SUSY (with the added criteria that they preserve finiteness) seems the only way to proceed.

(ii) spontaneously (SSB), by obtaining solutions which break SUSY, even though the Lagrangian does not do so. SSB may be sought for either in the classical equations of motion (tre,~-level) or dynamically, by radiative corrections. We will see later when superspace aspects of gauge theories are discussed that only the first of these methods is possible: if SSB does not occur at tree-level it cannot occur dynamically if SUSY is valid.

We have also to include the breaking of the associated gauge group G so as to contain at least the standard 3-2-1 scenario of QCD and electroweak; this may be done by SSB in a well-known fashion. The above discussion associated with positivity of the energy indicates that SSB is different in kind in SUSY as compared to an internal symmetry group G. This is seen in Fig. 3(i), where the potential V, regarded as a function of some scalar field ~b (or its expectation value) has minima for ~b ~ 0, corresponding to the breaking of G but not of


(i) (ii)

Fic . 3.

V

J

17

SUSY (since V is zero at its minima). In Fig. 3(ii) V has a minimum at 4' = 0, so G is unbroken there. However since V is strictly positive there is no SUSY-invariant ground state, so that SUSY is broken spontaneously.

When a global symmetry is broken spontaneously a scalar zero mass (Goldstone) particle is excited. In the case of SUSY we expect a similar particle, though now with spin ½, called the goldstino. Its existence can be shown by a similar proof to that in the global symmetry case (Salam and Strathdee 1974c). This can be seen from the requirement that the anticommutator of the total supersymmetry charge Q~ defined in Section 2.1 with its associated supercurrent J ~ (defined, for example, in the Wess-Zumino model by (2.1.4)) equal to the energy-momentum tensor T~v

Tuv = (7,)~#[Q~, Jr#] + (2.3.5)

as can be shown by explicit calculation. The (constant) vacuum energy density E has then the value

E~/~ = <01 T~ 10>. (2.3.6)

We have already noted that supersymmetry is spontaneously broken if E # 0, so requiring

<0l [Qc,,J~]+ 10> # 0. (2.3.7)

We may rewrite the 1.h.s. of (2.3.7) using the definition of Q~ as the spatial integral of J0a, and performing an extra differentiation under the space integral to replace it by a 4-dimensional integral, as

<01 [Q=,Jva]+ 10> = fd 'xd~ '<01T[J~(x) , Jv#(0)]+ 10> (2.3.8)

Since the r.h.s, of (2.3.8) is a total divergence, in order that it be non-zero the integrand (under the derivative) must fall off for large x at most as 1Ix 3 (in Euclideanized space-time). The only intermediate state in <01 [J~a(x), J~/~(0)] +10> which could do that would be a massless state, and since it has to connect the fermionic operator J,~ to the vacuum this new state must have spin ½; this is the goldstino. If we denote this state by I if>, with coupling s t rengthf to the supercurrent:

<01J,~ [@~> = f(71,)~# (2.3.9)

then from (2.3.6), (2.3.8) and (2.3.9) we have

E =f2. (2.3.10)

The presence of massless fermions in a theory is a prerequisite for spontaneous symmetry breaking. However such particles are necessary but not sufficient for SSB to have occurred; it is also necessary that the massless fermions be created from the vacuum, as indicated by

18 J.G. Taylor

F S F S

FIG. 4.

(2.3.9). In particular these fermions must all have the same quantum numbers as the supercurrent itself under any unbroken symmetries. Since the supersymmetry current is not itself supposed to transform under internal gauge symmetries this means that the goldstino must be a singlet of all such symmetries. Finally we note that iffis zero at the tree-level then it will be so to all orders in perturbation theory, according to the "non-renormalization theorems" to be discussed later (it may acquire a non-zero value by non-perturbative methods, though this is not easy (Witten 1981a)).

When supersymmetry is spontaneously broken we expect there to be associated mass splitting between the scalars and fermions. If we use the condition that the supercurrent is given solely by the presence of the goldstino intermediate state, then the matrix element of Jm between a scalar and fermion in a multiplet, as given by Fig. 4, (the pole approximation)

( p 2 B I J v~ [ p 1 F ) = fT v(i/¢l)(a +b/~2)u (2.3.11)

with q = (P2 - P 1). We may also write the matrix element (2.3.11), at lowest order, from the supercurrent (2.1.4) as

(p2BlJ~,~ I p l E ) = ipl~,u~. (2.3.12)

If we apply qv to both (2.3.11) and (2.3.12) and equate we obtain the relation

( m ~ - m ~ ) = af, b = 0 (2.3.13)

which relates the mass-splitting to the coupling strength a of the goldstino to the multiplet and f, the goldstino decay coupling constant. This relation will be relevant to our further discussions.

To answer the question as to whether or not spontaneous breaking of supersymmetry actually does occur we have to find if the auxiliary fields of the theory can have non-zero vacuum expectation values. This is because in order that SSB of SUSY do occur we need (01 [Q, x ] + I 0) to be non-vanishing for some operator X. X has to be fermionic in order that [Q, X] _+ be a scalar (and so have a chance of having a non-zero vacuum expectation value), and using the SUSY transformation laws (2.1.7) and (2.2.1) we have

<01Q~,X#]+ 10> = (75~<01D 10> (01 [Q~, ff#] + 10) = ( O l ( F + y s a ) ~ l O ) . (2.3.14)

Thus only if at least one of the auxiliary fields D, F, or G has a non-zero value from the field equations can tree-breaking SSB of SUSY occur. In general the auxiliary fields enter into the Lagrangian as

½(02 + F 2 + G 2) - O d(S) - F f ( S ) - G g(S)

where d, f a n d # are polynomials of degree at most 2 in the physical scalars S of the model being considered. The auxiliary field equations of motion are therefore

D = d(S), F =f(S), G = g(S)


so that SUSY SSB occurs if the equations

d(S) = f ( S ) = 9(S) = 0 have no solution.

There are two independent mechanisms by which this may occur, corresponding to the requirement that either D or (F, G) have non-trivial values. The first of these cases (Fayet and Iliopoulos 1974) requires the presence of a U(1) gauge field to which D belongs (since a non-trivial group transformation of D would lead to its having vanishing vacuum expectation value) with the term ~D added to the Lagrangian (2.2.5), for some constant ~. Elimination of D gives the scalar potential

1 2 2 2 2 ~m (A 1 + A 2 + B 1 q- B 2) q-½[~ q- 9(A 1B2 - - A2B 1 ) ]2 (2.3.15)

which generates SSB of SUSY but not of gauge invariance if 0 < ~g < m 2 and SSB of both SUSY and the gauge group if0 < m 2 < ~g.

The second case (O'Raifeartaigh 1975) uses the matter multiplets of Section 2.1. alone and special forms of the resulting equations of motion for F and G so that they have no zero solution. This can be specified for a general group if we recognise that the scalar potential arises from two sources (i) from the elimination of the gauge-multiplet auxiliary field D (ii) from the elimination of the auxiliary fields F and G. The former arises from a generalization

of the first term in (2.2.5) to ~.#,,D,,(c~ + T"49), where T = are the generators of the internal a

symmetry group in the representation, with indices j, to which 4aj = (Aj + iBj) belong, and the summation is over all generators with associated coupling constant 9=- The potential resulting from elimination of the D,'s is

12 = 2 = [g°(4~+ T=~b)]=" (2.3.16)

For the other term we may again use complex notation for the scalars, with f and g determined by what is termed the superpotential, which is at most cubic in the components ~b i as

W (dP a,.. .) = a~d A + aij(aick j + aijk~)i~)j~) k (2.3.17)

(we regain the discussion of Section 2.1, and in particular (2.1.11) if we take i = 1, W((~) = 1 0 ~ 3 +½m~b2). Then f and g are given by the derivatives of W w.r.t, q~i, and the further scalar potential is

V2 = ~IDW/04) i 12. (2.3.19) i

We note that the Yukawa coupling of matter spinors to matter scalars, generalising the last two terms in (2.1.10), is now ¢i02W/Oc~iO~bjqsj (where ¢i are now taken to be !eft-handed spinors constructed from the original Majorana spinors of Section (2.1.)). The condition for a non-zero value for the F or G auxiliary fields is thus that the equations

(~b + T°~b) = 0 (2.3.20)

OW/Od A = 0 (2.3.21)

do not simultaneously have a solution. Since (2.3.20) is always satisfied by 4) = 0 we thus require that

OW/O~)i 1~,=0 :~= 0. (2.3.22)

20 J .G. Taylor

A simple example for which this is the case involves three multiplets O1, 02, 03 of matter fields, with

W = a~b 1 -b~l~b ~ "1- C~2(~3 (2.3.23)

with positive values for a, b, c; the equations (2.3.22) require tk22 to take both the value 0 and a/b.

We note that the superpotential (2.3.23) changes only by a phase under the transformation ¢1 --' ei~¢l, ¢2 - ' ¢2, ¢3 --' ei~ ¢3. This overall phase is eliminated in V 2 of (2.3.18) and may be removed from the Yukawa coupling by a suitable chiral transformation ~b --, e -i~/2 ~k. The model, and all similar models, have therefore a symmetry called R-invariance, with an associated integer-valued quantum number which can be assigned to the scalars and spinors transforming non-trivially under it, as above. Gauge vectors will be invariant under R-transformations, but their associated spinors 2 must carry non-zero R-quantum number if they are coupled to matter through the ~[¢~b interaction of (2.2.5) with ~ and ¢ not being inert under R. The interpretation of the R-quantum number as fermion or lepton number has been considered, we will discuss that in the context of applications of SUSY to particle physics in the next section.

2.4. Towards a realistic SUSY gauge model

The next step towards unification of the forces of nature is the grand unification of SU(3) of chromodynamics (the strong interactions) and SU(2)x U(1) of electroweak theory. The SU(5) GUT model has been a promising candidate for this due to the good features that (a) it is the group of smallest rank which is possible to use (b) it explains the equality of electric charge of the proton and positron (c) it gives predictions for sin 20w and mb/m~ in good agreement with experiment (d) it leads to the testable prediction of proton decay and the possibility of explaining the origin of baryon number in the universe. However there are the difficulties of explaining (i) the origin of the ratio of the grand unification mass MGUM

2 2 10-26 and the W-boson mass, M w / M G u M ,~ (in SU(5) (the gauge hierarchy problem)) and (ii) the origin of fermion masses and generations. We will not discuss question (ii) but (i) has received a great deal of attention from the aspect of supersymmetry.

The gauge hierarchy problem is concerned with the origin of the very small number 10 -26 or even smaller ones if the fundamental unit of mass is taken to be the Planck mass mp = Ep/c 2. It arises most crucially if elementary (Higgs) scalars are present to bring about spontaneous symmetry breaking with the desired hierarchical form of some very large vector masses (such as MGUM) and much smaller ones (such as Mw) . Radiative corrections may not allow this (Gildener and Weinberg 1976; Gildener 1976), and in particular the quadratic divergences usually accompanying elementary scalar fields may require "fine- tuning" of parameters to many orders of magnitude to achieve large gauge hierarchies (Susskind 1979). As an example a dimensionless bare mass #0 may be defined in terms of a cut-off A and a bare mass m o as/~o = mo/A. If the physical mass has a quadratic radiative correction m 2 = m 2 + A2go then

#2 = _ 92 (1 - m 2/g2A2). (2.4.1)

Thus #2 must be specified to O(ra2/g2A2), and if m ,-~ 1 GeV, go 2 ,-~ 1, A ~ Mou M then /~2 must be "fine-tuned" to 1 part in 103° ! Even finer tuning is needed in the standard gauge model if we take the natural cut-off to be the Planck mass, being now to I part in 1038.


We may avoid this feature if elementary scalars are excluded, as, for example, in technicolour (Susskind 1979) but the problems associated with unwanted chiral symmetries have led to somewhat unappealing complications ("extended technicolour ') of the original scenario. If we accept the presence of elementary scalars then supersymmetry would guarantee the absence of quadratic divergences, as we discussed in association with the Wess-Zumino model in Section 2.1 and associated with (1.2.11) in a more general context. Since the gauge models considered in the previous section were gauge generalizations of the Wess-Zumino model (for the matter sector) we expect the absence of quadratic mass divergences to persist to all orders; this may be seen by superspace techniques (as we discuss later). Moreover the masslessness of scalars can be absolutely ensured to all orders if they belong to supermultiplets with chiral fermions for whom masslessness is guaranteed by a chiral invariance. If the fermion mass were zero at the tree level they will then acquire a mass of order MS, which is the mass parameter describing SUSY breaking. The hierarchy problem has thus been solved if M~ ~ Mw, so that Higgs scalars are no more massive than Mw. However there is still the question of why M~ << Mx. Modulo that, SUSY is attractive to apply to solve the gauge hierarchy problem in GUT's.

If the standard SU(3) x SU(2) x U(1) model is extended to a SUSY version, each particle will have an accompanying superpartner. These are shown in Table 1, where the partners of the various gauge vector bosons, called gauginos, are taken to have spin ½ and not since there is no satisfactory renormalisable theory for the latter, and the bosonic partners of quarks and leptons are taken to have spin 0 rather than 1 since only sensible theories for the latter would require them to be gauge particles, with places already filled by the photon, W and Z bosons. Moreover the internal quantum numbers of superpartners are identical to their originals (though this would not be true in extended supersymmetry, which we discuss later). We note that we have included the massless Goldstino (or Goldstone fermion), since the usual Higgs-Kibble mechanism absorbing this into the related gauge particle, which then acquires mass, cannot arise unless supersymmetry is a local symmetry. However this requires the introduction of supergravity, which we have deferred discussing till later. The theory of coupled SUSY and supergravity is, in any case, non-renormalizable, so is difficult to discuss when radiative corrections are being considered, as we are presently.

An attractive possibility that the Goldstino is actually a neutrino was shown to be unsatisfactory (deWit and Freedman 1975) by the application of low energy theorems to

Table 1.

Particle Superpartner Symbol SU(3) rep SU(2) rep QEM. Spin

quark 3 2, 1 ~,½ ½ squark 3 2, 1 ~,½ 0

lepton 1 2, 1 1,0 ½ slepton 1 2, 1 1,0 0

gluon 8 1 0 1 gluino 8 1 0 ½

photon 1 1 0 1 photino 1 1 0 ½

I. vector 1 3 + 1, 0 1 Boson

Wino, Zino 1 3 _+ 1, 0 ½ Higgs 1 2 _+ 1,0 0

Higgsino 1 2 _+ 1, 0 ½ Goldstino 1 1 0 ½

22 J .G. Taylor

r-decay. The resulting suppression occurs for processes in which there are no other contributions to the conservation equation [see eqn. (2.1.5)]

~3~ < I J~ I ) = 0 (2.4.2)

than from the neutrino (Goldstino) pole. Since there are no boson-fermion mass degeneracies this is so for processes which involve only one neutrino and no photon. The neutrino pole residue in (2.4.2) (the neutrino emission amplitude) must therefore vanish at zero neutrino momentum, so being in contradiction with known results on r-decay. One way out of this difficulty is that the low energy theorems are invalidated at the order of the present neutrino mass limit of ~ 6 eV for v e and 1 MeV for vu; this only seems likely if unknown particles were present, but since their masses would need to be at least several GeV they are unlikely to help the situation. The other method is by absorbing into the gravitino, which we defer discussion till later.

If we return to Table 1 of superpartners of the known particles we expect that the gauge coupling constants of all superpartners are known from the original gauge couplings, following our construction of the supersymmetric actions in Section 2.2. There will be new matter self-couplings which are introduced to achieve SSB of supersymmetry by the mechanisms of the previous section; all of these parameters will, in general, be unknown. An immediate experimental restriction is that the masses of all superpartners to quarks and charged leptons must be greater than 15GeV, otherwise they would have been seen at DESY.

We can expect a more complicated structure for the Higgs sector than the minimal version of the standard model, since to give mass to both the up and down quark requires a Higgs field H and its complex conjugate H*, with couplings Ha~un and H*dLdR respectively. It is not possible for both H and H* to appear in the same supermultiplet of N = 1 supersymmetry, so that at least two Higgs doublets H and H' must be present, leading to physical charged Higgs scalars.

In order to be precise concerning limits on the allowed parameters by comparing with experiment it is necessary to discuss the various types of models in more detail. These can be separated into three types:

(1) Softly broken models, in which mass terms are added explicitly so as to break supersymmetry. If these are scalar mass terms they generate at most logarithmic divergences by radiative corrections (Iliopoulos and Zumino 1974; Girardello and Grisaru 1982), and so do not necessarily destroy gauge hierarchy preservation. The Goldstino does not appear in them, and they just have the minimal set of particles of Table 1, without G. There are restrictions on couplings and masses so that large parity violation or flavour-changing neutral currents do not arise and fine tuning is apparently needed to achieve this (Ellis and Nanopoulos 1982). However they are not attractive models, and have not been pursued much in this context (but see, for example Lahanas and Taylor 1982); we will however, turn to them later when discussing finite quantum field theories of N = 2 and N = 4 super Yang-Mills.

(2) Spontaneous breaking by the auxiliary field D being non-zero (D-type), as discussed in association with eqn. (2.3.15). The mass-breaking formula (2.3.13) imposes an important criterion here, since in this case it can be written more precisely, at tree-level, as (Ferrara, Girardello and Palumbo 1979)

E (m~- m2e) = Z gO (2.4.3) U(1)


where the sum is taken over the D-terms of the broken U(1) subgroups (with f of (2.3.9) being the vacuum expectation value of D, the Goldstino being the associated gaugino 2, the super-current J ~ being D(7~,2)~ and a being the gauge coupling constant).

For the standard group SU(3)x SU(2)x U(1) there are only two such U(1) generators, weak hypercharge Y and the third component 13 of SU(2). Unfortunately both Y and 13 vanish when summed over quarks (or leptons), so that

m~ = E m~. (2.4.4)

In order to have squarks heavier than quarks and sleptons heavier than leptons the gauge group must be enlarged. The simplest extension is by addition of a further 0(1) sub-group, with associated gauge boson Z (the zum) and superpartner Z (the Zumino). Squarks and sleptons will then have a mass of about half the IV, so about 40 GeV. The low energy features of such SU(3)xSU(2)x U(1)× U(1) models have been analysed (Fayet 1977, Weinberg 1982, Hall and Hinchliffe 1982). The models must satisfy the following properties: (a) anomaly cancellation for renormalizability (since the 0(1) gauge boson must be axially coupled to quarks and leptons to violate (2.4.4); (b) U (1) and 0 (1) must be traceless to have no quadratic divergences; (c) SSB of supersymmetry must occur.

It has proved difficult to satisfy all of these criteria simultaneously; if they are satisfied then there may be a loss of asymptotic freedom or too great a degeneracy (Sakai 1982). One way of cancelling the Lr(1) anomaly is to introduce mirror fermions and their superpartners (Fayet 1982). Another is to push the anomaly to a high energy by choosing a heavy sector of matter to achieve the SSB of supersymmetry but decoupled from the light sector (Barbieri, Ferrara and Nanopoulos 1982); the effect is still to have light masses of O(Mw) for squarks and sleptons. There are additional difficulties of coupling consistently to gravity, since the gluino is then required to be massless to all orders of perturbation theory (Barbieri, Ferrara, Nanopoulos and Stelle 1982); this will be returned to later in this section.

(3) Spontaneous breaking by F being non-zero (F-type), as discussed in association with eqn. (2.3.22). To avoid the difficulties associated with the mass sum-rule (2.4.4) the masses of squarks and sleptons are generated by radiative corrections after SUSY has been broken by F terms involving new matter supermultiplets. Such a programme has been carried through by a number of groups (Alvarez-Gaume, Claudson and Wise 1982; Dine and Fischler 1982; Ellis, Ibafiez and Ross 1982). The new exotic sector only communicates its SUSY breaking to the remaining particles by radiative corrections. For example a mass can be given to a squark 4 by the diagram of Fig. 5, where the gluon propagator has the self-energy insertion from an exotic particle A and its superpartner _~ with mass difference of O(M~) and average mass Ma; the squark mass will then be O(%M~2/MA). It is even possible to arrange these and other diagrams to generate a negative (Higgs mass) 2, so triggering S SB of S U (2) x U (1). Particle masses are rather model dependent in these cases.

The common general features of all of the above models are that the interactions of squarks, sleptons, gluinos, etc are nearly completely the same as of quarks and leptons, but the superpartner masses are unknown. The most important unknown parameters are the

A,A

Fro. 5.

24 J.G. Taylor

couplings f of the Goldstino G to a particle-superpartner pair and the supersymmetry breaking parameter M~ :f ~ Am/M~, where Am is the mass difference of the pair (which may depend heavily on the details of the SSB mechanism), and where (01 V I 0) = (M~) 4, V being the potential. In most models there is a conserved quantum number Ctwiddleness") conserved in all interactions; a superpartner can only decay into another superpartner, so that ~ ~ e + ~ is allowed, but ~--, e + v is not.

Various analyses have been made recently of the possible experimental properties (production cross-sections, decay modes) of superpartners that would be relevant to searches for them at existing high energy accelerators such as ISR and the/~p collider, or those presently under construction, such as LEP, CESR II, SLC or HERA (Barbiellini et al. 1979; Hinchliffe and Littenberg 1982; Jones and Llewellyn Smith 1982; Harrison and Llewellyn Smith 1982). In general the conclusions are that SUSY partners will be detectable if their masses are of O(Mw). e~ machines should be able to detect squarks and sleptons, though more difficult for gluino detection, which should be easier to observe in ep machines. A successful result of such searches would be remarkable; absence of any superpartners would still be of value in restricting model-building considerably.

We have spent some time on this question of bringing low energy experimental criteria to bear on supersymmetric versions of the standard 3-2-1 model since the non- supersymmetric model is now so well accepted. In spite of the lack of convincing success in constructing such a supersymmetric version we can also consider supersymmetric versions of the conjectured GUT's, for which there are the additional data of sin 20w and the proton decay lifetime Tp. This programme will have all of the difficulties of the non-GUT theories as well as additional problems. In particular the extra spinors and scalars needed to achieve SUSY and its SSB will reduce the E-function, so increasing the grand unification mass MGu M and hence the proton lifetime, as well as possible changes in the previously successful predictions of sin 20w and mJm~. For example with SUSY SU(5) with two doublets of Higgses it was found (Ellis, Nanopoulos and Rudaz 1981) that MGU M increased by a factor of 40 compared to the non-SUSY value, whilst sin20w increased from 0.215+0.006 to 0.236_ 0.002; such predictions seem to be model-dependent however (Igarashi et al. 1982). There have also been troubles over the predicted lifetime for proton decay, which might be

4 2 thought to have been reduced from depending on MGUM to MGUM by the process of Fig. 6 in which only one inverse power of MGu M arises from the propagator for H (Weinberg 1982; Sakai and Yanagida 1982). However the q~H vertex has an extra factor of m~, so that with further factors of (16n2) - 1 the rate still seems compatible with the current experimental limits (Ellis, Nanopoulos and Rudaz 1981; Dimpoulos, Raby and Wilczek 1982). An important feature of Fig. 6 is that bose statistics and supersymmetry require that the squarks are from different generations; nuclear decay will thus proceed mainly into K's, though this is not inevitable (Salati and Wallet 1982). Since the situation concerning such decay is presently uncertain we will have to wait for more data on this decay to clarify the situation.

We note that there could be quite widely different mass-scales at which gauge-breaking

FIG. 6.


and SUSY-breaking occur. For example we may consider the gauge hierarchy (Rajpoot and Taylor 1983a)

M* G* M-+~SU(3)* x SU(2)* x U(1)* --+ SU(3) x SU(2) x U(1)~WSU(3) x V(1)

where G* is the grand unifying gauge symmetry and the asterisk denotes that the theory is in the supersymmetric state. In particular we could choose G = SU(5), with matter multiplets in the 5 and 10 representations of SU(5). There is thus an extra mass scale M* at which there is breaking of SUSY, beside M w and M~. A detailed Lagrangian leading to the above hierarchical descent was outlined in the above reference.

Analysis of the renormalisation group equations shows that the prediction of sin 20w are very similar to those for the simple SU(5) GUT model

O w ( M w ) = ~ + 50tem(Mw) sin 2 9~s(Mw)

but the relation satisfied by the SUSY-unifying mass-scale M~ is modified by the presence of the SUSY-breaking mass scale M*, and is

9 1 n ( M ~ ) ( M * ) = rc (1--8~em~ 1-1 ~-g +In Mww ll~tem \ 30q/"

If the proton-decay lifetime z e is taken to be 1030 years, and so M~ ,-~ 1015 GeV, we find M* ~ 1 0 1 2 GeV, whereas if M~ were close to the Planck mass (with z e ~ 1045 years) then M * .~ M w . Proton decay therefore seems an important parameter in determining possible restrictions on M*. A value of M* as high as 1012 GeV would be expected to give a radiative mass to the Higgs particles of O(M*), and then we might expect decoupling of the latter from low energy processes (Applequist 1980). The most readily observable effect of a heavy Higgs sector would then be in a shift of the W and Z masses.

There may also be indications of the need for supersymmetry from earlier epochs in the history of the universe. The application of GUT's to the early universe has been particularly appealing as giving a possible explanation of baryon excess and the value of the baryon to photon entropy ratio ns/ny ..~ 10 -s to 10-10 (Weinberg 1979; Nanopoulos and Weinberg 1979) as well as giving a new inflationary universe scenario (Linde 1982; Albrecht and Steinhardt 1982) which avoids the problem of excess production of monopoles and prevents baryons having too great an entropy. The embedding of these ideas in a supersymmetric framework has no difficulties in principle, though there are new technical problems. For example super GUT theories tend to have several degenerate minima (Dimopoulos and Georgi 1981 ; Sakai 1982), and the wrong vacuum choice may be made, causing difficulties in generating enough baryon asymmetry (Nanopoulos and Tamvakis 1982). There are also problems over tunnelling from incorrect vacua, though there are suggestions that supersymmetric GUT's may have some advantages over conventional GUT's as far as the initiation of inflation and the length of the inflationary epoch are concerned (Ellis, Nanopoulos, Olive and Tamvakis 1982).

Finally we should note that there is the very important limit on the total number N v of neutrino species with neutrino mass less than 1 MeV:

Nv ~< 4 (2.4.5)

(Yang, Schramm and Steigman 1979). This limit arises from the fact that more than the limit given by (2.4.5) would increase the expansion rate of the Universe and so generate an unacceptably large value for the primordial 4He abundance of about 25 ~, if one accepts

26 J.G. Taylor

nB/n ~ > 2 x 10-10. The same result applies to the total number of nuino species (where a nuino is a light neutral spin-~ superpartner, such as a photino, etc). The bound (2.4.5) can be extended if one assumes further neutral particles are only weakly coupled and are more massive than 1 MeV (Steigman, Olive and Schramm 1979). If the photino is light, and decouples at a lower temperature than ~ 100 MeV, then (2.4.5) indicates that no further light nuinos can be tolerated. This is an important limitation on super unified models.

The limit (2.4.5) is also an important one for the case of locally supersymmetric theories of nature to which we will turn in some detail in the next section. The gauge field of the local supersymmetry transformation e~(x) will be a massless spin ~ field ~u~(x), the gravitino. If local SUSY is spontaneously broken then the resulting spin-~ Goldstino can be absorbed by the gravitino to give it mass (the super-Higgs effect). To determine this mass we can argue by analogy. In the case of the-W-boson, the mass of the W is M w ~ g < ~b), where g is the gauge coupling and ( ~ ) the vacuum expectation value of the Higgs field. We therefore expect to replace # by fe and ( ~ ) by the vacuum expectation value (M*) 2 of the auxiliary field generating the SSB of supersymmetry:

real 2 ,~ Ep(M*) 2 = (M*)2/mp. (2.4.6)

The massive gravitino will also contribute to the expansion rate of the early universe, and will untowardly affect the primordial 4He abundance if it is too light. For its decoupling temperature T d (for the +½ polarization state) in terms of that for the muon's neutrino Ta(vl, ) is (Fayet 1982)

Ta/Ta(v ~ ) ,,~ (m3/2/10- 5 eV)2/3. (2.4.7)

The gravitino will only decouple at an early enough time provided that

m3/2 ~ 10 -2 eV. (2.4.8)

After decoupling the gravitinos will cool off at the normal expansion rate, but at a lower effective temperature than neutrinos. In order that they do not give an unacceptably large energy density to the universe this requires (Pagels and Primack 1981)

m3/2 <,~ i keY. (2.4.9)

However the upper limit (2.4.9) is only for stable or long-lived gravitinos with mass lower than 1 GeV. If they have a mass greater than 10 5 GeV their lifetime for decay into a photon+phot ino (and other channels) is less than 10 -2 sec. In that case the gravitinos would have decayed before the neutron/proton ratio freezes out, and would have had a negligible effect on 4I-Ie abundance. We conclude that there are the limits (2.4.8), (2.4.9) or

m3/2 ~ 105 GeV. (2.4.10)

There have been recently numerous works on the use of unextended supergravity to give an alternative approach to the breaking of unextended supersymmetric gauge theories (Nath, Arnowitt and Chamseddine 1983; Ellis, Nanopoulos and Tamvakis 1983; Cremmer, Fayet and Girardello 1983; Gatto and Sartori 1983). These have stemmed from the construction of the coupling of N = 1 supergravity to N - - 1 SYM theories (Cremmer, Ferrara, Girardello and van Proeyen 1982). The chiral R-invariance associated with conservation of the "twiddleness" number mentioned earlier, is broken by the gravitino mass term (2.4.6) occurring if the local SUSY is spontaneously broken. This latter mass then gives mass to superpartners of leptons and quarks of order [ ma/2 "1- me,q l, where me, mq are lepton and quark partner masses. If ma/2 > 15 GeV then these and the photino and gluino masses also


generated in a similar fashion may be large enough presently to have escaped detection. As was stated earlier these models can only be regarded as effective actions due to the non-renormalizability of the unextended supergravity sector.

The final remarks we have to make on global supersymmetry are concerned with the problem of the concept of temperature in a SUSY framework. We have assumed that there is no difficulty in using this quantity in discussion of the early universe, but it has been pointed out (Das and Kaku 1978) that a globally supersymmetric field theory at a non-zero temperature T automatically has its supersymmetry broken. The reason for this was clarified more recently (GirardeUo, Grisaru and Salomonson 1980) as arising from the requirement of summing over fermi fields which are antiperiodic in time in the interval (0, ifl), with fl = 1/kT. Ward identities, describing the global SUSY, are obtained from the requirement that the Green's function generating functional Z(J) is invariant under a change of integration variables given by a global supersymmetry transformation. However the requirement of antiperiodicity of the fermi fields used in defining Z(J) only allow transformation of variables by such non-constant fields, under which the action is not invariant. Masses of superpartners now become split by an amount O(g2T2), and whereas as T increases spontaneous breaking of global symmetries disappears above some critical temperature this is not true for the SUSY breaking. It was also claimed by the above authors that there is no necessity for a Goldstino to accompany this symmetry breaking. This has been challenged more recently (Teshima 1983), and if this latter result is true then a massless Goldstino was present in the early universe if the latter is described supersymmetrically. Independently of this, finite temperature effects on early universe super- cosmology could be different than usually discussed, especially above the W boson mass.

We conclude that there is yet no evidence from high energy physics that global supersymmetry is relevant to gauge theories, either for the standard SU(3) × SU(2) x U(1) or for the still incompletely justified grand unified theories involving S U (5) or larger groups.

2.5. Extended supersymmetric gauge theories

We may try to construct local gauge theories which are also invariant under N-extended supersymmetry transformations as described in association with (1.3.11). This was first done for N = 2 shortly after the construction of the N = 1 case (Ferrara and Zumino 1974) and a little later for N = 4 (Gliozzi, Scherk and Olive 1977; Brink, Schwarz and Scherk 1977). We will briefly describe these Lagrangians.

For N-SUSY there are 2N operators S~+ and 2N operators S~_ i defined in Section 1.3. For massless states with p2 = 0, we may choose a representation with vacuum state I - 2 ) so that S ~ - i l - 2 ) = 0. The "creation" operators then raise helicity (the appropriate quantum number for massless states) by +½. A multiplet of states would therefore have the set of helicities

( -2 , -2+½, . . . , - 2 +½N) (2.5.1)

obtained by acting on I - 2 ) with successive powers of S~+. Since these operators are mutually anticommuting the degeneracy of the state (S+)r[2) will be NC,. Thus the set of states with maximum helicity 1, for N = 2 and 4, will have (helicity) degeneracy as follows

N = 2:(02,½2, 1) N = 4:(06,½ *, 1) (2.5.2)

on combining the set of helicities (2.5.1) and their (PCT)-conjugates with opposite sign. We

ppNP-B

28 J.G. Taylor

note that N = 3 has the set (2.5.1) equal to ( -1 , -½3 ,03 ,+½) and conjugates ± n3 ±3 1), and so giving the same as N 4; the on-shell gauge theories for N 3 and - - 2 , v ~ 2 ~ ~

4 will therefore be the same. For N = 2 the set of fields will therefore be the gauge vector A~,, a doublet of Weyl

spinors or equivalently a Dirac spinor x, a scalar S and a pseudo-scalar P; all of these fields will be in the adjoint representation of the internal symmetry group G. The Lagrangian of the theory is

Tr { -¼F~,vF ~'v + ½(D/aS) 2 q- I(D#p)2 + i:~Dx + ig~[x, s] + gx75 Ix, P] -½g2([S, p])2} (2.5.3)

where D~ is the gauge-covariant derivative D~, = a~+ ig[A~, in the adjoint representation, and F~v is the usual gauge-covariant field strength in terms of A~,. The N = 2 SUSY transformations under which (2.5.3) is invarient

tSAu = i ( gT~x - ~Tue) 3P = ~75e-g75x 6S = i(~e - gx) fix = [ a ~ F ~'~ + 75 [P, S'] + iDPys - DS]e (2.5.4)

where e is a singlet under G. The formulation in terms of a doublet of Weyl spinors ~i has an SU(2) invariance, together with an additional U(1) chiral invariance obtained by combining chiral transformations of ~b ~ and a complex phase transformation of (S + iP) (Fayet 1976).

The case of N = 4 SYM is similar to that of N = 2, but with the scalars S and P replaced by a 6-plet of S U(4) A o satisfying the reality condition

A.*. = .4 ij = e, OklAk, (2.5.5) tj

and a quartet of Weyl spinors x i transforming in the 4 of SU(4); all of the fields (A~,, x ~, Ao) are in the adjoint representation of G. The invariant Lagrangian is now

L = r r { -¼F, ,vF +½D A jO"A,j+ - ½ g ( ~ [ x j, Ao] - ~,[x~, A ti] _ )+ ¼02 [A,~, A u ] [A '.i, A k'] _ } (2.5.6)

with N = 4 SUSY transformations

6A~ = i(giT~x i - ~iT~e i) ~ Ai j = i( gu X ~ -~- I~ijkl~kx 1)

¢~xi "~ tr lavFl~V si -- ~ AiJs~ - 2 g [ Aik' Akj] - 8j (2.5.7)

where e c = C~ r is the charge-conjugate spinor to e, with ~ = e+T ° the Dirac conjugate. The representation of N = 4SYM has explicit SU(4) invariance but not U(4); the reality condition (2.5.5) prevents the extra U(1) invariance that could be achieved for the N = 2 case; this problem for N = 4 is caused by the terms ~ x A in (2.5.6).

The great interest in these two cases is that N = 4 SYM is the first quantum field theory discovered to be completely finite to all orders (Mandelstam 1982; Howe, Stelle and Townsend 1982), following early indications that this might be so by explicit calculation of the fl-function to three-loop order (Avdeev, Tarasov and Vladimirov 1980; Caswell and Zanon 1981; Grisaru, Ro~ek and Siegel 1980) and an argument in terms of anomaly multiplets with certain unproven assumptions (Sohnius and West 1981a; Ferrara and Zumino 1981). Such a remarkable result achieves one of the "holy grails" of quantum field


theorists which has been searched for ever since the introduction of quantum field theory over 50 years ago and the ensuing realization of the ultra-violet divergence problems which beset any non-trivial theories. We will present these proofs in due course (in Sections 5.5 and 5.6).

Having such a beautiful quantum field theory available would lead us to expect that nature would use it to avoid these u.v. divergence questions. Various attempts have been made to apply it to high-energy physics along the lines of the N = 1 applications of the previous section, thus either to the standard S U ( 3 ) x S U ( 2 ) x U(1) model or to GUTs. Since N = 4SUSY contains 4 N = 1 supersymmetries combined in a very special manner then all of the difficulties discussed in the last section will still be present. However we noted that there seemed to be too few constraints on the available models. We concluded that new experimental data is crucial to reduce this plethora of models. An alternative method is to impose N = 2 or N = 4 SUSY as a stronger symmetry than the case of N = i. We will deal with N = 4 SYM first, especially because it is the maximal N-SYM theory that can be constructed (N t> 5 SYM would have propagating fields of helicity 2 1> 3/2, as is clear from (2.5.1)).

Besides this attractive feature of N = 4 SYM and its finiteness to all orders, is the question of interpretation of the global symmetry group SU(4) associated with (2.5.6). Barring breaking this group in a very drastic fashion, the SU(4) symmetry may be conjectured to be observed in the family structure, and so leading to the expectation of a fourth family. A more economical model of this sort would be N = 3 SYM, but we have seen that this latter theory has the same physical fields as the N = 4 case, and so the same expected family structure.

In order to relate to the known spectrum of particles we have to overcome some difficulties that appear at the outset. All fields are in the adjoint representation of the group, so that SSB of any internal gauge symmetry will produce massless neutral currents in an SU(5) or SO(10) gauge group, an unacceptable result from low energy phenomenology (Rajpoot, Taylor and Zaimi 1982). This is because the vacuum expectation values of the scalar fields used to give masses to the gauge vectors by SSB, and of the gauge fields responsible for the neutral currents, will all lie in the Cartan sub-algebra of the symmetry group; the typical vector meson mass term [A~, (~b)] 2 will thus be zero in this case. This is a serious problem which does not seem easily solvable without using a larger group than SU(5) or SO(10). Provided there is at least one extra non-electromagnetic U(1), descent to 3-2-1 may avoid the above problem.

A second difficulty is that, even starting from N = 2 SUSY, multiplets suitable to contain fermions will have both left-handed particles and their oppositely-chiral (or helicity) particles (Witten 1981a). These particles will therefore be in the same representation of the internal symmetry group. For N = 2 this follows from (2.5.1), where helicities (-½, 0) and (0, +½) must be combined together, and a similar, but stronger, argument works for N = 4, where only the adjoint representation is present. There is, therefore, the problem of the existence of mirror fermions (Fayet 1981). It does not seem possible to give large masses to the unwanted mirror partners of known fermions as in the SU(2)L x S U(2)R x U(1) models (see Rajpoot 1982, and earlier references therein). We will have to consider this problem in more detail when symmetry breaking mechanisms are being constructed.

A further problem (Witten 1981a) is that all supersymmetries must be broken at once if one is. This follows from the anticommutation rule (1.3.11) being used in a similar manner to (1.3.10) being used to deduce (2.3.1); (2.3.1)is then extended to being valid when S~, S t on the r.h.s, are replaced by i + S~,, S~,i for each i. Then if ( I H I) > 0 for some state I ) we must

30 J.G. Taylor

necessarily have S~ I ) ~ 0 for each i, and I ) will therefore break each of the supersymmetries. We note that this difficulty is not present if explicit breaking of SUSY is used, and no spontaneous symmetry breaking.

Having recognized these problems we must now consider how we may break N = 4 SUSY. The scalar quartic potential in (2.5.6) is zero for non-zero (Ao), say along a generator T a of the internal symmetry group which is independent of (ij). However the value of the potential there is zero, as is true for any constant solution of the scalar field equations. Thus we cannot break N = 4 SUSY spontaneously from (2.5.6). The only mechanism, other than by non-perturbative effects (which appear unlikely: see Witten 1981 a) is by explicit breaking. This can be achieved, without loss of finiteness to all orders, by addition of purely scalar mass terms (Taylor 1983a) or fermi mass terms (Parkes and West 1983; Namazie, Salam and Strathdee 1983), the latter requiring an extra scalar cubic interaction term with coupling constant proportional to the fermionic mass. All of the mass terms must satisfy the mass sum rule (1.2.11) (with the vector mass set to zero) in order for finiteness to occur. The internal SU(4) is also split by the explicit breaking terms. The resulting Lagrangians then allow spontaneous breaking of the internal symmetry, though the (mass) 2 sum rule (1.2.11) must still be satisfied, though now with non-zero vector mass (Rajpoot and Taylor 1983b). This generalises a similar (mass) 2 formula (though not involved with finiteness) given earlier (Ferrara, Girardello and Palumbo 1979). Thus to ensure all scalars have sufficiently high mass it would appear necessary to have suitable massive fermion partners.

If we turn to N = 2 SYM, the attractive feature of finiteness persists to all orders provided that the l-loop fl-function is set to zero by a suitable choice of representations for matter multiplets besides the adjoint gauge multiplet (Howe, Stelle, West 1982). This may be achieved, for example, by taking 2N matter multiplets in the fundamental representation N of SU(N). We need to extend the Lagrangian (2.5.3) to include such matter multiplets, and this can be done directly for matter described by the N = 2 relaxed hypermultiplet (Howe, Stelle and Townsend 1982). SSB of the N = 2 SUSY is not possible in this case, (Rajpoot, Zaimi and Taylor 1982) and again explicit mass terms preserving the finiteness must be, and have been constructed (Parkes and West 1983; Rajpoot, Taylor and Zaimi 1983). These terms will allow SSB of the internal gauge symmetry, but should now avoid the first difficulty mentioned above since matter and gauge fields may be chosen to be in different representations. The possible application of these N = 2 SYM models look much more hopeful than for N = 4 SYM, in spite of the maximality of the latter. Clearly more work on this question is needed.

The situation is particularly exciting with finite quantum field theories as possible models still to be explored and much higher energy data becoming available in the next few years. It is clear that the final word in the presence or absence of global supersymmetry in nature is still a long way from being uttered.

3. SUPERGRAVITY

3.1. The construction of supergravity

If we follow the gauge approach we would expect to be able to construct a theory of fields invariant under local supersymmetry transformations, in which the transformation parameter of (1.3.4) is now space-time dependent. The commutator (1.3.7) of any two SUSY transformations will be a translation by the vector (~2y'el), which is a non-constant co-ordinate transformation. The gauge approach would then lead us to require the presence


of gravity in terms of a curved space-time, so leading to supergravity. The compensating gauge field for the local SUSY transformation e~(x) will the spinor-vector ¢~(x) already introduced in Section 2.4. The gauge fields for co-ordinate transformations will be the vierbein eua and the spin-connection w~ab, with 0 ~< a ~< 3, the latter field being the gauge field of local Lorentz transformations.

In second-order form, when ~C~ b is expressed in terms of e~, and ¢/1~ by means of its field equations, the multiplet (¢/1~, d,) is expected to transform as a multiplet of N = 1 SUSY with spins (5, 2). The transformation rules for such a multiplet, for constant e~, is

tSe~ = ig~'a¢ /1 (3.1.1)

6¢/1 = O'~Yae )t3 ~ e~. (3.1.2)

We may covariantize this w.r.t, general co-ordinate transformations and local SUSY transformations by replacing the r.h.s, of (3.1.2) by the covariant derivative of e

i b 6¢~ = D/1e, D~ = d~ + ~ w ~ trabe. (3.1.3)

This choice is clearly appropriate since it has the usual form of derivative of the gauge parameter for the variation of the gauge field. We write down the covariant Lagrangian expected to be invariant under (3.1.1) and (3.1.3) as the sum of the Einstein-Hilbert action and that of Rarita and Schwinger for ¢/1 :

1 e R ( e , w ) z ~v,z~ LSG R -- 21¢2 --~e ~/1?s~vD~¢. (3.1.4)

The first term is defined in terms of the SL(2, C)-field strength of the spin-connection w~ b:

R ~ = t3[~ w ~ - w'[~w~

R = "~/1ovl'~ab /1 b b " (3.1.5) • -a~b,,m, eae/1 = 6 a, e = dete/1

The second is the Rarita-Schwinger Lagrangian in which the gravitational effects are only through the spin-connection co-efficient w~ b entering in Da as in (3.1.3).

The invariance of (3.1.4) under the SUSY transformations (3.1.1) and (3.1.3) can be proved directly; we refer the interested reader to an analysis of this in van Nieuwenhuizen (1981). We note that the closure of the SUSY algebra (3.1.1) and (3.1.3) is only complete, when evaluated on fields, provided that the field equations are used; in particular

1-6(/~ 1 ), 6(/32)]¢/1 = ¢~ CK2T(~I~2) + (~ L(~l/t214~ b) + t~( -- ~1/~2¢/t) + (terms proportional to R,) (3.1.6)

where ~2 = ~2~'e~ and t~C_,CT, 6 L denote a general coordinate transformation and Lorentz transformation with the parameters in brackets; R , = e~,vaa~D~¢ ~ is the Rarita-Schwinger field strength. It is clear that there is need for auxiliary fields in order that the algebra close completely off-shell (without use of the equation of motion R/1 = 0), since the graviton has 16 degrees of freedom, reduced by 10 to 6 through the co-ordinate invariance of (3.1.4),

= • - d ~ %, and the local-SL(2,C) invariance

= e %,3¢/1 = ~ oo~w,

with e~b = _ebo. On the other hand the gravitino field has only 1 6 - 4 = 12 degrees of freedom, using the gauge invariance (3.1.3). Thus we have the mismatch between ~°(B) and

32 J.G. Taylor

d°(F), the bose and fermi degrees of freedom respectively, of value

~° (F) - t~°(B) = 6. (3.1.7)

We will return to this problem later in our discussion of auxiliary fields and superspace supergravity.

3.2. Extended supergravities

The simplest extension of the N = 1 supergravity constructed above is by addition of further fields. These latter at least must be in N = 1 supermultiplets; in order to obtain N-extended supergravity the total multiplet including the graviton must be in multiplets of N-extended supersymmetry. We will not consider the coupling of ordinary matter to N = 1 supergravity at this point, since it is more convenient to analyse by using superspace arguments, but limit ourselves to the attempts to construct N-SGR's.

We proceed by analysing the possible multiplets of fields appropriate to N-extended supergravity. Using a similar analysis to that of Section 2.5, especially associated with equation (2.5.1), the suitable multiplets with maximum helicity 2, for various N, are:

N = 2:(1,23-2,2) N = 4:(02,½, 1,34,2) N = 8 "raT° 156 12s as 2) (3.2.1) . t v , ~ , ~ , ) - ,

(where PCT-conjugate states have been included). Thus N = 2 supergravity will have a Maxwell field A~, and two gravitini ~k~ as well as the graviton. It was first constructed (Ferrara and van Nieuwenhuizen 1976) by coupling the matter multiplet composed of a spin 1 and a spin 3 fields--the (3, 1) multiplet--to the (2,3) graviton multiplet discussed in the last section. The method to achieve this was by means of the Noether technique, which uses the fact that a globally supersymmetric action I will have a SUSY variation proportional to

f d4 x( O t, e)j~v

for a non-constant infinitesimal SUSY parameter e(x), where j~ is the so-called Noether current, as we described in Section 2.1. One may start an iterative procedure by adding extra terms to the action and transformation laws to ensure that the complete action is invariant; the first additional term will be that from couplingj~ to the associated gauge field, which is the gravitino in this case. The resulting process stops after a finite number of steps; for N = 2 SGR the action is

e R e ~i F~VpD d~i e F2 4- K ~ i retF~V + ff~v) + y5 (~i'v + ~'v)] (3.2.2) -~ - - -~ .~ w~-~ ~ - ~ - - ~ . ~ •

where ff~ = ½ei,~aF ~ and ff~ is the supercovariant curl

1 i i ij (3.2.3)

(so called because 6P~ does not involve the derivative of e~). The action (3.2.2) is invariant under two local SUSY transformations, local Lorentz and general co-ordinate transformations as well as gauge transformation tSA~ = a~A. The theory has a manifest global


S U (2) invariance, under which the ~b~+ and ~b~-i transform as _2 and ~_; on-shell there is a further chiral U(1) invariance, with

6~k ~ = - i~s~k ~, t~ P ~,v = iee~,v~,~F p'r (3.2.4)

(where the equation of motion for the photon field is necessary for the U(1) invariance to be valid).

This N = 2 theory may have the photon field coupled directly to a non-zero charge # on the gravitino doublet (Das and Freedman 1977, Fradkin and Vasiliev 1976). An invariant action can only be constructed in this case provided a cosmological constant and a gravitino mass-like-term be added. The extra terms are

6e# 2 + 2 e ¢ ~ t # ' v ~ (3.2.5)

with modification of the covariant derivative D~ in (3.2.2) so as to be gauge-covariant; the mass-like term is needed to make the gravitini massless in de-Sitter space. These questions, and the related one of removal of the cosmological constant, will be discussed in more detail later.

N --- 3 SGR was constructed a little later in terms of its physical fields, with three gravitini, three Maxwell fields and a single spinor (Freedman 1977; Ferrara, Scherk and Zumino 1977), with a similar cosmological term and mass-like term for the gravitini if the vector fields are used to have a non-Abelian gauge invariance under 0(3); this invariance extends to a global U(3) if the gauge coupling constant # is set to zero.

N = 4 SGR has been constructed in two versions, one with explicit global SO(4) invariance (Das 1977; Cremmer and Scherk 1977) or with SU(4) invariance (Ferrara, Cremmer and Scherk 1978); both versions have also been gauged (Das, Fischler and Rocek 1977; Freedman and Schwarz 1978). These theories are of interest in their own right, but since they can be obtained by contracting the N = 8 version this is more important to consider. This latter theory originally proved impossible to construct directly in four dimensions and is most natural to analyse by dimensional reduction from 11 dimensions. We will therefore turn to consider supergravity theories in higher dimension. There has been a great deal of interest in such theories recently, and there are reasons to consider higher dimensional versions of supergravity in their own right, as we will see.

3.3. Supergravity in higher dimensions

Spinors in higher-dimensional space-times are themselves of higher dimension than are those in four dimensions. This can be seen from the dimension of the associated Dirac matrices FM, where 1 ~< M ~< d, with the usual anti-commutation relation

[FA, FB] + = 2r/A~ (3.3.1)

where ~/AB = d i a g ( + l , 7 -1 , - -1 , - - - ) is the Lorentzian metric. The dimension of the irreducible representation of these matrices is 2 [d/2], where [d/2] denotes the integral part of d/2. This dimension is 32 for d = 10 or 11, so that exactly eight 4-dimensional spinors can be accommodated in one l 1-dimensional spinor; this would correspond to N = 8 supergravity in d = 4 arising from N = 1 supergravity in d = 11. Any higher number of dimensions would not seem to be allowed, since we have already noted that N = 8 is the maximum extension number for 4-dimensional supergravities to contain particle spins no higher than two.

The expression of N = 8 SGR in d = 4, as N = 1 SGR in d = 11 should be expected to

34 J.G. Taylor

produce a more elegant formulation dueto lack of external indices. This was demonstrated explicitly by the construction of the d = 11 theory of simple supergravity (Cremmer, Julia and Scherk 1978) in terms of the gravitino e~ (the "elf bein" equivalent of the vierbein of d = 4), the Rarita-Schwinger field ~ku~ and an antisymmetric potential A u N e with related field strength FMNe = 4d[uANe] (all capital indices run from 1 to 11, tangent space and world indices being chosen from the beginning and end of the alphabet respectively). As usual the spin-connection field w~a gauging the local SO(I, 10) transformations does not propagate since it has no kinetic term. The action density is remarkably simple, being

1 + ~ E(f fMFMNWXYZ~bN + 12~kWFXr~kZ)(Fwxrz + F w x Y z )

2 e,M~._.MHF r, .~ ( 1 ~ MIMzM3M4"CMsMtiMTMs l't MgMIoM1 x

where E = det e~t, wuaB = w~aB(e) + KMaS where the contorsion tensor

I NP KMaZ = ~ { -- f-PNFuAB~be + 2(ffurn~'a - ffuFA@n + ffSrM~ba)}

(3.3.2)

with the usual 2nd-order value

W~AB=½(-- f~AB+t2~bU--~A) with ~ = 2 O t N e ~ .

Tangent space indices are raised and lowered by qaB and its inverse gas whilst world indices A B raised and lowered by the metric gun and its inverse gun with gun = eueNr/AB. The

matrices F A, -A,~ = FEA~_ Fat,] whilst the quantities P and k are supercovariant forms of F and w. The SUSY transformation laws under which (3.3.2) is invariant are

6e~ = - i~T A ~I M

3 T 6AMN = -~e [MN~IIp]

i N P Q R FeQRfN ^ 6~bU = DM(V~)e+-~(FM --8 M)FIvpQR8 (3.3.3)

with

DM(W)~ = aM~ +¼WMaBFaSe _.~ 1" NP ~'MAB WMAn +~NF MAS~'e

F MNPO. = F MNpQ -- 3t~[uFNe~QI.

The supercovariantizations ensure that 5~ and 6i~ do not contain ~u~. The form of (3.3.2) and (3.3.3) was derived by starting with the linear terms in 5E and 5~#; to cancel the variation ~ F 2 from F 2 in (3.3.2) the coupling term ~X~F and a ZeF variation term in 6~u were needed, with some unknown matrices X and Z. These latter are fixed by requiting the equations of motion for ~ to be supercovariant, and the remaining terms are fixed by similar techniques.


3.4. N = 8 Supergravity and further symmetries

The above theory of N = 1 SGR in d = 11 reduces to the expected set of fields for N = 8 SGR in d = 4 in the case of assuming trivial dependence of all fields on the extra dimensions x S , - - - , x 11. Thus if we denote the world indices 1 , - - , 4 by #, 5 - - - 1 1 by m and corresponding tangent space indices by ~ and a we can decompose the fields as

( ~ ~ ) (3.4.1) e~ = e~

(3.4.2)

(3.4.3)

A MN L = (A#v~, A#vt, A,nl, Amnl)

~' M~ = ( ~' ~ , ~' m~ )

where we have chosen a special "triangular" gauge for e A as shown, and assumed e~ and b e a are symmetric in this gauge. We may regard ev as the degrees of freedom describing the

graviton in d = 4, whilst ~ denotes a set of 7 vectors, ~ a further set of 28 scalars. Similar interpretations of the other modes can be given in (3.4.2) for AMNL and (3.4.3) for SM~. If we decompose the latter into 8 spinors in d = 4 we can write the (spin) multiplicity content of the above field decompositions (3.4.1)-(3.4.3) as

e~ = 2+ 17+028 (3.4.4)

AMNL = 121 + 07 + 35 + ~b (3.4.5)

~kM~ as ± !s6 (3.4.6) ~ - 2 T 2

The total content is therefore the expected set (from Section 3.2)

(2,~ s, 128,156,070) (3.4.7)

these being the fundamental fields of N = 8 SGR in d = 4 (though with different parities than naively expected).

The remarkable feature of the d = 11, N = 1 SGR presented above is that all of the fields are gauge fields, with no superfluous "matter" fields whose number and internal symmetry properties may be arbitrary. This property of the theory has led to the suggestion that the correct number of dimensions for the Universe is not 4 but 11, and that the reason why we only observe 4 is that the extra 7 have been "spontaneously compactified" to dimensions of order of the Planck length, 10- 33 cm. This is, indeed, the logical outcome of taking seriously the Kaluza-Klein approach. The idea was that of unifying gravity, regarded as the curvature of 4-dimensional space-time, with other forces also considered as curvature, but now in further dimensions. These extra dimensions are then expected to become observable at some energy, expected to be O(me), since that is the only dimensional constant available. Such an approach is clearly of great interest, and has led to a great deal of recent activity. We will discuss that in the next section, but here consider the four dimensional aspects a little further, both from their intrinsic interest and from their relevance to the spontaneous compactification programme.

Dimensional reduction of the above theory (3.3.2), (3.3.3) by the assumption of triviality in the higher dimensions (xS, - - - , x 11) was performed in detail (Cremmer and Julia 1979). They demonstrated that the theory does indeed have a satisfactory 4-dimensional limit, but moreover has a large "hidden" symmetry, which may be deduced by using the residual

PPNP-B*

36 J.G. Taylor

symmetries of (3.3.2) after trivial dimensional reduction. These remaining symmetries are

(i) general co-ordinate invariance in 4-dimensions (ii) global GL(7, R)

(iii) Abelian gauge invariance for A~ (iv) local Lorentz invariance in d = 4 (v) local SO(7) invariance

(iii') Abelian gauge invariances associated with A~vl, A~nl, A~vx, A rant.

We note that (i), (ii) and (iii) arise from the original general co-ordinate invariance in d = 11, whilst (iv) and (v) from the local SO(I, 10), being the Lorentz invariance in d = 11. In order to agree with the usual parity assignment of N = 8 fields, duality transformations similar to (3.2.4) are performed, which eliminate A~va, and transforms A~vt into a set of scalar fields. Rescalings and rediagonalizations are also performed, leading to the spectrum (3.4.7) and residual symmetries, besides the usual ones of (i) and (iv) and the extra ones (ii), (v), (iii') Abelian gauge invariances for the vector fields A~a , B~,b (the latter the duality transform of the axial vector fields A~ab). The SO(7)1o¢ x GL(7,R)slo b is then extended to SO(8)1oc x SL(8, R)glob, where the extension of the SO(7) to SO(8) is seen from the combination of the vector fields (A~a, B~b ) into the 28-dimensional adjoint representation of SO(8) and the scalars gab and ~a (the duality transforms of Al~va ) into the 35 of SO(8). Simultaneously GL(7, R) can be extended to SL(8, R). The SO(8)1o¢ may be made explicit by recognizing that the 35 scalars (gab, ~a) = Sa'b', (a', b' = 1 , - - -8 ) enter the Lagrangian as proportional to d~,Sa~b,~'S ~'~'. A square root of SaT may then be introduced as a local SO(8)-bein in analogy to the introduction of the vierbein e~ in the global group in GL(4, R) to give local SO(l, 3) invariance. This is only defined up to a local Lorentz transformation, so belongs to GL(4,R)/SO(1,3). Similarly the new square root v~ will belong to GL(8, R)/SO(8), corresponding to 35 parameters. The scalars must then be described by a non-linear a-model, where the gauge connection is non-propagating, but eliminates negative kinetic energies arising from the non-compact group GL(8, R).

It is now possible to consider the scalars and pseudo-scalars together as the elements of some coset space G/H with G and H larger than GL(8, R) and SO(8) respectively. If the maximal linear group acting on the physical fields is taken to be SU(8)= H, with 63 generators, then G must have 63 + 70 = 133 generators (and that SU(8) may be relevant arises from the fact that the multiplicities in (3.3.10) are dimensions of the totally antisymmetric represent ations of S U (8)). But 133 is exactly the number of generators in E 7, whose action on the vectors of its 56-dimensional fundamental representation x = (x°,yij), with x ij = - x j~, Yij = -Yji, 1 <<. i,j ~< 8 is

(~xiJ A~i xkfJ .a_ 1__ "~ijklmnopv ,, ~ 2 4 - ~ lJmnopYkl

k ~Yij "= A[iykfJ q- ~ijkl Xkl (3.4.7)

with A~ = -A~, A I = 0, and A~ is an SL (8,R) subgroup and Zijkl is totally antisymmetric in its indices.

As in the vierbein case in d = 4, where fermions are only acted on by the local SO(1,3) so we expect in this case that the fermions are only transformed by the SU(8) and are inert under the E 7. On the other hand the vector fields will transform under the E 7 a s the representation of dimension 56, composed of both the 28 of vector field strengths and the variation of the action density w.r.t, them. The further coupling of all fields is now consistent with the required S U(8)loc x E7~o b symmetry.

We may consider the SU(8)loc as the chiral symmetry of the N = 8 supera|gebra (1.3.11).


Table 2. Off-shell assignments of fields under SU(8)1o¢ x SO(8)

Spin 2 2 a i ½ 0

SO(8) 1 8 1 56 28 +~28

SO(8) 1 I 28 1 28 + 28

E 7 1 1 56 1 56

37

The scalars are to be described, from above, by the multi-bein matrix, which is in the fundamental 56-dimensional representation of E7:

( u~S °OrL~ (3.4.8)

where i,j and IJ are antisymmetrized SU(8) and SO(8) index pairs, and transforms as

Y(x) -* U(x)Y(x)O- 1 (3.4.9)

under UeSU(8)lo¢, OeSO(8) (indices in (3.4.8) are raised and lowered by complex conjugation). Then clearly the physical fields in Y lie in ET/SU(8), so are 70 in all. The vector gauge fields are real, so cannot transform under SU(8), but only under SO(8). The resulting assignments of the fields are given in Table 2, as are their transformations under the E T introduced above.

The possible physical application of the above symmetries will be discussed in a later section (Section 3.6). Truncation of the N = 8 theory in d = 4 can also be performed, with the resulting symmetry for N = 4, for example, being the known SU(4) x SU(1, 1). The d = 11 theory may also be reduced to supergravities in lower dimensions in a consistent fashion and the resultant symmetries discerned (as discussed more fully, for example, in Cremmer 1982).

The preceding discussion of N = 8 supergravity indicates that the requirements of local N = 8 supersymmetry, general co-ordinate invariance and local Lorentz invariance lead to a unique theory in which, however, there are enough vector fields to have a local SO(8) invariance. This possibility of adding the property of local gauge invariance to that of the other symmetries had first been accomplished for N ~< 5 (Freedman and Das 1977; Fradkin and Vasiliev 1976); Freedman and Schwarz 1978; de Wit and Nicolai 1981a) but more recently the complete construction of gauged N = 8 supergravity has been constructed (de Wit and Nicolai 1981b, 1982). To achieve this the global 0(8) invariance of (3.3.12) is promoted to a local one with extra terms in the action and transformation laws depending on the SO(8) coupling constant O. This theory cannot now be extended by duality transformations to E7, s o this is lost if g ~ 0. The hidden SU(8) symmetry must still be realized by composite gauge fields as in the previous situation. We will not give all of the details here, except to remark that the scalar field potential is the simple expression

V ( Y ) = i i 2_¼ ~lA2jktl IA~Jl (3.4.10) where the tensors A~ and As arise in the SU(8)-decomposition of the tensor

T! kl = / u kl -I- oklIJ'~IuJK u jrn t IJ ) t irn Kl - -OimJK OJmKl)

a s

jk, =

38 J .G. Taylor

It is this scalar potential which must be analysed for possible SSB of N = 8 SGR. Because of the local SO(8) invariance it only depends on 7 0 - 2 8 = 42 parameters. Its stationary points are not all known; we will discuss them later.

3.5. Spontaneous compactifications

The possibility of giving a geometrical interpretation of internal quantum numbers such as electric charge has led various authors to propose that the extra dimensions are available to us at a sufficiently high energy to overcome the "spontaneous compactification" otherwise making them inaccessible (Scherk and Schwarz 1975; Cremmer and Scherk 1976). By this latter is meant that the vacuum solution of the field equations involves compactification in the extra dimensions. Thus if a field cib(x,y) depended on both external and internal variables x and y and is expanded as (assuming y is 1-dimensional for simplicity)

cib(x, y) = Z e2niyn/t'~n(X ) (3.5.1)

then the equation of motion

t 2 ¢ = 0 (3.5.2)

(where [] is the d'Alambertian in both x and y variables) leads to the separate equation, for each n,

[I--12 + (2nn/L )Z]ck,(x) = 0. (3.5.3)

Each mode qb,,(x) on the interval (0, L) in y has therefore acquired a mass (2nn/L). Provided L < O(10- is cm) the associated energy of > 104 GeV would not present be observable.

Models which possess such spontaneous compactification have been developed (Horvarth, Palla, Cremmer and Scherk 1977; Horvarth and Palla 1978), though involving non- geometrical fields coupled to the metric. All fields are geometrical for N = 8 SGR, so that it is most natural to search for spontaneous compactification in that framework. This search has already led to several interesting solutions to the field equations for N = 1 SGR in d = 11 as well as various other cases, which we will now discuss.

The field equations of N = 1 SGR in d = 11 may be deduced from the Lagrangian (3.3.2). We are concerned with classical solutions, for which the fermion field ~k~ will vanish, so that the equations of motion are

1 1 [ 1 F FPQRS 1 F MeQRF~QR ---~gMN PQRS (3.5.4) R MN --~ Rg MN = -~

1 oM___MsPQRF F (3.5.5) V MFMPQR = 579 o MI___M4 Ms___Ms.

We are especially interested in solutions which allow the 11-dimensional space-time M to be considered as a product of a 4-dimensional M4 and a 7-dimensional manifold MT, so that M = M 4 x M 7. Thus we are interested in solutions with

O pm ~-- O, F.unpq = F l~vpq = F uv~ = O. (3.5.6)

The trivial solution guv= Anus, Or,,,, = --6,,,,,, Fup,,,r = F~m,,p = 0 is compatible with compactification in the extra dimensions, and so is compatible with the choice M = R___~ 4 x T~ 7, where T is the one dimensional torus as used in (3.5.1). If the massive modes of form (3.5.3) are discarded we obtain the trivial dimensional reduction used in the previous section.


A non-trivial solution is given by the choice (Freund and Rubin 1980)

Ft,,,,, p = 0 (3.5.7)

where c is a constant. Then (3.5.5) is trivially satisfied, whilst (3.5.4) becomes

R~,v = -4c29~ ~, Rm. = 32-cZg,,. (3.5.8)

Since c 2 > 0 then M 7 is compact, and so spontaneous compactification has been achieved. There are many solutions of (3.5.8), of which the maximally symmetric case is (Nahm 1978a; Duff 1982) anti-de-Sitter (ADS) space times the round seven sphere with curvature

R #voa = - ( 4 c 2 / 9 ) (g lJpgvcr -- g #~rgvp ) R..pq = ( c2 /9 ) (g,.pg.q - gm~gnp) (3.5.9)

In this case the vacuum is supersymmetric, since there are eight covariantly constant spinors with

t ~ / M = /)M/3 = 0 (3.5.10)

where

L)# : - D# +½cT#ys , / )m = Dm -~cFm

and the d = 11 F-matrices have been decomposed in M4 x M 7 as

FA = (y, ® 1,y,®F,). (3.5.11)

The space-time M is thus [SO(3, 2)/S0(3, 1)] x S 7, where S 7 is the seven-sphere with radius (3c)-1. Since S 7 = S0(8) /S0(7) the resulting bosonic symmetry is SO(3,2)x SO(8); this may be combined with the fermionic symmetry to give (d'Auria and Fre 1983) the supergroup 0Sp(4/8) (we will discuss supergroups in Section 4.1).

It is important to note that the above two solutions, R 4 x T 7 or AdS x S 7, are the only ones for which there is N = 8 supersymmetry. For (3.5.10) has solution if

[DM, DN]-5 = 0. (3.5.12)

Detailed analysis of this condition (Biran, Englert, de Wit and Nicolai 1982; Duff and Pope 1982) leads only to the above two solutions as allowed. It is also very relevant that the massless sector of the solution on S 7 is precisely the gauged SO(8) theory, with coupling constant

g = 3xc. (3.5.13)

This may be proved by considering the covariantly constant spinors r/on S 7 which satisfy the ST-part of (3.5.10). The SUSY parameter can then be decomposed into ~/'s. The fields hay , hum , hmn , ~1#, ~lm, Amn p may then be shown, by comparison with the SUSY transformations, to be decomposable into different powers of ~/'s. The number of different powers of ~/'s that are resulting can then be shown to lead to the spectrum (3.4.7).

Other solutions of the equations (3.5.8) can also be considered which do not preserve the full N = 8 supersymmetry. Thus we may consider the "squashed" 7-sphere (Awada, Duff and Pope 1983), which may be described as the distance sphere in the quaternionic projective place P2(H), with one parameter describing the degree of distortion, which is fixed by (3.5.8). The symmetry group of this distorted sphere is found to be SO(5) x SU(2), whilst there is only one covariantly constant solution, not eight as in the previous case.

40 J.G. Taylor

There is thus only an N = 1 SUSY for the vacuum solution, with the masses of seven of the gravitini expected to be O(me). The nature of the resulting four dimensional Lagrangian is not as yet known.

A further solution of no remaining supersymmetry was discovered recently (Englert 1982) in which there is an $7 sphere with the torsion required to define an absolute parallelism. This solution involves taking account of non-trivial values of Fm,pq instead of the choice (3.5.7). If we keep the value for F~,v,~,, of (3.5.7) then the non-trivial part of (3.5.5) becomes

C Vt F t'~"P = --12 ~.,,p~,st̂ Fqr~t (3.5.14)

with the constants in front of the two curvatures in (3.5.9) being changed to ( - 5cZ/6) and c 2 respectively, with the further constraint from the 4-dimensional part of (3.5.5) that

Fmpq,F ~" = 24C20m.. (3.5.15)

The equation (3.5.14) may be solved by the choice

F,,.p~ = 2S[.pq,,,] (3.5.16)

where 2 is a constant and S.p~ is the totally antisymmetric tensor which, if added to the Riemannian symmetric connection gives a "flat" space with zero curvature, i.e. it is parallelizable (Cartan and Schouten 1926), The new solutions now have (Fuv~a) 2 = 32~, (Frnnpq) 2 = -56~, ~1= __].V, where now Ruv = ~ l g u v , R.,. = YOre.- We note that the flattening of the $7 has not been accompanied with a disappearance of the cosmological constant V ~. This constant will be far too large, but a possible cure for this has recently been suggested (Duff and Orzalesi 1983) in terms of non-vanishing fermionic bilinears as defining a Cartan-Schouten parallelism with zero cosmological constant. However there are a number of questions to be answered about this assumption, as well as justification that fermionic bilinears would naturally arise.

Finally, we note that the above solution breaks N = 8 supersymmetry completely (D'Auria, Fre and van Nieuwenhuizen 1982; Biran, Englert, de Wit and Nicolai 1983).

Solutions have also been discovered for simple supergravities in d < 11. Thus for d dimensional SGR there is a solution which is AdS in ( d - 3) dimensions times a three sphere $3 for d = 7, 8, 9 and 10 (Duff, Townsend and van Nieuwenhuizen 1983). However we note that d = 10 SGR, with or without d = 10 Yang-Mills theory, appears to have no maximally symmetric solution about which small fluctuations are stable (Freedman, Gibbons and West 1983). There is also an interesting situation for supergravity in d = 7: the number of vector fields is equal to that required to gauge SO(5), and it is expected that both a gauged SO(5) x SO(5)loc theory can be constructed in this case, as well as interesting possibilities for spontaneous compactification (Sezgin and Salam 1983).

3.6. Supenlnifieation

We have so far discussed supergravity from a theoretical viewpoint, and emphasized the attractive feature possessed by N = 8 supergravity that all fields are gauge fields and the whole theory is uniquely specified by x, or x and 0 if the theory possesses local SO(8) gauge invariance. We now wish to see what relation may exist between the various versions of N = 8 supergravity described so far and the known properties of elementary particles.


The attractive feature mentioned above of having a strongly restricted range of interactions in N = 8 SGR implies that it is not possible to impose an arbitrary internal symmetry on the theory. Such arbitrariness does exist for N = 4 SYM, so making that theory intrinsically less appealing as a possible ultimate theory of nature, in spite of its finiteness to all orders. Thus despite the difficulties yet ahead in constructing a sensible quantum theory of supergravity this inherent restrictiveness should lead one to continue the attempt until it clearly succeeds or fails. However the allowed symmetries arising from N = 8 SGR do, on first sight, indicate further problems.

The theory of N = 8 SGR is based on the fundamental preonic multiplets (3.4.7), and there is a range of possibilities according to the number of these preons which are supposed to be observable at energies much below m e. One extreme is to suppose that the 3 known families of particles are indeed in the preon multiplet. This is appealing since after removal of eight fermionic states into the gravitini, to give them mass, there are exactly 48 fermionic states remaining. This would therefore allow 3 families of 16 fermions each, (with both right and left-handed neutrinos) of an SU(5) type of symmetry. It would be difficult to explain the standard SU(3) x SU(2) x U(1) model, even if the SO(8) were gauged, due to the fact that SO(8) cannot accommodate all of the observed particle states. This is mainly due to the fact that SU(3)x SU(2)x U(1) is not a sub-group of SO(8) (Gell-Mann 1977; Gell-Mann, Ramond and Slansky 1979). This might be avoided by compactifying the d = 11 version of N = 1 SGR to one with other than the round S 7 in the extra dimensions. Such possibilities were briefly remarked on in the last section, such as the "squashed" S 7 or the "flattened" S 7. In the former case the symmetry group is SO(5) x SU(2), in the latter G2, both of which would be unsatisfactory to fit the three families. One could instead consider the internal seven-dimensional manifold, and in this way obtain, for example, SU(3)x SU(2)x U(1) (Witten 1981b).

A more basic difficulty of any such approach is that the fermions in the theory, after an internal symmetry has been broken suitably, will be vector-like, being in real representations of the remaining low-energy gauge group. This is because the internal symmetry is not related to the four-dimensional space-time manifold in terms of which chirality is defined. Thus the preons of N = 8 supergravity must be considered not as presently observed particles (except for the graviton) but truly as the elementary particles out of which presently observed particles are to be constructed as composites. The possibility of using Regge-pole theory to implement this idea has been investigated (Grisaru and Schnitzer 1981) with the indication of Reggeization of the preon (3.4.7) together with the generation of composite fermions and other states.

An independent approach to this problem has also been attempted in group theoretic terms (Ellis, Gaillard and Zumino 1980; Derendinger, Ferrara and Savoy 1981). This supposes that the "hidden" S U(8) symmetry of N = 8 SGR discussed in Section 3.4 becomes dynamical with the appearance of 35 further (composite) vector bosons to augment the original 28 of (3.4.7) to the full 63 of the adjoint representation of S U (8). Possible "current" supermultiplets were then considered for which the anomaly-free sub-multiplets contain only a sequence of (10+3) families when classified by SU(5). Moreover all states with helicity i> 1 (except for the 63 SU(8) gauge fields) are assumed to acquire a mass O(me), since they would be governed by non-renormalizable effective field theories with cut-off at m e, so giving them such a large mass (Veltman 1980; 't Hooft 1979). These attempts have so far proved unsuccessful using a finite number of supermultiplets (Zumino 1981). This difficulty might also be regarded to cast doubt on the results of the dynamical analysis through Regge theory. However there are still some grounds for hope if a suitably strong

42 J .G. Taylor

dynamical framework could be given to show that the required infinite set of multiplets could decouple at low energies except for a small sub-set of the observed particles.

It is possible that the kinetic term for the SU(8) gauge fields is made as a composite, as it is in two-dimensional CP N- I models for a compact group manifold (D'Adda, L/ischer and Di Vecchia 1978; Witten 1979). However for the non-compact group manifold SU(n, 1) /SU(m)x U(1) in 2 or 3 dimensions there is no mass gap generated, nor a propagating gauge vector (Davis, Macfarlane and van Holten 1983). Since this latter is more analogous to that for N = 8 supergravity, with a non-compact E 7 as group manifold, propagating composite gauge vectors are not expected for the latter. The situation for the case of gauged SO(8) is not known. The simplest possibility of this would have preon binding produced by some unbroken sub-group of SO(8) (de Wit and Nicolai 1981). If this were SO(8) itself all fields carrying SO(8) indices would be expected to be confined (if SO(8)-colour confinement occurred like SU(3)-colour quark confinement, which need not be true in the gravitational context). Since the fermions and graviton would then be unconfined, from Table 2 there may still be difficulties of embedding the known fermions into the 56. This is especially so if the role of Higgs scalars in breaking the S U(8) were taken by a non-zero value of the function A~jkt in the N = 8 SGR scalar potential of (3.4.10). This allows an unbroken SU(5), but the decomposition

56 = 1 @ (5 ~) 10) ~) (5 ~) 10) @ (5 ~) 10) ~) 10

does not have the correct set of three copies of (5 + 10). There is the further difficulty that a non-zero value 0 of the scalar potential will lead to a cosmological constant

A = O(-geo /x4) . (3.6.1)

For any natural value of o of O(1) this takes a value O(m~,), which is very large indeed. It is possible that integration over A, as in the "space-time form" approach, (Hawking 1978) would be more natural, since that would correspond to summing over the parameter R (or g = x /R) in the description of the background solution; as such the averaged value of the cosmological constant will now be zero. However the inflationary scenario needs a cosmological constant, though possibly not one as big as (3.6.1).

A more detailed analysis of the stationary points of the potential (3.4.10) is clearly needed. Some non-trivial stationary points have been discovered (Warner 1982) other than the ones obtained by analysis in d = 11 as described in the previous section. It is an interesting but unproven conjecture that the set of all spontaneous compactifications M 1 x M2, with M1 a four-dimensional non-compact 4-manifold and M2 a compact 7 manifold (as solutions of the classical N = 1 d = 11 field equations) are indentical with all of the stationary points of the N = 8 supergravity Lagrangian in 4 dimensions. The proof or otherwise of this will depend on further analysis of the properties of the scalar potential and of solutions.

We must recognize that we have been led to the fundamental question: do we live in a four-dimensional space-time or an eleven-dimensional one? At energies much below me there may be expected to be no difference between these two possibilities of N = 8 SGR in d = 4 or N = 1 in d = 11 as the alternative models and the above conjecture is true. However the difference would show up at the Planck mass; pre-historic cosmology (before the Planck time) should have very different features, as that shown by the spectrum (3.5.3) indicates. The quantum features will also be of considerable difference. We can only begin to analyse such features by means of superspace techniques; we will turn to that next. We conclude that N = 8 supergravity in d = 4, or N = 1 SGR in d = 11, are both elegant theories with little flexibility. Presently there is little evidence that they are related to low


energy particle phenomenology. However the unified electro-weak model of Glashow, Salam and Weinberg was only accepted as a possible theory of electromagnetic-weak interaction unification when the feasibility of it allowing a renormalizable theory of weak interactions had been shown by 't Hooft (so allowing sensible calculations to be made). It is very likely that a similar situation might be occurring in the presence of gravity, so that the quantum features of supergravity must now be explored. In order to do that we will have to develop powerful enough techniques, both at classical and quantal level, in superspace.

4. GLOBAL SUPERSPACE

4.1. The super-Poincar6 algebra

The extended super-Poincar~ algebra Se n was defined in terms of its generators i {Pu, Juv, S~+,S~_i} with commutation or anticommutation brackets given by (1.3.2),

(1.3.3), (1.3.11) and the usual commutators between Pu and Ju," We should also include the generators Tfl of SU (N) with Tfl commuting with Pu and Juv whilst

[T~, ' ( 1 ) (4.1.1, =

where T; = 0 and

. b = 6[, T~]. (4.1.2)

Then the set 5°n = {Pu, Juv, T], S~,+, S,_i} can be decomposed into a bosonic set L 0 and a fermionic set L 1 :

= s = {S~+,S~_i} (4.1.3) Lo {P~,Juv, T~}, L 1 i

and the commutation and anticommutation relations (1.3.2), (1.3.3), (1.3.11), (4.1.1), (4.1.2) can be encompassed in the conditions

[Lo, Lo]_ ~ L o [Lo, La] e L 1 [LI,Lx]+ ~ L o. (4.1.4)

The set of objects {Lo, Lx} with the condition (4.1.4) involving both commutators and anticommutators is called a graded algebra (or a superalgebra). There is a large body of mathematical literature analysing the properties of such objects and their related supergroups (Kac 1977; Freund and Kaplansky 1976; Corwin, Ne'eman and Sternberg 1975; Rittenberg 1979; Scheunert 1979) but we will not discuss these generalizations and extensions except to mention that superalgebrae have similar properties to Lie algebras and in particular can be partly classified along similar lines to them. There are the matrix superalgebrae of form

(A B):(0A O)eLo,(O Bo)eL ' (4.1.5)

where A is an m x m, D an n x n and B, C are m x n and n x m matrices. The case when

AT=--A, DTG+GD=O,B=CTG, G=( ~ (4.1.6)

44 J.G. Taylor

where G is the 2p x 2p antisymmetric matrix, has A in the algebra O(m) of the m-dimensional orthogonal group, D in sp(2p). The resulting algebra is denoted osp(ml 2p), with associated group OSp(m 12p). We noted earlier that OSp(4[ 8) is the symmetry group of the compacti- fled solution for N = 1 d = 11 SGR on the round 7-sphere S 7.

It is natural to ask where Sel and SeN fit into the above classification. This may be seen for Sel by considering osp(114), which decomposes so that Desp(4) ... S0(3, 2), which is the 10-parameter algebra generated by the rotations Jab in a 5-dimensional space (1 ~< a, b ~< 5). This is a simple algebra, but by contracting, with the choice

P# = R - 1J5/~ (4.1.7)

we see from the usual angular momentum commutation relations that

[P~,Pv]_=-iR-2j~v---.O as R - - ~ . (4.1.8)

This is the Wigner-In6nu contraction of the de-Sitter group to the Poincare group (In6nu 1962). By taking a factor R- 1/2 in "odd" elements B and C we rederive the relation (1.3.9). A similar contraction may be shown to occur for osp(ml4) to give SeN, though the details of the resulting superalgebra depends on the exact details of this contraction; in particular central charges may be generated as arising in (1.3.11) (Lukierski and Rytel 1982).

Since the process of contraction is somewhat cumbersome it is useful to proceed to analyse Se N directly. We wish, in particular to obtain a complete picture of the irreducible representations (irreps) of SeN which appear appropriate for constructing supersymmetric theories. It is these irreps which can be regarded as the basic building blocks of such theories, and their knowledge will prove of value in such constructions.

We expect that the useful irreps of SeN will have labels given by the eigenvalues of the Casimirs, of the algebra, which extend those of the Poincar6 algebra. The two for the latter algebra are the mass m = (p#p~)l/2 and the helicity 2 (for massless representations) or spin s (for massive representations) with s = 0,½, 1 . . . . . In both cases these are determined by the Pauli-Lubanski vector W~, defined by

W~ = ½%v~oPVJ ~ (4.1.9)

where Wu = 2p~ for p2 = 0 and Wu W ~ = - 2p2s(s + 1) for p2 > 0. The vector W~ does not commute with S~ due to (1.3.2) but may be extended to a vector C~ which does (Salam and Strathdee 1974a; Sokatchev 1975; Jarvis 1976)

C~ = t - . ± (4.1.10) W~, -~Sty~sS

where Y~ = T~-P~,I~/P 2 is perpendicular to p~. Then C ± satisfies the SU(2) algebra in the rest-frame if p2 > 0, (with C~- = 0) so (C~) 2 = -2p2y(y+ 1) where Y = 0,2, . . . , ± • super- helicity is defined by analogy to the non-SUSY case if p2 = 0. The operator Y is called the superspin, and is an important label for irreps. In the case of extended supersymmetry C~ may be modified simply from (4.1.1) by inserting the same indices on the two factors in the second term, whilst a similar extension of the SU(N) generators may be given to that for T,~; the resulting eigenvalue of the SU(N) Casimirs will be the same as for SU(N) (Taylor 1980a, 1982a; Ferrara 1982; Rittenberg and Sokatchev 1981; for a general discussion see Nahm 1978b).

We may see how to characterize these irreps as follows (using Wigner's method of induced representations). If we first take N = 1, then S~+ and S~_ separately anticommute, and their mutual anticommutator is that of a c-number (on states with P~ diagonal). We may therefore take a Fock-space representation with S~+ an annihilation operator, S~_ a


creation operator, and a vacuum state 10) with

Sa+ 10) = 0. (4.1.11)

The possible states will therefore be obtained by applying powers of S~_ to ] 0), to give the states

10), S~_ 10), S_ S_ 10). (4.1.12)

Thus states have spins 0 (twice) and ½ if l0) has spin zero, so we obtain the (complex) multiplet (½,0z). We can then build upon this fundamental irrep by taking the vacuum state I Y), still satisfying (4.1.11) to have spin Y, and by ordinary vector multiplication of spins

y ® (½,0 2) = (y+½, y2). (4.1.13)

A more detailed analysis shows that Y is indeed the superspin of the multiplet, being the expected eigenvalue of Wu 2 on f Y), the states with spin Y + ½ being created from ] Y) by the action of S~_. Thus the basic multiplet (4.1.12)

Oo = (½, 0 2)

was used to describe supersymmetric matter in Section 2, the Y = ½ multiplet

Oa/2 = (1,½, O) (4.1.14)

was used to construct unextended supersymmetric gauge theories in Section 2, whilst the multiplet with Y =

~3/2 = (2,3, 1) (4.1.15)

was used to construct N = 1 supergravity in the previous section. We note that these multiplets have additional fields automatically present to preserve the equality

0°(B) = 0°(F) (4.1.16)

between bose and fermi degrees of freedom in an irrep. Thus ¢o has O°(F)= 4 as a Majorana spinor and 2 complex scalars, not the single one of Section 2. *a/2 has an extra scalar beyond the gauge vector and spinor. We have already noted that such auxiliary degrees of freedom are needed beyond the physical fields describing the on-shell states in order to preserve (4.1.16) off-shell (p2 4= 0). The irreps we are discussing automatically contain these auxiliary fields, since they always satisfy (4.1.16).

We may proceed in the same way for higher N. For general N we may classify states by the Young tableaux of irreps (Howe, Stelle and Townsend 1981). We can use that the complete antisymmetry of products of S~_i's and that the antisymmetrized product of three spinor labels is zero, so that the SU(N) tableau [--V--V-] cannot appear. Thus the S U(N)-tableau of any product of S,_i's on l0) can only be of form

p boxes

I boxes. (4.1.17)

46 J .G. Taylor

where the corresponding product of S_'s is p

H (Sir- Smr) H Set-j IO) (4 .1 .18 ) r = l [j]

with antisymmetrization over the set o f / S U ( N ) indices j. Since the first factors in (4.1.18) have spin 0, and antisymmetrization in SU(N) indices implies symmetrization in spinor indices, the spin of (4.1.18) is 1/2. We can thus count the states (4.1.18) to give the fundamental Y = 0, SU(N) singlet irrep for any N. For N = 2 these states have (spin) (representation of S U(2)) content

(1, 12, 0 1 + I + 3 ) (4.1.19)

for N = 4 have content

(2,34, lXS +6 +~,½,+2o,01 +a-+ lo+ ~-+ 2o,) (4.1.20)

whilst for N = 8 the Y = 0 multiplet is

A 7 "~119 5272 ")1700 31904- 1 6 1 8 8 , 1 3 5 3 6 , 0 4 8 6 2 ) (4.1.21) ~, ~,-" , 5 , ' - , 5 ,

(where we have only given the total degeneracy of each spin state for N = 8 and not its detailed SU(8) content). We may construct irreps with higher values of Y and/or SU(N) irrep by direct multiplication of the Y - - 0 singlet irreps of (4.1.19)-(4.1.21) by the appropriate spin, as we did for the N = 1 case above. A similar construction can also be given for the massless case, though now with only one instead of two creation operators S~_i per each SU(N) index i (these are discussed more fully in the references quoted above).

Finally we may construct component functions and deduce their transformation rules and invariant quadratic Lagrangian for any irrep, for all N, by the use of a suitable set of basis functions (Bufton and Taylor 1983a). In particular for N = 1 we can introduce the vacuum state

10) = S+S+ = w_ (4.1.22)

which clearly satisfies (4.1.11); we denote this state as w_. The other states in (4.1.12) will be denoted

S~,_S+S+ = u,,_, S_S_S+S+ = e_. (4.1.23)

We can then consider the Y = 0 multiplet as being described by the expression

~o = e_ A(x) + ff~- qJ~,_ (x) + w_ B(x). (4.1.24)

Thus ~o ~ (A, B, ~b,_ ). We may evaluate the action of a SUSY transformation on ~o as

6~ ~o = (gS)~o = - [e_ 6A + fi~- 6~,_ + w_ fiB] (4.1.25)

and we may re-express the middle term in (4.1.23) as an expansion in the basis functions e _, w_, ~ ' - ; for example we clearly have

S ~ _ w _ = u s _ .

We may thus determine the transformations 6A, fiB, 6qJ 2_ in terms of A, B, ~b and e with the same results as (2.1.7) obtained originally by direct analysis. The advantage of the basis functions is that they can be used for all N, where these functions are obtained by replacing

N 10) by ~ (S~_S"+) in (4.1.18). Suitable algebraic manipulation allows the same

n = l


method as for N = 1 to give the exact form of the SUSY transformations of components, and quadratic Lagrangians, for the Y = 0 SU(N) singlet irrep for all N. This may then be used to deduce the same features for any irrep, though further decomposition of components into irreps of spin and S U(N) would still be necessary in these cases.

For example, the basis functions are p

e t -k = ei'"'i~'"JPl~'"tN-P-LS(~,-i,... S~L--)iL 1--I (S_jS -k~)JO) r = l

and the components A~ --k are defined contravariantly by

~o = Y, ~-_k~'-k(x-)

(4.1.26)

(4.1.27)

If we use that S~_i multiplied by the Young tableau (4.1.17) can increase l or p by 1 and S~+ can decrease 1 or p by 1 (with an extra factor of p), we obtain the general transformation law

~ t - k _ 6A~ - = I . ~ ' - iA ~"')k + ii~t~- [kle~_~_ A r--~-ks] + III~ + dO (~1 - A~-)ik

+ iVan+ ( ~ r -~ -k i0 )#+r-Ar, (4.1.28)

where I, II, III and IV are suitable co-efficients. We note that there is an obvious problem in using these irreps for N > 2 to construct

N-SYM theories or N > 4 to construct N-SGR's. In such cases higher spins than those of the physical particles will be present, and so will have to be made auxiliary. This could be a difficult task to achieve, and we will see shortly that it cannot be achieved satisfactorily for N ~> 3 for either N-SYM or N-SGR. This feature is the N = 3 barrier, and we will shortly see how difficult it is to penetrate.

4.2. Superfields

It is reasonable to argue by analogy concerning the structure of the supersymmetry algebra 6P 1, and so to introduce an anticommuting spinor variable co-ordinate 0~ as the fermionic partner of the space-time co-ordinate x ~ :

Poincar6 algebra ~ --- i0(1, 3) ~-* SuperPoincar6 algebra 6e 1

Elements of ~ = (P~, J~v) ~ Elements of 6 a i = (Pu, J~v, S~)

Poincar6 Group ~ = 10(1, 3) ~-~ SuperPoincar6 group S ~ = SI0(1, 3) # v Elements o f ~ = (x z, A~) ,-~ Elements of S:~ = (x , Au, 0 ~)

Product: (x, A) (y, E) = (x + Ay, AE)~--, Product: (x, A, 0) (y, Z, ~b) = (x + Ay + ½0-),Zu(A)~k, AE, u(A)W + 0)

Space-time: x = I0(1, 3)/0(1, 3) .-~Superspace: SI0(1, 3)/0(1, 3)

In other words superspace is composed of the set of co-ordinates (x ~, 0~) in the same way that space is composed of the co-ordinates x z. We note that the product formula for two elements of the super-Poincar6 group involves the spinor representation u(A) of the Lorentz transformation A, so that 0~ is truly a spinor under the Lorentz group. Attempts have been made to construct physical theories based on the use of super-algebras in which the anticommuting parts are inert under the Lorentz group (Ne'eman 1979; Fairlie 1979; Taylor 1979c; Dondi and Jarvis 1979), but these are somewhat unsatisfactory due to the presence of states with the wrong connection between spin and statistics necessitating their removal by unnatural restrictions.

48 J. G. Taylor

Table 3.

N 1 2 4 8

2 aN 16 256 65,536 4,294,967,296

It is natural to introduce the concept of a superfield ~b(x, 0) as a complex (or more generally Lie-algebra) valued function on superspace (Salam and Strathdee 1975a). Since the O's anticommute

0~0t~ + 0#0~ = 0 (4.2.1)

the expansion of tk in powers of 0 must terminate at the fourth power:

1 1 1 +10002 + I ( o 0 ) 2 D . (4.2.2) ~b(x, 0) = A(x) + Off/+-~OOV +~OTsOG +-~@iTu),sOAu

The superfield ~ is thus a compact shorthand for the multiplet of component fields (A, F, G, D, A u, if, 2). A similar construction may be given for superfields with external Lorentz indices ~bs (x, 0), where J corresponds to an irreducible representation of the Lorentz group, such as a spinor q~, or a vector ~b u with O;'~p u = O.

We may generalize the notion of superspace to extended superspace in which there are a set of N anticommuting spinors 0~+i, 0~_ (using chiral notation with chiral projectors ½(1 + iys) , as in the definition of S~+, S~_~ in Section 1.3). The general superfield will now be of form ~bj (x, 0~ + i, 0~_ ), and its expansion will involve up to products of 4N O's. In eqn. (4.2.2) there are 8 fermi and 8 bose degrees of freedom; for general N there will be 24n-1 fermi and an equal number of bose degrees of freedom. We note that the total number of degrees of freedom increases rapidly with N, as seen in Table 3. It will therefore be necessary to reduce these degrees of freedom by suitable constraints, and this will be discussed shortly.

It is clear from eqn. (4.2.2) that the scalar superfield for N = 1 contains both the Y = 0 and Y = ½ irreps, the first corresponding effectively to the component fields (A, F, G, A~, ~,), the latter to (a~, 2, D), where A~ and A~ are the components o f A u parallel and transverse to Pu. To separate out these components it is appropriate to introduce the covariant derivatives (S, +, S,_ ~) by requiring that the S's and D's anticommute (D~+,D~_i) relatedto i with each other. We may solve the relations (1.3.2), (1.3.3) and (1.3.11) by the representations.

= IO~,S,,+ Pu " ~ = i[O/0~ + + i(¢Ot)~,+],S~-i = (S~+)* (4.2.3)

(where O/d0 is defined as left differentiation; on a function of one O,f(O) = a + Ob = a - bO, the left derivative acts as alOOf = b), and then D~'s anticommuting with the S~'s are given by

D~+ = [0/00~ + - i(¢0~)~+ ], D~_~ = (D~+)*. (4.2.4)

From (4.2.3) and (4.2.4) we obtain

[ o ; + , + - - [ O ; + , S a _ j ] + = [O,_,, S a _ j ] + = IDa_,, St+ ] + = 0

' 2~C)~+#_S~. [D~ +, D#_j] + =

(4.2.5)

(4.2.6)

We may use products of D's on a basis state I 0) of the previous section to produce a new representation, since the SUSY transformation will commute through this product of D's. We can clearly classify the superspin Y and SU(N) content of these different irreps as we did the spin and SU(N) content of the states obtained by acting on 10> with products of


S~'s. If the basis state has spin j we expect its superspin content to be (j___½N), ( j + ½ ( N - 1 ) ) . . . . j, with related multiplicities. Thus for N = 1 and j = 0 we will have superspins (0 and ½), as contained in the scalar superfield 49; for N = 1, and j general we will have the same superspins (j, j + ½) as in the superfield with external spin j.

This equality of superspins in irreps contained in a superfields with external spin j with all irreps obtained by acting with powers of D on a vacuum state with spin j is understandable as follows. We will identify the components of a superfield as the values of the various powers of D~ on 49~, evaluated at 0 = 0:

Aj(x) = 49j(x, 0)1 0=0,-.-- (4.2.7)

However we may first construct suitable vacua from which these components may be obtained by (4.2.7). We may take these vacua to be

N

I-I (D~+D~+ )[-[D-Oj (4.2.8) i = 1

where the indices on the product of D's and 49j are to be combined to have a given spin and SU(N) transformation character. These states may now be used to define components by application of further powers of D_, evaluated at 0 = 0; since S_ and D_ are equal at 0 = 0 these components are similar to those of the previous section. The vacua we have constructed in (4.2.8) have precisely the same set of superspin and SU (N) characteristics as those given by the construction of the preceding paragraph, thus justifying that identity.

As an example of this the Y = ½irrep will be given by the vacuum state D+D+D,,_49, and will have components

D+D+Da-49 10=0 = 2,_ + ~ b ) , _ D#_D +D +D,,_ 49 10=0 = C,,_#_ (D+ p2A)+ (aUVpC)~,_#_At~

D_D_D+D+Da_49 [0=0 = (02)a_ +p2~O~_

where the r.h.s, of these expressions are in terms of component fields proportional to those in (4.2.2).

In order to reduce the large number of degrees of freedom in a given superfield we can attempt to project out certain of the supersymmetric irreps it contains. Thus for N = I the simplest case is the removal of the Y = ½ and one of the Y = 0 irreps by the chirality condition

O~ + 49 = 0. (4.2.9)

The solution of (4.2.9) is called the chiral superfield. This equation can be solved explicitly, using the representation (4.2.4), to give

49 = 49(x ~ -i6+ yuO_, Oe,- ). (4.2.10)

This solution specifies the manner in which the chiral part 0~+ of 0 enters into 49. The chiral superfield has component expansion in powers of 0_ as

49 = A +O_ ~b_ +½Fl~_O_ (4.2.11)

where A, F and q/are evaluated not at x u but at (x ~ -iO+yuO_). Thus it corresponds solely to the Y = 0 irrep of 5:1. The oppositely chiral S.F. will be defined by D~_ 49 = 0.

For higher N the analogue of (4.2.9) is

O~+ 49 = 0 (4.2.12)

50 J .G. Taylor

for which a similar 0_-expansion can be given as in (4.2.11), with only up to 2N powers of 0_. For N = 2 this will have maximum spin 1, so would appear to be appropriate to describe N = 2 SYM, as we will see later. For N = 4 the chiral superfield (4.2.12) will have maximum spin 2, so be expected to describe N = 4 SGR. However this turns out not to be so, but it only describes N = 4 conformal supergravity due to the spin 2 field (and other lower spins) being described by constrained fields.

The general method of projecting out a representation of a particular superspin from a superfield was first done using the Casimir operators for N = 1 (Sokatchev 1975) and N = 2 (Taylor 1980a) and N = 4 (Pickup and Taylor 1981). This used the projector onto a given eigen-subspace constructed from a Casimir, say the superspin C~ z. The projector FIr onto the subspace with superspin Y for a superfield is defined as

E(Cu ~ 2) + 2p22(2 + 1 )] (4.2.13) 1-I = 1-I [Y (Y + l ) - 2(2 + l ) ] ( - 2p 2) Y 2 # Y

where the product is over all superspins 2 not equal to Y. These projectors are, for N = 1, for example,

- 1--~-O'+ O_D_D,+ (4.2.13a) ri0,+ = -D-D-D+D+/4[- I , Ho,- = I-I*+,EI1/2 - 2r-q

where rio. + is the projector onto the irrep with Y = 0 and positive chirality. Expressions for irreps of SU(N) may now be written down in a more compact form by means of the expansion of a superfield in terms of its chiral irreps (Siegel and Gates 1981).

An alternative approach (Sokatchev 1982) to the problem of projection of irreps from a given superfield is to use the previous definition (4.2.8) of vacuum state, and set this to zero for all but the required irrep. This method turns out to be of great value in investigating the irrep content of solutions to torsions and field strength constraints used in the construction of superspace versions of N-SYM and N-SGR's, which we will discuss later. For example for the scalar superfield in N = 1, the Y = ½ irrep is defined by removal of the two Y = 0 irreps, so by the conditions

/5+D+~ = 0 = / ) _ D _ ~ . (4.2.14)

These conditions are equivalent to the single condition

/)D~ = 0 (4.2.15)

since this constraint can be turned into

075Dd p = 0

by multiplying (4.2.15) by bi /~ s D and use of the product rule

Dilk~, sD . DD = -- 2ilk2Dr sD. (4.2.16)

This and other product rules arise from the fact that there are the same set of invariants for products of D's as enter in the expansion (4.2.2) for products of O's (these products are given in Sokatchev 1975 and Salam and Strathdee 1975a). Similar constraints that are simpler than the full projection of operators (4.2.13) are important for higher N, and will be discussed in relation to the construction of N-SYM and N-SGR in superfield form in due course. There is a reality condition we should mention here (Firth and Jenkins 1975; Pickup and Taylor 1981 ; Gates and Siegel 1982) which imposes reality on the component field of


highest spin of an irrep if that spin is an integer. Thus for N = 1 this condition is

D~+ V/~_ = D~_ ~'/~+.

There are similar constraints for higher N, which we discuss later. We finally discuss the construction of quadratic Lagrangians in superficial form. This is

performed by using the integral over the anticommuting variables 0 defined (Berezhin 1966) for a single anticommuting variable by

f dO = O, This definition looks somewhat strange, f(O) = a + Ob then

fdO. = (4.2.17) 0 1.

but gives a translation invariant integral; if

fdOf(O) = = ]dOf(O + e). (4.2.18) r

b This property corresponds to supersymmetric invariance of the resulting theory, as we shall see. Moreover the integral allows for integration by parts in 0-space:

fdO df/dO = (4.2.19) O.

We also note that integration is identical with differentiation; since by (4.2.18)

fdOf(O) = (4.2.20) Of~dO.

We extend the Berezhin integral to the set of 4N O's, though will only discuss N = 1 explicitly here. In that case the measure on superspace will be dgx d40, where we will use the normalization so that

~d40(O0) 2 = (4.2.21) 1.

For the chiral superfield we may also use the measure d4x d20 _ in chiral superspace with

fd20_ _ _) = (4.2.22) (0 0 1.

These correspond to differentiation similar to (4.2.20), and expressed in a supercovariant form (since D, and O/O02 are identical at 0 = 0 to within a total derivative)

L(x) = fd4OL(x,O)= I (DD)2LIo=o

L_(x) = fd20_L(x,O_)= -1D_D_LIo=o. (4.2.23)

The supersymmetric invariance of an action defined either as

A= fd4xL(x) or A_= fd4xL_(x)

of (4.2.23) follows immediately by the translation invariance of the Berezhin integral, and more particularly by (4.2.19):

= a fL( )d4 = fd'xd'0 'EO/00"-i( 0),3L(x,0) = 0. (4.2.24)

52 J.G. Taylor

Finally we define 6-functions in 0-space by

f d 4 0 1 6 ( 0 - )f(O 1) = f(O) (4.2.25) O 1

so that by (4.2.21), 6 ( 0 - 0 x) = [ (0 -01) (0 -01) ] 2. The 6-function in chiral superspace, 6_ ( 0 - 01), may be conveniently defined as

62_(0-0 x) = D +D +r4(0-01) . (4.2.26)

These 6-functions allow for the functional derivatives

6dp(x, O)/ rdp(x 1, 01) = 64(0 - 01)64(x - x 1) (4.2.27)

6 ~ -- (g, 0 - ) / 6 ~ _ (Z 1, 01 ) = 6 2 (0 - - 01 )64(Z - - Z 1 ) (4 .2 .28)

where z ~ = x i' -iO+vuO_ is the variable introduced in (4.2.10). We note finally the change of variable rule

d(aO) = a- 1 dO. (4.2.29)

Since 0 has length dimension L 1/2, as we see in (4.2.11), then [dO] = L-1/2, from (4.2.17) or (4.2.29).

4.3. The Wess-Zumino model in superspace

We may now construct a superfield action for the spinor-scalar model of Section 2.1. The component fields will be contained in the chiral superfield tk_ satisfying (4.2.9). We

note that the powers of 4)_ are also chiral, though (~b_)* = 4)+ has opposite chirality to 4)_. The kinetic term (2.1.2) can be seen to be contained in

Lo = f d'*Odp + tCp_ (4.3.1)

since this can be evaluated, using (4.2.23) as the 0 = 0 contribution of

D +D +D_D_(dp + q~_ ).

On use of (4.2.9) this reduces t o / ) + D + [q~ +/) _ D _ ~b _ ], which can be written as

D+D+dp+ .D_D_dp_ +D~+q~+D~+D_D_dp_+r~+D+D+D_D_r~_. (4.3.2)

If we use the anticommutation rule (4.2.6) and (4.2.9) the second and third terms reduce further to terms proportional to b~+q~+ifi~+Da_q~_, ~b+p2~b_. With the identification of components given by (4.2.7) the total expression is equal to (2.1.2).

To construct the mass term (2.1.8) we consider the expression

L , = [d20_t~ 2 _ +h.c. (4.3.3) ¢ 1

I /

We note that since the length dimension of (k-, [4)-] = L - i, L , has dimension [L , ] = L 3 ,

as required for a mass term. Furthermore the 0+-dependence of 4)_ only give a total derivative, since we express any chiral field ~b as exp l- - i0+ ~0_ ] ~b (x ~, 0_ ). We may evaluate (4.3.3) directly by using (4.2.11 ) and (4.2.22) to give the requisite expression (2.1.8).

Finally we may consider interaction terms such as (2.1.10). This itself can be constructed as the chiral integral

t ~

Lint = g [d20-t~ 3- +h.c. (4.3.4) J


which again may be determined directly by expansion of ~b 3_ in powers of 0_ using (4.2.11) and then explicit integration by use of (4.2.22). Further powers of ~b_ could be considered, but dimension arguments require they are to be expressed using dimensional coupling constants. They would not be expected to be renormalisable, so will not be considered further. We may extend the quadratic and cubic interaction terms to a set of chiral multiplets &_ to be of form, with inclusion of possible linear terms,

,~ a ~) a .~_ m ab O a f~ b ..t_ f abc O a._ (Oh_ (~ c_ . (4.3.5)

In component form these expressions (4.3.5) were used in the discussions of application of N = 1 SYM to high-energy physics in Section 2.4.

For N = 1 supersymmetry we cannot construct any further Lagrangians without introducing irreps with Y >I ½. These contain spins of at least 1, so will only have potentially satisfactory interactions and quantum field theories provided they correspond to gauge theories (super Yang-Mills) or supergravity theories. We will turn to these after a brief description of the perturbation theory rules for quantum field theory of the Wess- Zumino model in supercovariant form, the so-called supergraph rules.

4.4. Supergraph rules

Perturbation rules were developed in a superfield form in order to give a more powerful technique of handling the cancellations discovered in component calculations in lowest order (Capper and Leibrandt 1975; Salam and Strathdee 1975b). This cancellation occurs effectively due to the fact that there are powers of the anticommuting variables in the 6-functions 64(0) and 62_+ (0), so that products of a set of these will cancel if they correspond to variables taken around a closed loop in a Feynman diagram. To discuss this in detail we have to construct the supergraph rules arising from the Wess-Zumino model of the last section.

To be specific we take the total action as

A = Aki n + Ain t (4.4.1)

Aki n = fd4x[d40q~+~b_ +d20_~b_j_ +d20+t~+j+] ,J

(4.4.2)

t" Aim = Jd4x d20_ V(~_ ) + h.c. (4.4.3)

where chiral sources j_ and j+ = j*_ have been introduced. Then in the usual fashion we introduce the Greens function generating functional z(]_ ,j + ) by

z(j_,j+) = f d [ ~ _ ] d[q~+] exp [iA] (4.4.4)

where d[4~_ ] d[4~ + ] is a short-hand for functional integration over all of the components in ~b_, q~ +. We may write, again in the usual fashion

1 6 z( j_, j + ) = exp i{ f d4x[d20_ V(- i~ j f ) + h.c.]} zo(j_,j + ) (4.4.5)

54 J . G . Tay lo r

where z o is the free-field generat ing functional

Zo(j_,j+)= fd[4'_]d[4'÷]expifd'*xd40[4'_4'÷ +(D2 /D)j-'4'-+(D2 /[-q)j÷4'÷]

(4.4.6)

Then by comple t ing the square we obta in

Zo(j_,j+)=exp{-2fd4xd4OjTNj } (4.4.10)

where

N = MT-1M = - 4 D - 1 T . (4.4.11)

Inclusion of a mass t e rm ¼m(4, 2 + 4,2 ) in Akin changes T of (4.4.9) to

1 (-mD2_/217 1 T , = ( D 2 7 2 [ - ] D2_.m/2[_]), TZ,,I=(I+m2/[-1) - ' , ~ _mD2/2D)

and N of (4.4.10) to

(mD2+ /2[[] 2 -1/I-1 Nm = 4 ( 1 + m 2 / D ) -1 \ _ 1 /D mD2_/2I'q2]" (4.4.12)

The following F e y n m a n rules therefore arise in m o m e n t u m space:

( T(4, _ 4,1+ )) = _ 4([-q + m 2) - 164(0 _ 01 )64(x _ x 1 ) <T(4,+4,~+)> = 4re(V] + m 2 ) -1 "62(0+ -O*+)64(z-z 1) < T(4,_ 4,[ )> = 4m(r-] + m2) - 1. ~2(0 - _ 01__ )di4(z- z 1) (4.4.13)

F r o m the funct ional derivat ive rule (4.2.26) we see that at chiral vertices there is an extra power of D2+ or D 2_ for each 4' - or 4' ÷ line respectively. We will discuss the ultra-violet divergence aspects at a later section (Section 5.4).

= 2 0 (4.4.9)

where we have used that

20÷(De-/vI~]- "4,- = - 4 (D2+D2-/F1)J-4,- = J - 4 , - (4.4.7)

with D 2_ = / ) _ D _ and D÷D_j_2 2 . = - 4 [ ~ _ , so tha t the source te rms in (4.4.2) are now coupled to 4, _+ th rough an integral over the whole of superspace. We can evaluate (4.4.6) as a Gauss ian integral if we take

and the variable in the exponent of (4.4.6) to be the superspace integral of

½urTu + urMJ (4.4.8)

where


5. G A U G E D SUPERSPACE

55

5.1. Supersymmetric gauge theories

A gauge field theory in ordinary space is described in terms of a connection potential A~. We would expect a similar theory in superspace to be described by a connection superpotential AM = (At,, A i+ , A=_i), where M denotes the pair (p, ct) of bosonic and fermionic co-ordinates. AM will be a function of the superspace variable z M = (x ~', O,,+i, 0~_). It is natural to introduce field strengths FM2V so that if A M transforms under a supersymmetric gauge transformation as

A M --* A ~ = e - i A A M eiA + e - i A t ~ M eiA (5.1.1)

where A is in the Lie algebra of an internal symmetry group, then FMN will transform in a covariant manner. Such an approach has been set up (Grimm, Sohnius and Wess 1977, Wess 1977, 1981) using the supercovariant derivatives D= introduced in the last section. If we define D A = (t3,,,D=) we can introduce an achtbein e~ and its inverse eAM with DA = e~t3M. The connection 1-form

A = dzMAM (5.1.2)

will then be taken to transform under a gauge transformation x = e iA as

A 1 = x - l A x + x - 1 dx. (5.1.3)

The field strength 2-form will be defined, using the language of superdifferential geometry in which d z M ' d z N = (--1)~'(M)+c'(N)+ldzSdz M with ct(M)= 0 for M bosonic, 1 for M fermionic (see Wess and Zumino 1977; Downes-Martin and Taylor 1977)

F = d A + A . A (5.1.4)

so that under (5.1.3) F transforms covariantly:

F 1 = x - l F x . (5.1.5)

The covariant derivative of a 1-form a transforming as a I = ax is Da = da + aA. If we take the components of F to be defined by F = -~dz n dzAFAB we may evaluate the r.h.s, of (5.1.4) by means of expressing A = eBAB, e B = d m e ~

F = eA(eBDBAA) + (eBDBeC)A c. (5.1.6)

(o) C If we define the torsion of the background superspace TAB a s

then

(o} C TAB M C (5.1.7) = e[BDA]e M

(o) C FAB = D[AAB] -- [AA, AB] + TABAc. (5.1.8)

The same formula arises if we take the covariant derivative ~A as

~ A = DA - - A A

and evaluate the commutator, expressed as usual in terms of thc field strength FAB and (o) C

background torsion TAB as

(o) C [ ~ A , ~ B ] = -- F AB -- T A B l e . (5.1.9)

56 J .G. Taylor (o) C

If we use that TAB is defined from (5.1.9) with ~A replaced by D A then the only non-zero value of the background torsion will be

(o). T~Y~#_j = - 2i(Tac)~+#_ 5~. (5.1.10)

We have therefore obtained complete expressions for FA8 in terms of the components of AA (o) C

and the background torsions TAB. At this point we recognize that there are many supermultiplets contained in AM. Thus

for N = 1, the superspin content will be:

A~ :Y = 0,½, 1 (5.1.11) . 1 3 Aa " Y = -i, 1,-~

From the discussions of spin content of irreps given in Section 4.1 we see that component fields will have Poincar+ spin up to ~ in A~ and up to 2 in A a. An even more disastrous situation occurs for N = 2:

A~: Y= a 3 0,3, 1,3 Aa : Y O, a 3 (5.1.12) = ~, 1 ,~ ,2

so there will be component fields with Poincar+ spins up to 3. For N = 4 the range of Poincar6 spins is up to 4. These highest spin component fields cannot occur as gauge modes, since they are not contained in the scalar superfield gauge parameters A of (5.1.1). There must therefore be constraints which exclude the irreps with higher Y in (5.1.11), (5.1.12) or the N = 4 case. These constraints must be chosen so as to be gauge covariant, otherwise the gauge symmetry will be broken. They must therefore be chosen by setting some of the components of the field strengths FAB to zero. This cannot be done with impunity, since the Bianchi identities deduced from the Jacobi identity

Y~ [~AE~B, ~c]] = 0 (5.1.13)

where the summation ~ is over the cyclic interchange of A, B, C, relate the field strengths and their derivatives. These identities, using (5.1.9) and (5.1.13) are

(o). I AB c ~ ~ [~AFBc -- TI~CF AE] = 0 . (5.1.14)

It turns out that these are sufficient to allow satisfactory constraints to be constructed for N = 1 and N = 2 SYM but are too restrictive to allow N = 4 SYM constraints to lead to anything but field equations. We will turn to these questions now.

5.2. N = 1 S Y M in superspace

It is natural to suppose that the fermi-sector field strengths may be set to zero with impunity. Indeed if we wish to be able to gauge the chiral matter superfield 4)- we expect that the gauge-covariant version of (4.2.9)

~ + q ~ = 0 (5.2.1)

have a non-trivial solution. But (5.2.1) leads to a non-trivial integrability condition, by (5.1.9)

[~+,~+]+q~ = F~+~+4).


Since this is true for any solution of (5.2.1) we require

F~+t~ + = 0. (5.2.2)

The constraint

F~+~_ = ~ + A~_ + ~ ~_ A~+ - 2i(Tac)~+~_ A a = 0 (5.2.3)

can be seen to be a conventional constraint, in the sense that it allows us to solve for the vector potential A a in terms of the spinor potential A,+ without any equation of motion appearing.

We may use the Bianchi identities (5.1.14) to obtain all of the components of FAB in terms of covariant derivatives of the spinor superfields FAB. Thus the identity I~+t~+r + and its complex conjugate are solved by

Fa~+ = ~#a~+ W~_, F,,o,- = y~+ W~+ (5.2.4)

whilst Ia,+#+ requires the W~ are chiral:

~ + W~_ = ~ _ Wt~ + = 0 (5.2.5)

and I~+/~_ allows F,, b to be obtained

Fab = l ( ~ + 6 a b W + + ~ _ a a b W _ ) (5.2.6)

with the further condition

~ + W + + ~ _ W _ = 0. (5.2.7)

We note that (5.2.7) is the reality condition (in covariant form) mentioned in Section 4.2, so that there is only one covariantly chiral Y = ½ irrep allowed by (5.2.5) in W~, this being suitable to describe N = 1 SYM.

It is possible to solve (5.2.2) directly. Writing out (5.2.2) in full it has the form

D(~+A#+)- [A~+, At~+] + = 0 (5.2.8)

and the solution is

A~+ = - i e - VD~+e v (5.2.9)

for an unconstrained scalar supertield V. Similarly we obtain

A~_ = -ieVD~+e -v .

These solutions transform correctly under gauge transformations, with

eV~eS+eVeA, e - V ~ e - T e - U e A, D ~ _ S = D ~ _ T = O

or

V ~ V + A + S + + . . . , U ~ U - A + T + ....

We can fix the gauge to give A = S = T, U = 0, A + = A a + gauge transformation, V = V ÷ and then

W~+ = (D_D_)e- VD~+ e v (5.2.10)

as obtained earlier by a direct construction (Salam and Strathdee 1975a). Finally a gauge invariant Lagrangian with the correct dimensions may be constructed from Tr (W+ W+),

58 J .G. Taylor

for which

Thus the action will be

D~_ Tr (IcV+ W+) = ~ _ Tr (I~+ W+) = 0.

f d 4 x + ( W + ) + ( 5 . 2 . ) d20 Tr h.c. 1 1

A further gauge transformation can be made to take V into the Wess -Zumino gauge with V 3 = 0, so that V = Oi~,u~,5OAu+OOO2+(OO)2D. The components of (5.2.11) can then be evaluated, and agree with the component version described in Section 2.2.

The N = 1 SYM gauge multiplet may be coupled to N = 1 chiral matter of Section 4.3 by means of regarding e v as a compensating field, with gauge transformations (for ~b_ in the fundamental representation of the group)

~b_ ~ eA+q~_, ~b+ ---} ~b+e A. (5.2.12)

Then the kinetic energy (4.3.1) can be extended to the gauge invariant term

L = ck+e-Vck_ (5.2.13)

which is also reduced, in the Wess -Zumino gauge, to the component values given in Section 2.2.

5.3. N = 2 S Y M in superspace

The first question to discuss here is the constraints to be imposed on F A B . We cannot take all components of F # to be zero (where a = ~+,~-i) since there would then be no gauge invariant object left of dimension L-1 , where [A~] = L-1/2, [ F ~ ] = L-1. But then no dimensionless action could be constructed, since for N = 2 the chiral measures d4x d40 + or d40 d*0_ have dimension L 2 and so a second power of an object of dimension equal to L - a is required. A weaker constraint than for N = 1 was suggested (Grimm, Sohnius and Wess 1978) to be

i F~iJ)+#+ = F,,_(i#_j) = F~,+#_j = 0. (5.3.1)

These constraints were later supported on the basis of a similar requirement of "representation preservation" to that described for N = 1, but now involves the use of central charge representations (Sohnius, Stelle and West 1980a). In any case there is in all such approaches no assurance that the representations to be preserved are the correct ones. This is especially true for the maximal case of N = 4 SYM (and similarly for N = 8 SGR) where no other "matter" multiplets can occur. As to be expected those situations are far more complicated.

The constraints (5.3.1) can be used, in conjunction with the Bianchi identities I_~+t~+r+ of (5.1.14) to show that

ij .. F~+tj + = e%~+#+ W (5.3.2)

with F o~_ifl_ j = 13i j~e~_ ~ff _ ITV

~ + W = ~_~I7¢ = 0. (5.3.3)

so that an appropriate invariant chiral action is

f d 4 x [ d 4 0 W 2 d40 l~r2]. (5.3.4) + +


The Bianchi identities may be used to solve for the other components of FAn as for N = 1, with l~+#+r_ being solved by

i Fa~_+ = ~ ( ~ a ) ~ + ~ - W (5.3.5)

(where raising and lowering of SU(2) indices is performed with e ° = -e~j) , la~_+ ~_ being solved by

i Fab ---- - - ~ ( ~.~ + O'ab,.~ + ~ " -'1- ~ _ G ab,.~ _ W ) (5.3.6)

and the further (and final) constraint

~ + _ ~ + I~ = ~ _ ! ~ _ W. (5.3.7)

This latter constraint is a reality condition similar to (5.2.7) for the N = 1 SYM case. From the discussion in Section 4.1 we know that the superspin content of a scalar superfield W is y = (1,½2,01 +a-+3), where the SU(2) representations m of the associated irreps have been included as Y-~. The condition (5.3.3) at the linearized level singles out Y = 01 in W, and the linearized form of (5.3.7) relates this to Y = 0 T in go'. The Y = 01 multiplet has spin content (1, ½2, 01+a-+a) with correct dimensions for the scalars so that t he / - sp in 1 scalars are auxiliary. The components may be given gauge-covariantly as the values at 0 --- 0 of (W, ~ c , _ i W , ~ _ z ~ _ W , ~ _ a , , b ~ _ W ). Thus the theory has the correct linearized spectrum; since it is c~nstructed in a gauge invariant fashion it will therefore be the complete superspace version of N = 2 SYM.

In order to discuss the quantum properties of this theory, as for the N = 1 SYM case, we need to obtain solutions of the constraints in terms of unconstrained superfields, the so-called "prepotentials". This was already achieved for N = 1 SYM in the last section in terms of the prepotential V in (5.2.10). For N = 2 SYM we only need to achieve this for the constraint (5.3.7), which may be solved at the linearized level (Mezincescu 1979) by

n 4 n~J t." ITV = n 4 niJ ~ (5.3.8) W = J t . . + u _ r i f t ~ - ~ + , i j

where D _ 0 = e ~ - # - D ~ _ i D # _ j is symmetric in i a n d j and Vii is real: 11/* = V ij = eikeJlvu. This uses the identity rlij r~41) = !~ r~4 rliJ ~ + u - ~ + ~ j ~ - i j ~ + ' - ' - , which can be verified directly by the

i j 4 commutator ID +, D_ ] = _ (Cl~)~ +r- I-D(d+, n3j~ 1 = 0, where we define _ u ~ _ j +

with D3r - = ~_-h-_r,-r~- D r,_D_~_D_r ~_, i ' 1

being the completely alternating symbol of S L ( 2 , C ) x SU(2), with indices raised and lowered by the metric.

The solution at the non-linear level may be obtained by a perturbation expansion in powers of the coupling constant g, or equivalently the potential A~ (Howe, Stelle and Townsend 1982). The constraints (re-introducing #) are

.~. D(~+ A~+) - g l A d + , A_~+ ] + = 0 (5.3.9) | t J

( D ( a + A # _ ) - g [ A ~ + , A ~ _ _ ] + ) ~ = 0 (5.3.10)

where (Tji) ~ is the trace-free part w.r.t, the SU(2) indices i, j : ( T j ) ~ = "J'ri--~v~lxi"rk" k" The

P P N P - C

60 J.G. Taylor

linearized constraints (5.3.9) (with O = 0) may be solved by the general solution

A~_+ = D~+ U+Da+ V (5.3.11)

where W = D~ V, U being purely a gauge part of A~+ not entering the Lagrangian. The final constraint (5.3.1) is then solved by V = D°V~i, as before. The presence of U is necessary for the gauge parts of V to be non-zero, as may be seen by going through the procedure of projecting out the various irreps in (5.3.10) by means of multiplication by D51rD~_+ and D4zrD~_ ; for example the Y = ½ irreps in V are given by those in U - [7 (where (U+ U) is a pure gauge transformation in x-space).

We may extend the solution (5.3.11) to the non-linear case

A_~+ = ~_~+ U + ~ + V (5.3.12)

3 using the identity ~(~+ ~#+) = e~/~e~ 4, so that W = ~4V. It is thus necessary to solve the last constraint in (5.3.1). This may clearly be solved in perturbation theory, since if

the constraint becomes

A~ = E 9"A~ ") n = O

n - 1

D(~+ A~)_) ~ = ~ [A(~')+, A~ "-')] + (5.3.13) r = 0

with the r.h.s, of (5.3.13) only involving lower orders of A~ than the l.h.s. Since (5.3.13) is a constraint limiting only the 0-dependence of A~)+ then a superfield solution will be possible. This can be expressed in terms of B~_+ = d/dg(oA~+ ), for which the constraint becomes

~ + B ~ _ + ~ _ B ~ + = O. (5.3.14)

If we use (5.1.9) and (5.3.1) we obtain a covariantization of (5.3.9), and the resulting action as an integral over the full N = 2 superspace as

ff dt fd'xdSg(W V, +h.c.). (5.3.15)

The gauge invariant properties of this form of the action have been analysed (Howe, Stelle and Townsend 1982); we will discuss its quantum properties in Section 5.5.

5.4. The N = 3 Barrier in gauged su~rspace

We now turn to discuss the maximally extended version of supersymmetric Yang-Mills theory, N - - 4 S Y M , in superspace. We will attempt to use the same method as for N = 2 SYM described in the last section, and take as the most appropriate constraints, the N = 4 extension of (5.3.1):

F~l j+ = F~_(i~_j) = 0 (5.4.1a)

F~+/~_/= 0. (5.4.1b)

1978a) that these constraints, together with the Bianchi

tj F~+#+ = e~+#+ W ij (5.4.2)

It has been shown (Sohnius identities (5.1.4) lead to the equation of motion for W i~, defined by


a s

[] W ° = 0. (5.4.3)

We might expect that weaker constraints than (5.4.1a,b) might avoid this difficulty, but that this is not the case was shown by simple counting arguments for N = 4SYM in 4 dimensions (Rocek and Siegel 1981) and by a different method for extended SYM's in various dimensions (Rivelles and Taylor 1983a). The latter proof is also applicable to the problem of constructing extended supergravities, to which we will turn later, so we will describe it briefly here.

The analysis is performed solely with the fermionic component fields of the extended SUSY algebra ~ 4 and the related super Yang-Mills theory, and then only at the quadratic level in the Lagrangian. This latter is to be of form

4

Lo = Z ,~jlb2j + ~ :gcrl¢ (5.4.4) j = l

where the spinors x¢ and r/¢ are a set of auxiliary spinors with a total of 2A degrees of freedom (where f may run over spin labels as well, though with no differential constraints allowed). We wish to construct the component fields {2j, x¢, tie} from a set of irreps of 6a4 and L 0 is to be the sum of the quadratic invariants for these various irreps. Some of these invariants will enter with negative signs, so that we may use the field redefinition rules (Rivelles and Taylor 1982c)

where x = ~b~ -~b2, r/=/~(~k 1 + ~2)- This redefinition of spinors must produce the auxiliary spinors in (5.4.4) and leave the four physical propagating Majorana spinors 2 I.

We now use that any massive irrep of S~N has 22N- 1 fermi (complex) degrees of freedom, and we suppose that L o is to be constructed from m irreps of 6a4 with positive and n with negative energy. We can now equate the degrees of freedom of positive and negative energies in (5.4.4) to those arising from the sum of the quadratic invariants of the irreps. If we use the reality property of the N = 4 irreps, each irrep will have 26 such degrees of freedom, so that for the positive energy terms

(8 +A) = m" 26 (5.4.6a)

and for the negative energies

A = n. 26 (5.4.6b)

(where we have used (5.4.5) and its generalizations to split the auxiliary terms in (5.4.4) into equal numbers of positive and negative energies). The difference of (5.4.6a) and (5.4.6b) gives

( m - n) = ~ (5.4.7)

which is not possible. A similar analysis in dimension d ~< 10 gives the results shown in Table 4.

We may extend these results up to d = 10, since then a spinor has 16 components, corresponding to 44-dimensional spinors. Thus N = 1 SYM in d = 10 corresponds to N = 4 SYM in d = 4, a feature used (Gliozzi, Olive and Scherk 1977) in the construction of an SU(4)-invariant form of N = 4 SYM. From Table 4 we see that the N = 3 barrier in d = 4 becomes a barrier at N = 1 for all higher dimensions. In particular the N = 3 barrier

62 J. G. Taylor

Table 4.

d 4 5 6 7 8 10

N 1 2 4 1 2 1 2 1 1 1

1 1 (m-n) 11 ~ 1 1 s 1 ~ ~ ~

cannot be avoided by going to higher dimensions and then performing trivial (or non-trivial) dimensional reduction.

The basic reason for the existence of the N = 3 barrier is the exponential increase in N of the fermionic degrees of freedom in 5aN but the only linear increase in N of the number of propagating spinors in N SYM. The extra degrees of freedom in 6e N must therefore be removed, and the only way of doing this appears to be by the use of field redefinition rules (5.4.5) and its analogues. It is like trying to get an ever increasing number of quarts in the same pint pot; ultimately it cannot be done. The same situation will arise for extended supergravities, though the same N = 3 barrier is further remote from the holy grail of N = 8 than for SYM.

Since there appears to be no easy way to avoid the monogamy of the field redefinition rule (5.4.5) we must attempt to displace the barrier at N = 3 to beyond 4 for SYM, and beyond 8 for SGR. This will be possible if we can reduce the number of independent generators S_~ in 6DN by a factor of two. If that is done, then the factor 26 o n the r.h.s, of (5.4.6a), (5.4.6b) becomes 23, and the condition (5.4.7) now reads

(m-n) = 1. (5.4.8)

The N = 3 barrier has now been displaced to N = 5, and there is a distinct possibility that a superspace version of N = 4 SYM will be possible.

There appear to be three ways in which the reduction of the number of independent fermi generators S~ in S~N can be achieved to build a superspace version of N-SYM (and N-SGR's) beyond the N = 3 barrier. One of these is by explicit use of N/2-superfields, wherein the full SU(N) symmetry has been reduced to SU(N/2). This is the method used to build an N = 2 superfield version o f N = 4 SYM (Howe, Stelle and Townsend 1982), which we will describe shortly. The second method uses the light cone gauge, which allows for explicit reduction of all spinor degrees of freedom by a factor of two on removal of the non-propagating spinor components. This was used to construct an explicitly SU(4)- invariant, but not explicitly Lorentz covariant superspace version of N = 4SYM (Mandelstam 1983) which we will turn to in Section 5.6. Finally there is use of the central charges introduced briefly in (1.3.11 ). These allow a Dirac constraint

Si~+ = l ~ l f l - ZiJSfl_j (5.4.9)

which makes the generators Sg+ redundant if the further condition

z * i J z ik = t~kp 2 (5.4.10)

is imposed (which, in fact, results from (5.4.9) and its complex conjugate). This constraint will cause a "spin-reduction" in the component fields of an irrep of ~ N , apparently first recognized by Haag (see Sohnius 1978b) and considered more fully from the viewpoint of representation theory more recently (Rands and Taylor 1983). It is possible that only by use of central charges will the full symmetry of the maximally extended supertheories be possible; we will return to that problem in the last section.


We consider, then, in more detail the use of N = 2 superfields to construct N = 4 SYM. The construction of the latter theory in terms of N = 1 superfields had already been achieved (Fayet 1977), since under N = 1 SUSY, the N -- 4 SYM gauge multiplet (1,14,06) splits into the N = 1 gauge multiplet (1,~) and three matter multiplets (_~, 02). This can be written in N = 1 superfield form as the N = 1 SYM term (5.2.11) together with the extension of (5.2.13) to matter in the adjoint representation for three independent N = 1 chiral superfields ~i

3

Tr (eV qb~-e- v ~ i ) i = 1

together with the cubic term

(ig 13 !)E ijk Tr (~ i [~ j , g}k] - ) + h.c.

We note that certain auxiliary scalars are present here, though no auxiliary fields of higher spin, as there would have to be if N = 4 SUSY were manifest. However the theory has an additional explicit SU(3) invariance, which in combination with the N = 1 SUSY, leads to a satisfactory symmetry structure on-shell.

We may also regard the N = 4 SYM gauge multiplet decomposing under N = 2 SUSY into the N = 2 SYM gauge multiplet (!, ½2, 02) and an N - 2 matter multiplet (_~2, 04). This latter multiplet cannot be constructed as an off-shell irrep of N = 2 SUSY, since all such irreps have component spin at least one. A matter irrep with spin reducing central charge has been constructed in superspace by Sohnius (1978b), and even coupled to the N = 2 SYM in superspace. However the matter superfield has the degeneracy constraint (5.4.10), which has the appearance of an equation of motion in the higher dimensional space of co-ordinates (x t', z,j), where the central charge

z iJ = ~ /~z i j . (5.4.11)

To avoid having to solve such a constraint a "relaxed" hypermultiplet has been introduced (Howe, Stelle and Townsend 1982). This starts with the real external I-spin-1 multiplet gJ ~ ~ and considers the constraint

[-T'F] = 0. (5.4.12)

Here we use the super-tableau notation described in Section 4.1, with a dot in a box i denoting the presence of D~+. The resulting components of L ° would be that of the

N = 2 SYM multiplet with Y = I = 0 with a constrained vector defined as ~ ] . To remove this constraint it is natural to introduce an/-spin 2 multiplet L ° u ,,, ~ with relation

I I I I I ' I - - - - - 0

ffVrq = ~3ZE] (5.4.13)

The last constraint leads to Y = 0, I = 1 as the only irrep of S~ 2 in L ~jkl. These constraints lead to a multiplet of (32 + 32) degrees of freedom, now with no conserved vectors. The constraints may be solved in terms of a prepotential Pal :

z i j k l 21.~(ij l)kl){ r l i f~ot + i ~t -- = S ~ + ~ - ~ a + F i ' b D ~ - p i )

~ J 3--1)ij [)3k r ~ + -'k l l)(i I'~J)kl)l rb~ t+ +h.c . (5.4.14) 2 ~ + ~ t . t . / . , k T 2 ~ " -- kJ~.," + ~L~at + y !

A further multiplet is required to remove (24 + 24) degrees of freedom in the above (32 + 32) relaxed hypermultiplet, and this is taken to be the Y = 0, I = 1 irrep contained in the real

64 J .G. Taylor

scalar superfield V, with constraint

D~+lj+ V i = [O,,+ , D ¢ _ i ] _ V = 0 (5.4.15)

where D~+¢+ ~ " = e~jD~+/~#+. The solution of (5.4.15) is

V = n~J nkl,. (5.4.16) u + a..,, _ ,.a, i j k l

with x~ik~ real and totally symmetric in i,j, k, I. The appropriate superspace action is then

fd4x + +/2JkZxoJ (5.4. 7) dSO[~p h.c. 1

where 2_~+ = D~+j/J j ; the field equations

L ijkl = O, 2_~+ = D~_+ V (5.4.18)

lead on-shell only to the N = 2 matter multiplet. The N = 2 relaxed hypermultiplet may now be coupled gauge covariantly to the

N = 2SYM theory of the previous section by replacing D~_ by the gauge-covariant ~a = D a - A s in the constraints (5.4.13), (5.4.15). The solutions (5.4.14)-(5.4.16) also suffer the same modification except for the addition of an extra commutator propor t ional to

i j ra o r + ~ _ [ W , ~ + p , , ] in the second equation of (5.4.14). The coupled action (5.4.17) remains unchanged, with 2~+ = ~+j / )~ . An N = 2 superfield version of N = 4 SYM has therefore been constructed. Again there are not the full set of N = 4 auxiliary fields, since in the N = 2 version only auxiliary fields with spin up to 1 have been included, whilst spins up to and including 2 would be needed from full irreps of 6e4.

5.5. Quantum properties of supergauge theories

The quantization supergraph rules for the chiral superfield were derived in Section 4.4. To extend these to the case of SYM theories we have to add gauge fixing terms. This has been developed in detail (Ferrara and Piguet 1975). The quadratic part of the action (5.2.11) may be written as

L 2 = - t r VD~+D2_D~,+ V (5.5.1)

where the operator between the superfields V in (5.5.1) is proportional to the projector 1I 1/2 of (4.2.13a) onto the Y = ½ irrep in V. This is gauge invariant under an addition of an attitional irrep with Y = 0. If we add to this the Y = 0 irrep as gauge fixing term then we have

L½ = -½ tr VD(~II o + 1-I1/2)V. (5.5.2)

In particular for ~ = 1 we have, with 1-I o + 111/2 = 1,

L~ = - ½ t r V[--IV. (5.5.3)

The propagator in this case, following the method of Section 4.4, is

( T( V V x ) ) = [] - a f'*(x - x a)64(0- 0 ~). (5.5.4)

The quantum rules of a gauge theory are only complete once the Faddeev-Popov ghost contributions are also obtained. These are given in terms of the gauge transformations of Section 5.1 in the standard fashion, with the gauge transformation of the prepotential V


being

6A V = -- iL v/2 [A + A + coth (L v/2) (A - J~)] (5.5.5)

where Lx Y = [x, Y'] _. The resulting ghost contribution is thus

tr ~d4x d40(c ' + 6')L v/2[(c + 6) + coth L v/2(c- 6)] (5.5.6)

with chiral and anti-chiral ghosts c, ~. In order to discuss the ultra-violet divergences in a sensible manner we must next define

a method of regularization of the contributions from supergraphs so as to be consistent with supersymmetry. The method of dimensional regularization ('t Hooft and Veltman 1972; Bollini and Gambiagi 1972) which preserves gauge invariance does not preserve supersymmetry in superspace. For if only the bosonic dimension is continued from 4 to d the quantity S ddxD2D~L is not 0-independent except in d = 4. If the Berezhin integral itself is continued in both bosonic and fermionic dimensions to S ddx dvO dr0' where v = 2 [d/2], then the propagators will behave like p-~ purely on dimensional grounds, so will violate unitarity due to higher order poles being present. An alternative method of dimensional reduction (Siegel 1979), using continuation in all momenta and space-time co-ordinates to d < 4 but leaving the number of components of all other tensors in 4-dimensional form, was expected to be SUSY-invariant, but was later shown to be inconsistent on general grounds (Siegel 1980) and also in explicit cases (Avdeev, Chochia and Vladimirov 1981). The most favoured method which preserved supersymmetry explicitly is that of addition of higher derivatives. Thus addition of the term to L 2 of form

- t r VD~+ D2_D~,+ fq 'V (5.5.7)

is immediately seen to be gauge invariant (under addition of a Y= 0 irrep to V) in the Abelian case; its form can easily be generalized to the non-Abelian case (Ferrara and Piguet 1975). The gauge breaking term can be taken to be the appropriately modified version of the Abelian gauge breaking term. For r a positive integer the propagators have behaviour as (p2)-r, but there are also extra vertices introduced by these higher derivative terms which give rise to new divergences.

Power counting for N = 1 can be performed to give the overall degree of divergences of a general supergraph as follows (Grisaru 1982). Each vertex has 4 powers of D~+ both for the chiral or gauge contributions. If there are L loops, with P propagators, V vertices and E external lines, each vertex will give a factor 2 2 p2 v. (D+D_), so a total of Each propagator behaves as p-2, but (~b_ ~b_ > and (q~ ÷ ~b + > have an additional factor of 1/p (unless m = 0, when they are zero); we take C such lines. Each loop integral contributes p4 from the momentum integration and p-2 from the 0-variables. Finally each external chiral line has been over-counted by D~: ~ p, so that the overall degree of divergence is

t3 ° = 4 L - 2 L - 2 P + 2 V - C - E c = 2 - C - E c . (5.5.8)

Thus graphs with two or more external chiral lines have O ° = 0, and are even finite if the same chirality occurs on the two external chiral lines when E c = 2, due to S d40q~_ = 0. Thus self mass and coupling terms (m~b2++Oq~3++h.c.) require no renormalization. Supergraphs with two external V-lines apparently have a ° = 2, but this is improved by gauge invariance to 0 ° = 0.

If the higher-derivative regulators are included the r.h.s of (5.5.8) is modified, with all

66 J .G. Taylor

contributions being finite except at 1 loop:

d ° = 2 + 2r(1 - L) - E c. (5.5.9)

Thus diagrams with any number of external gauge lines and no more than two matter lines are divergent. Since these diagrams can be investigated directly they cause no problem in principle.

As a result of the above analysis we can deduce the non-renormalization theorem preventing radiative corrections breaking of N = 1 SUSY if there is no tree-level breaking as stated in Section 2. This follows since all diagrams give an expression involving a full superspace integral ~ d40. But for the integral to be non-zero the integrand must have a suitable number of powers of 0, and this is only possible if some auxiliary field D or F be non-zero at the classical level.

Very powerful non-renormalization theorems have been proved for extended superfield versions of extended SYM theories (Grisaru and Siegel 1982). This depends especially on the use of the background gauge approach (for references see Abbott 1981), in which explicit gauge invariance is not lost since only the quantum part of the gauge field has a gauge fixing condition applied to it. Resulting counter-terms are then only expressible as integrals over full superspace S d4N0 of expressions involving the gauge potentials A~ and not the prepotential V itself. In this case any counter term of form (at the linearized level)

d4x TA= (5.5. 0) d4N O A • 1

can only be constructed with a local operator T (of dimension U with t ~< 0), provided

4 - 2 N + 1 + t = 0. (5.5.11)

Thus (5.5.11) cannot be satisfied if N = 2 or 4, and for these values of N there can be no counter terms for N-SYM beyond one loop. In particular this proves the finiteness of N = 4 SYM as constructed in N = 2 superfields in the last section, since for that case the l-loop divergence also cancels as is known by explicit calculation. Thus N = 4 SYM is a complete finite quantum field theory! (Howe, Stelle and Townsend 1982). An alternative proof of this finiteness will be given in light-cone co-ordinates in the next section (Mandelstam 1982).

This property of finiteness can be extended to a class of N = 4 SYM theories (already discussed briefly in Section 2.5) in which explicit mass terms are added which are soft enough to preserve finiteness (Taylor 1983a; Parkes and West 1983; Namazie, Salam and Strathdee 1983). We will consider these also in the next section in the context of the light-cone gauge.

Finally we note that a further class of N = 2 SYM theories with other N = 2 matter representations than the adjoint can be made finite if these matter representations are chosen to make the l-loop fl-function (and so the l-loop divergences) vanish (Howe, Stelle and West 1983). For this to occur we have that the total fl-function function for N = 2 SYM and m~ matter representations in the representation R~ is (Ferrara and Zumino 1974)

fl(g) = (2g a [ 167r2)[~,m~T(R,) - C 2 (G)] (5.5.11)

where T(Ri) is defined by Tr (T°T b) = T(Ri)6 °b, with T", T b being the generators of G in the representation R~. The r.h.s, is zero, for example, in the case of 2N fundamental matter representations for G = SU(N), when T ( R ) = ½, C2(G ) ----N. The argument has been


completed by extending the N = 2 relaxed hypermultiplet discussed in Section 5 to the case of a general representation R~.

It appears difficult to extend the class of finite N = SYM theories to those which, when massless, possess only N = 1 SUSY. Thus the suggestive argument that N = 4 SYM is a completely finite theory (Sohnius and West 1981a) can be extended to N = 2 SYM from the existence of a global U(2) invariance (Howe, Stelle and Townsend 1982) so that the anomaly multiplet containing the trace of the energy momentum tensor T~ also contains the divergence of a conserved axial current j~. Since T~ oc fl then the theory is finite (modulo l-loop effects). For N = 1 SYM the component Tff still belongs to an anomaly multiplet along with the divergence of the axial R-current. However both of these can suffer radiative corrections to all orders, in general, and the only current which satisfies the usual Adler-Barden non-renormalization beyond one loop belongs to a different N = 1 multiplet (Clark, Piguet and Sibold 1979). Only the use of N = 2 supersymmetry seems able to bring strong enough constraints to make the anomaly multiplet have no renormalisation beyond one loop by bringing these anomalies together. This is a question in which further research will undoubtedly be of value.

5.6. Super Yang-Mills in light-cone superspace

The second method of scaling the N = 3 barrier erected in Section 5.4 was by means of using the light-cone gauge. This was used very effectively (Mandelstam 1982) to construct a superspace version of N = 4 SYM with explicit global SU(4) invariance, though with loss of explicit Lorentz invariance. We will discuss that construction here, though use the Brink formalism (Brink, Lindgren and Nilsson 1983a) who also gave a later proof of finiteness of the N = 4 SYM theory (Brink, Lindgren and Nilsson, 1983b). We will also consider N = 1 and N = 2 SYM as well since they may also be constructed in light-cone superspace (Taylor 1983b; Hassoun, Restuccia and Taylor 1983).

The method uses the light cone co-ordinates x + = (1/x/~)(x°++_x a) and transverse co-ordinates x i (i = 1, 2) (Kogut and Soper 1970). The similarly defined Dirac matrices y+, y- satisfy y+y- + y - y + = 2, so P+ = ½y+y-, P_ = ½y-y+ are orthogonal projections. In a standard representation for the Dirac matrices these projectors extract either the 1st and 4th or 2nd and 3rd components of a spinor, and so clearly give spinor reduction. However the total internal symmetry associated with the extension indices i (1 ~< i ~< N) is unchanged, so that S U(N) symmetry is still preserved.

Light cone dynamics is achieved by taking, for N-SYM, the vector gauge condition A_ = 0, and elimination of A + and ~_ by the equations of motion (or equivalently by Gaussian integration). The resulting component action involves solely physical modes. These may then be put into superfield form using chiral superfields.

Similar expressions arise for N = 2 SYM (Taylor 1983b) and N = 4 SYM (Brink et al. 1983a) each with only cubic and quartic interaction terms, though with only two or zero powers of derivatives in the interaction respectively. Superfield quantization can now be performed and the degree t3 ° of ultra-violet divergence computed for a general supergraph. We find (Taylor 1983b)

a ° = 4 - N . (5.6.1)

Reduction of this quantity can be achieved by extraction of powers of momenta on to external lines, an idea first suggested by Salam and Strathdee for N = 4 (Namazie, Salam

PP/~P-C*

68 J.G. Taylor

and Strathdee 1983), giving

d ° = 4 - N - E ( N = 1,2) (5.6.2)

< 0 (N = 4)

The finiteness of N = 4 SYM is thus proved by explicit calculation, as first suggested by Mandelstam (Mandelstam 1982).

To do that in detail, the covariant derivatives d i and dj are defined in terms of the associated Grassmann varianbles 0;, 0~ as

d ~ = -8/O0~-(i/x/2)O~d_, dj = ~/OOJ+(i/x/~)OjO_, with O_ = (1/x/~)(O0-03);

the indices i,j take the values 1 . . . . , N of the internal symmetry of the N-extended supersymmetry algebra. The gauge multiplet for N = 1, 2 and 4 are contained in the corresponding (complex) chiral superfield 49(x,0,t7) for which di49 = 0, and whose first component is

O7_lA(y). Here y is the chiral variable y = (x+,x - -ix/~.OJOj, x i) and contains all the dependence on 0, in the fashion well-understood from the fully supersymmetric case, whilst

A = (1/w/2)(A 1 + iA2), Ai being the transverse components of the gauge vector field Au in the light-cone gauge.

The non-Abelian Yang-Mills gauge theory may be written in the light cone gauge as the sum of terms (Brink, Lindgren and Nilsson 1983a)

Lv = -½ Tr A[-q-A + 2ig Tr (~- ~ OA[ A, c3-A]- +c3-I¢TA[A,~-A]-)

+ 292 Tr (8- ~EA, ~3__A]_ 8- ~[A, c3_A]_ ) + spinor and scalar contributions. (5.6.3)

We will treat the terms of O(9°), 0(9) and 0(9 2) differently, and construct separate N = 1 and N = 2 superfield expressions for them that, after integration over 0 and 0 agree with the component form (1).

For N = 1 the chiral gauge superfield is

49 = O-IA(y)+O-lO~(y).

From the anti-commutation relation for d and d we have therefore that

491o=o=o = ~-_1A, d4910=0=0=~-1~,, dcTa49[o=o=o = - i v /2A .

(5.6.4)

(5.6.5)

Then the kinetic contribution in (5.6.3) is seen to be reproduced to within a divergence, by

the action of d~ on (i /2x/~)Tr ~ []d_ 49, evaluated at 0 = 0 = 0, the O(g) term by a similar

action on x/~yTrO_~b[~3_49,~49]_ +herm.conj. , where O = 1/~q/2(¢31 q-ig32) and the O(g 2) term in the same way on -g2Tr(~3-1[Sd49,O_~p]_aT_l[8_dc~,O_49]). Thus the total N = 1 SYM superfield Lagrangian in the light-cone gauge is

fd4x dO dO Tr (i/2x/~)q~ ~3 49 + x/~g(O ~p[¢3 ~49] + ~ 4910 c~, Oq~] ) { [] 49,

-920--l[~3_d49,O_{p]_O-_l[O_d~-p,O_49]_} (5.6.6)

which we may also obtain from the N = 1 SYM theory of Section 5.2 by integrating out the non-light cone parts of the prepotential V (Hassoun, Restuccia and Taylor 1983).

We may proceed in a similar manner for N = 2, using the N = 2 analogue of the gauge multiplet chiral superfield (5.6.4) as

49 = 0-_ i A(y) + 0 7_ loil],ti(y)"k- OiOJBij(y) (5.6.7)


where Bij is an antisymmetric tensor in i, j. We may use the N = 2 analogues of (5.6.5) to see, after some algebra, that the superfield Lagrangian extending (5.6.6) to N = 2 is

fd4x d02 d02 Tr -¼~[-q¢ ig(c~[8 O, Jc~] - c~[~ (9, ~q~] { )

-½9z(O- '[O_ ~o,~qS]_aT.'[O_ ¢,d2~o]_ )} (5.6.8)

where we have defined d E = ½eudid J, and similar expressions for ~ , dO 2 and d0 -z in (5.6.8). We note that for both N = 1 and 2 the O(g z) term is simpler than that for N = 4, which involves two terms. We add also that we have chosen to identify the component content by (5.6.4) and (5.6.7) so that, as for N = 4, the superfield is dimensionless. In this way we achieve as close as possible comparability to the N = 4 case.

Finally the N = 4 analogue of (5.5.4) and (5.6.8) is

fdgx { - [] ¢ + ~g (c9- _ + ) dO 4 d0 4 Tr }a 2 1 4;I-¢, a¢] h.c.

-g2(,~:-l[¢,,~_¢]-~--1[,~,a_~]_ +½1-¢,~]_[-¢,~]_)}. (5.6.9) The theories (5.6.6), (5.6.8) and (5.6.9) may be quantized in the now-standard manner by

the addition of chiral sources j and f a s the additional terms

fd4x + (5.6.10) dNOj¢ h.c.

with dj = d e = 0, and N = l , 2 or 4. We rewrite the integrals in (5.6.10) over the Grassmann variables so that the total kinetic energy term from (5.6.6), (5.6.8) and (5.6.10) has the form

fd,~x dNO dN{7(@TK@ cysT j ) 1 + (5.6.1 )

where

( 0 TN) jT = [(dN/pN_)j,(d~V/pN)f],p_ = ix//~a_, ¢T=(~b , 4~), K = TN

and T N is the differential operator

I (i/2x/2)[:]O_ (N = 1)

T N = --¼1-] (N = 2) -~:_2V3 (N = 4). (5.6.12)

The Gaussian integration over ¢ and /~, which are independent fields for N = 1 and 2 (though not for N = 4 where the additional reality condition d*q~ = d'*¢ must be used) gives the propagators

A4~ ¢ = A ~ = 0, A¢~ = (1/TNpN_)~4+2N(z--z 1) (5.6.13)

where t~4+2N(g - - g 1 ) = ¢~'$(X - - X 1 )t~N (o - - 01 )(~N ( O - - ~1 ).

The propagator A¢,~ has the common value for N --- 1, 2 or 4 equal to

Acq~ = - ( 2 / [ ] t~ 2 _ ) 3 4 + 2N ( 2 - - Z 1 ). (5.6.14)

We may now proceed to the superfield perturbation rules arising from (5.6.6) for N = 1. There are three and four-line vertices, described by the O(g) and O(g 2) terms of (5.6.6). The

70 J .G. Taylor

factor 6J(z)/6J(z i) = d 6 6 ( z - z 1) gives a factor of d(d) at the end of each internal ~b(q~) line. Each vertex is integrated over x, 0 and 0, since it is non-chiral.

Let us now calculate the degree of divergence of a primitively divergent supergraph, which we effect by simple dimension-counting of all of the factors in the supergraph. Each (~b~) line will contribute L 1 by (5.6.14), (for some length dimension L) with a further factor of L- 1/2 from each of the factors of d and d at each of its ends. Each vertex will give a factor of L 3, whilst the momentum carried by either three or four-line vertices scales as L - 3 from (5.6.6). If there are n 3 three-line vertices, n 4 four-line vertices and p propagators the total length dimension of the resulting supergraph will therefore be

0 ° = ~na + 2n4 - p - ½ E + 3 (5.6.15)

where the last term of 3 in (5.6.15) corresponds to an overall integral over superspace still to be performed in the effective action and E is the total number of external lines. Using the topological relation

p +½E = ~n 3 + 2n4 (5.6.16)

we find from (5.6.15) that d = 3. We may go further by using the fact that each external line leaving from one of the O(o) or (92) interactions in (5.6.6) always has a further momentum operator ~_ or 0 attached, so reducing the degree of divergence by E, giving finally

0 ° = 3 - E . (5.6.17)

This gives logarithmically divergent charge (E = 3) and wave function (E = 2) renormalisation constants.

A similar analysis may be performed for N = 2 using (5.6.8) and N = 4 using (5.6.9). In the latter case since the measure d4x dSO, dp, ?~ and all vertices in the action are dimensional, every diagram is logarithmically divergent. These can all be made finite again by the extraction of a momentum onto each external line. The final result is the universal formula for the ultra-violet divergence degree for N = 1, 2 and 4 SYM

d = 4 - N - E . (5.6.18)

We see the increased softening of ultra-violet divergences as N is increased. We might conjecture a similar formula for supergravity, so explaining the ultra-violet finiteness of N > 4 theories at one loop, though this is clearly an area to be explored much further. There are also topics in the above analysis about which we have been very cavalier: what form of regularization is appropriate; how may infra-red features be best dealt with, and how does the existence of a divergent wave function renormalizition constant for N = 2 relate to the //-function for this theory, which is non-zero at one loop but zero at two loops and higher? Indeed the problem presented for N = 2 is an important one as a clue to the power of light cone gauge superspace formulations.

We can now see what additional mass terms may be added to (5.6.9) to preserve finiteness. The gauge superfield 4~ is dimensionless, as is the measure d'*x dSO and the propagators (5.6.12) for N = 4. Additional mass terms will give only finite radiative corrections if they are built from a constant spinor ~b o to be of the form

#2~boqSSq~ (5.6.19)

where S is a differential operator of total dimension L' with t = 0. Since the quadratic mass terms obtained from a scalar or fermion covariant mass has/~ of the dimension of mass, q~o has to have no more than four powers of 0 in it. The reduction of covariant spinor mass


terms to the light cone gauge produces cubic terms which lead to a cubic superfield interaction term (Namazie, Salam and Strathdee 1983; Parkes and West 1983) and give masses at tree level which satisfy the super-trace condition (see also Rajpoot, Taylor and Zaimi 1983) even after spontaneous symmetry breaking brought about by the total scalar potential including the quadratic, cubic and quartic scalar interactions (Rajpoot and Taylor 1983b).

6. LOCAL SUPERSPACE

6.1. Super-differential geometry

It is natural to extend the concept of superspace so as to allow for curvature. We might expect that the component versions of the various supergravities of Section 3 would thereby be lifted into a superfield form, so automatically including the elusive auxiliary fields needed to close the gauge algebra. Such has proved possible for N = 1 supergravity and nearly so for N = 2 SGR. However the N = 3 barrier, whose existence we discussed for supersymmetric gauge theories in Section 5.4, enters again and prevents immediate construction of N-SGR's in superfield form beyond N = 3. We will discuss the reasons for this in Section 6.6, but to begin with we must consider precisely what we mean by the notion of curvature in superspace and the idea of a graded or supermanifold, some of whose co-ordinates are anticommuting.

The exact mathematical formulation of a graded manifold has been considered by numbers of authors (Berezin and Leites 1975; Kostant 1977; Downes-Martin and Taylor 1977; Crawedzki 1977; De Witt 1977; Batchelor 1982; Rogers 1980, 1981). The most suitable of these is that of Rogers (1980) since it allows for the most natural inclusion of fermionic component fields. A Grassman algebra BL is composed of anticommuting generators fix . . . . ,ilL, with a basis for BL equal to the 2 L different products 1, fix . . . . . flz, fllfl2 . . . . , f i t . . , fit.. The even and odd parts BL,0, BL,1 are composed of products containing even and odd numbers of factors. The product of m copies of BL0 and n copies of BL,1 is denoted by B~'"

B"~" = (BL,o) ®m (~ (BL,1) ®n. (6.1.1)

A topology can be given to BL, and so to B~'" as an fX-sequence space in the basis of products fl¢u) of the generators, so that if

(u)

for real coefficients x~,) then the E x-norm on x is

Ilxll = ~ Ix~)l. (u)

With this definition of"closeness" it is straightforward to define the notion of a differentiable function of elements of B~'" with values in BL. A supermanifold ./t "~L'" is then defined as a topological space with an atlas, or a collection of mappings ~k~ of open sets U~ covering ~t~'"), into open subsets of B~'" which have the usual infinitely differentiable property on overlaps of open sets. In other words a supermanifold is locally isomorphic to a subset of B""L, and in the usual fashion the local co-ordinates can be denoted by

(X 1 . . . . ,xm, o l , . . . , O n) = (X, O0)

72 J.G. Taylor

where _x ~ BL,0, _0~ BL, 1. The cases we have considered in the last section for N-extended global superspace had m = 4, n = 4N. Our present situation does not allow us in general to regard the O's as spinors; the action of the Lorentz group will only be defined on the tangent space to ~,~m.,) at each point.

The value L of the number of generators of the Grassman algebra B L must be allowed to be infinite in order to have an unrestricted supply of different values of O's. For otherwise no product of N superfields, for any value of N, can contain terms with more than L factors with odd elements:

N

I-I ~bi(x(°,- 0(i) = 0 (N > L ) (6.1.2) i = 1

if ~bi(x(°,_0(°)~BL1. The problems of convergence of infinite series in Boo is completely controlled by the ~l-topology, so that we really need to consider the graded manifold ./¢~'"). We remark that superfields ~b(_x,_0) can now be expanded in the usual fashion in powers of the anticommuting variables 0, with inclusion of anticommuting fermionic component fields as elements of B®. We also note that the above notion of a supermanifold allows for non-trivial global structure in the fermionic variables; for example B~'" /D is compact, where D consists of elements of B~ '" with integer coefficients (Rogers 1981).

The simplest extension of Einstein's ideas about curved space-time is to construct a theory which is invariant under general co-ordinate transformations on J¢~'"). In local co-ordinates z M = (x,_0) we thus require invariance under

z M ~ z M' = ~M(z). (6.1.3)

An important question to be settled is the precise form of tangent space group which extends the Lorentz group occurring in Einstein's general relativity.

One approach was to take the tangent space group to be the whole of O S p ( N / 4 ) when n = N, m = 4 so that tangent space rotations may mix bosonic and fermionic indices. The result is "super-Einstein" theory (Arnowitt and Nath 1975) in which there is a metric gMN (z) and invariant line element

ds 2 = dz M gMlV dz N (6.1.4)

We are here using the Chang summation convention (Chang 1976) where fermionic variables are contracted with a positive (negative) sign when they are raised on the left (right); the raising and lowering is achieved by gMN and its inverse g ~N with gMLgLN = 61~.

The natural extension of Einstein's field equations will thus be

1 R M N - - ~ R g M N = 2gMN (6.1.5)

where we have included a cosmological constant 2. This field equation would arise from the Einstein-Hilbert superspace action

(R +

This theory was analysed extensively and on quantization about a flat background solution shown to have finite radiative corrections to all orders (Taylor 1978; Arnowitt and Nath 1979a). This very attractive result hides two fatal drawbacks of the theory. Firstly the background solution had to be chosen to contain a dimensional constant k and be in the form

ds 2 = (dx u - i~y ~ dO) 2 + k dO dO. (6.1.6)


The first term in ds 2 is the metric of global symmetry, as we see from the fact that it is invariant under the SUSY transformation given by the action of US~ on a superfield. From the representation (4.2.3) this is

6x ~' = ig?~O, ~0~ = e~. (6.1.7)

The second term is also invariant under (6.1.7), but due to the presence of the constant k of dimension L 2 it is excluded from discussions of global supersymmetry. It is essential to include such a term in our present analysis since otherwise the associated metric 9~N is singular, so 9 <°)MN does not exist. Thus the metric of global supersymmetry is not a solution of the equations (6.1.5). The danger is then that ghosts may arise, since, for example the Dirac equation can be changed from ¢~ = 0 to (0 + kl/2[ --])~1 =- O, with a ghost mass k-1/2.

The second problem is that the metric 9 Mu will have irreps with superspin up to 2½ (for N -- 1), and so Poincar6 spins of component fields up to and including 3. It is necessary to impose constraints on the metric 9 MN and connection FNM beyond that of zero torsion, F~LM] = O.

The first difficulty may be avoided by accepting that the background metric is singular, and is given by (6.1.6) with k = 0. It may also be obtained by taking k -~ 0 (Arnowitt and Nath 1979b) though this appears difficult to analyse for N > 1. To proceed using the techniques of super-differential geometry it is then necessary to work with multibeins E~

M B = 6]. As usual the variables A denote tangent space and their inverse Ea u, with E a EM indices, M world indices, with A = (a, ct), M = (m, p), the Latin (Greek) letter denoting bosonic (fermionic) co-ordinates. We may take for the background multibein ea u the value given in Section 5. The second problem will also have to be resolved, so that we will have to find constraints on the curvatures ROABC and torsions TACB in order that irreps with spin higher than two be excluded; this is completely analogous to the discussion on superspace versions of gauge theories in Section 5.

We may start immediately to reduce the irrep content of the spin-connection field f~ACB, which gauges rotations in the tangent space by taking the tangent space group to consist only of Lorentz transformations (Zumino 1975b; Wess and Zumino 1977):

f~cB = 0 (B = bosonic, C = fermionic, or vice versa)

~')~a 1 ct b fl = ~-f~AB(aa)~. (6.1.8)

The conditions (6.1.8) still allow superspins up to 2½ (in f~]b), SO that further constraints must be imposed. We will discuss various approaches to discovering possible sets of constraints for supergravities in the next section. We will finish this section by a brief review of the definitions of torsion and curvature tensors, and of the infinitesimal transformations of various objects. We will do this in terms of the language of forms introduced in Section 5.

The basis of forms is the co-ordinate differential dz M, or in the tangent space E a = dzME~t, so that a p-form ~b may be expanded, in a local co-ordinate frame, as

C~ = E A" A . . . A EA'~gA,...A . (6.1.9)

The tangent space transformations LA n satisfying (6.1.8) act only on the local indices. For these transformations to be gauged the spin-connection one-form flA 8 is introduced, with infinitesimal transformation law under the tangent space group

, ~ f ~ 8 c s c 8 = - d L ~ --k~AIA C --EA~2 C

= --DL] (6.1.10)

74 J .G. Taylor

where D is the covariant derivative w.r.t, f~, and d = dzMt3M = EADA . The torsion two-form T 4 is defined as

T a = DE A = ½ E C E B ~ c (6.1.11)

whilst the curvature two-form R.~ is defined by

R~ dtq~ c B l ~ o ~ c o B = + ~ a f ~ c (6.1.12) -2"~ *~ *~'CDA"

The Ricci identity is thus

D 2 ~ C B C B = - - R A ¢ c + ~bAR c (6.1.13)

and differentiating (6.1.11), (6.1.12) and using (6.1.13) we obtain the Bianchi identities

I A = D T 4 - E B R A = 0 (6.1.14)

I~ =_- DRSA = 0. (6.1.15)

These latter identities are of great value in allowing other torsions and curvatures to be obtained from a predetermined set with simplification arising from suitable constraints, and also in checking if an assumed set of constraints is consistent and does not imply any field equations (differential equations in x-space). In particular they allow the curvatures to be obtained from the torsion and its derivatives (Dragon 1979). For completeness we note the components of the above forms:

2~sc = - (D[sE~])E A + ~'~c] (6.1.16)

R B D A M S F B = E [ ~ D D ] ~ M A (6.1.17) - - ~'~[CA~'~D] F

E B n E n B n E ¢~B = T ~ D ~ E A + D [ C f ~ g ] A (6.1.18) - - a'%CD] ~" ~EA - - ~ t'[CA ~'~D]E

where the last version of (6.1.18) is useful for analysing the value of the curvature at the linearized level.

Finally we have that under infinitesimal co-ordinate transformations (6.1.3) and tangent space rotation A~A, E A and f ~ A transform as

B B IA fiE A = DM~ A +~ ~M + EMAB (6.1.19)

with

and

DM~ A = ~M~ a + ~ B f ~ B

~ ' ~ B A ~ N R B M A _ _ ~ M A 1 B , r~C A I B A I C n B .~- --r ~,,~ MAZ,,.C -- *l A a,c, MC

(6.1.20)

(6.1.21)

6.2. Discovering auxiliary fields

Many methods have been suggested to discover suitable constraints on the superspace geometry described in the last section in order to lead to a satisfactory superspace version of N-extended supergravity. There presently seems no fully super-geometrical approach which assures the existence both of suitable irreps containing the graviton and gravitino and the possibility of constructing a superspace Lagrangian. The most satisfactory is that which starts from component versions and non-linearizes by using the resulting linearized


superspace constraints at the non-linear level; we will describe that method here. It involves the process of "gauge completion" (Arnowitt and Nath 1976; for supergravity see Brink, Gell-Mann, Ramond and Schwarz 1978). This method builds a bridge from components to superspace by first recognizing the components and transformation laws of component versions of supergravity as the 0 = 0 parts of suitable superfields and secondly "completes" these transformation laws to higher powers of 0 by using the geometric structure in superspace described in the last section. The difficulty about this approach is that it needs as input knowledge of the auxiliary fields and their local SUSY transformation laws. This appears to be a vicious circle, but we may use the technique of compensating fields (de Wit 1982) or field redefinition rules (Rivelles and Taylor 1982c) to discover what possible or actual candidates for such auxiliary fields may occur for N-SGR. We will do that in this section and discuss gauge completion and its results in the next section.

We will consider first the case of N = 1 supergravity, and use our knowledge of the spin and degeneracy of irreps with a given superspin Y as described in Section 4.1. We have already remarked that the irrep containing the graviton is that with Y = 3, with spin content (2j,, a, 1A). We have added the suffix P or A to denote physical component fields (dimension L -1) or auxiliary (dimension L-Z). The extra component field beside the graviton and gravitino is an auxiliary constraint vector A t :

t3uA~' = 0 (6.2.1a)

and the graviton huv and gravitino d/u~ are also constrained, with

hb,v] = ~'hu~ = h~ u = ~ud/u: , = ~t~d/ut~ = 0. (6.2.1b)

The degrees of freedom in the various component fields are thus O°(huv ) = 5, g°(At) = 3, O°(d/u:`) = 8. The linearized Lagrangian invariant under the N = 1 SUSY transformations is

i AUA u +-2 ~u~d/u - h u~ [-]hu~. (6.2.2)

The non-locality may be removed from (6.2.2) by insertion of a further [] in each term, the resulting Lagrangian being that for linearized conformal supergravity. The use of the superconformal groups has been much emphasised by de Wit and collaborators (de Wit 1982), where compensation is performed at the fully non-linear level. However in order to explain the underlying features of gauge completion and the range of possible supergravities we will initially restrict our discussion to the linearized level (6.2.2). In any case conformal ideas cannot help broach the N = 3 barrier.

To remove the constraints (6.2.1) so that we can describe N = 1 supergravity with no differential constraints on any of the fields At , d/t:,, huv but only the expected gauge invariances for d/u:` and hu~ and A u auxiliary we require the ultimate degrees of freedom to be O°(hu~) = 6, O°(Al,) = 4, O°(d/u~) = 12. Such counting indicates the need for a further spin ½ and two extra spin zero fields to be added to the Y = ~ multiplet. This is to be effected so that recombination of fields can explicitly occur so as to remove the differential constraints in (6.2.1); the SUSY transformation laws can then be obtained for the redefined fields from the known ones for the original fields.

One of the field redefinition rules to achieve this is

A 2 - (t3uq~)2 = B 2 (6.2.3)

so that B u is unconstrained, with

A t = B u - [ ] - lSuS~B~ , c k = D - I O ~ B u.

76 J .G. Taylor

We may write this so-called "annihilation" rule (because it combines a spin 1 and a scalar field into a purely auxiliary field, vanishing on-shell) as

1 a - 0 p ..~ 0. (6.2.4)

We have already met the fermionic annihilation rule

1 1 ~-~- ~ 0 (6.2.5)

in Section (5.4). There are also the "creation" rules for recombining fields so as to form the linearized Einstein or Rarita-Schwinger Lagrangian, these being

2 e ( h u 0 - 0p(b) = 2LEinst(hm) (6.2.6)

b = x / ~ ( h ~ - [] - l~at~bhab )

3(~],u)--½(,~, ) = 1LRs(I///~ )

2 = ? . ~k - ¢- 1_0. _~ (6.2.7)

where 0p(~b) denotes the Lagrangian ½(Ou~b) 2, etc and LRS(~Ou) is the Rarita-Schwinger Lagrangian for pure spin 3 of (3.1.4).

We may now use these annihilation and creation rules to add a compensating multiplet to the Y = a Weyl multiplet. Two chiral multiplets (with opposite chirality) allow us to construct the suitable subtraction sum, using (6.2.4)-(6.2.7) as follows:

Y

3_2 (1~ 2~ 3) !* 0,~) 0 + - ( %

o - - (o~ ½- o ~ )

Bm h~, v ~bm, S,P (6.2.8)

We have included the parity assignments on the bosonic fields and chiralities on the spinors. The net effect of (6.2.8) may be written more succintly as

Ly=]_(Lo+ +Lo_ ) = 1 2 2 LEinst(h#v) + LRS(~#~ ) --~(B m + S + p2). (6.2.9)

This is the minimal auxiliary field set (Stelle and West 1978; Ferrara and van Nieuwenhuizen 1978) with the smallest possible set of (12 + 12) bose and fermi degrees of freedom. The total non-linear Lagrangian is given by taking the non-linear versions of Lvins t and L R S in (6.2.9) (in terms of the vierbein (e~), and an extra factor of det (e~) in the last three terms. The non-linear local supersymmetry transformations may then be obtained from the linear SUSY transformations by the technique of supercovariantization of derivatives. This removes derivatives of the local SUSY parameter e~(x) from entering in SUSY transformations. The supercovariant derivative D~ °v of the derivative O~,A of a field transforming under SUSY as 6A = gB is obtained by adding the connect ion-- - i~B: DC~°VA = D~,A - ~B. Then 3(D~ °v A) is free of any term containing due. In this manner the fully non-linear local SUSY transformation rules for the minimal set of auxiliary fields may be obtained as:

=V2Du+ysfB~L_~ ~ 1 v'X 1 -] 61p ~, --~7~,7,B ) - S T ~ ( S + ?5P)Je


. _ _ leys(g~ v 1 "~ v ~ 1 fiB s = ~te \ - ~ ' ~ y v ) R +ilcgy B A b t , - ~ i x g 7 0 ~ B u

1 1 b d - ~ i x ~ (S + r5 P)~'5 ~ - ~ iKe~ ~ Bb gY5 ~'~ ~a

1 1 1 V 1 6S = - ~ ie- gy~ R ~ --~ ixgy5 B v ~v - itcgy ~v S + -~ itcg(S + Y5 p)ru ~O u

1 l 1 ~ ~ 1 u 6P = - - ~ i e - g),syuR~' + ~ i x g B ~b~-ixg~, ¢ ~ P - ~ i x g y 5 ( S + ysP)y tPl,. (6.2.10)

where R ~ = e~W~ysyvDp$ ~ is the gravitino field strength. The algebra of local SUSY transformations may then be found to close without use of the equations of motion as

[3 ~, 32] _ = 6susv + 6L.T. + 6G.C. (6.2.11)

where the parameters of the supersymmetry, Lorentz transformation and general co- ordinate transformation are respectively

_ _ / £ ~ 2 ~ja , 2 2 l --~Keab2pB ~ 1 2 +4ixg~tr.b( S+ Y s P ) / 3 2 , ~ 2 = ig2yagl" (6.2.12)

Other auxiliary field sets may also be found. Historically the first such (Breitenlohner 1977) had (20 + 20) degrees of freedom and used two Y = 0 and two Y = ½ irreps, with the following subtraction table:

Y 3 2

0 2 + (0/, 1 2

_1~ - ( 1 ~

or, in concise form

(1] 2~ 3+)

!+ o~) 0~; 2 (1~ 0~ ½+)

O~ 1+ ~- )

V m A m n m hu~ 21,22 ~k~ S ,P (6.2.13)

L3/z+Lo+ + L o _ - L x / 2 + - L 1 / 2 _ = L E i n s t + L R s + ( S 2 + p 2 + A 2 + V,,,-B,,,).2 2 (6.2.14)

Again the linearized transformation laws may be obtained by the field redefinition rules, and then non-linearized by supercovariantizing derivatives; the non-linear version of the Lagrangian (6.2.14) involves the full non-linear forms of the first two terms and an extra factor of e = det e~ in the last four terms. We note here the fact that some auxiliary fields enter with positive, some with negative signs in (6.2.14); all have negative sign in (6.2.9).

A third set of auxiliary fields may be obtained (Sohnius and West 1981b) by using the new field redefinition law

1] - 1] ,~ 0. (6.2.15)

which arises by the recombination

U ~ - V~ = - OuAv~UvaaW~ (6.2.16)

where U~,, V u are both constrained and

A u = U u - V u +g.t., U u + V u = e ~ , v ~ V w ~a.

78 J.G. Taylor

The compensating multiplet now has Y = ½, and the subtraction sum is Y 3 (2~, 3 + 5 - 1~)

- ( 0 ~ x+ 1~) 2

hu~ ~b~ Au, Wae (6.2.17)

and

L 3 / 2 - - L 1 / 2 = LEinst(h/~v) + LRS(~Ot~) + F uv(A ) W ~* (6.2.18)

where W uv* = ½eu~W~,. We note that there is a gauge invariance of this (12 + 12) "new minimal" set under 6A# = OvA, 6W~a = 0[aA~] which must be taken account of when we turn to discuss the superfield aspects; it corresponds to the conservation of the R-current of Section 2. This is because the Y = ½ compensating multiplet, when regarded as containing the possible anomalies TuU, aujf, c~uj ~'5 (where Tuv is the energy-momentum tensor, Jt,~ the supercurrent and j5 the chiral or R-current), has no room for Ouj u5, which must therefore remain conserved.

We can analyse other possible sets of auxiliary fields for N = 1 supergravity by use of the field redefinition rules suitably extended to incorporate annihilation rules for higher spin. In the case of auxiliary fields with spin only up to one, but obtained by combining all possible multiplets, only the above three can be shown to be irreducible (de Wit 1982; Rivelles and Taylor 1983c, 1983d; Kugo and Uehara 1983). If higher spin auxiliary fields are allowed then potential candidates for compensating multiplets have been discovered at the linearized level (Rivelles and Taylor 1983c) such as a set with compensating irreps with superspins (0, 0, ½ +, 1, 1) and auxiliary fields with spins up to 3. This and other candidates require further analysis to see if they can be non-linearized.

N = 2 SGR may be treated in exactly the same manner as for N = 1. The Weyl multiplet containing the physical fields is now that with Y = 1 and has spin and SU(2) content

'3+ 32 1~+~-,,3 12 ,~+~ (6.2.19) z , p , ~ , XA , - ~ , v p ) .

There are two physical vectors present in (6.2.19) of which one must be made auxiliary. To do that we must introduce the new annihilation rule

1~ - 1~ ~ 0 (6.2.20)

which is proved by the identities

Tuv = ODV~ ] + e~,~a~'~A ~, T~2 = Fro(V)2 _Fro(A)2. (6.2.21)

In order to remove all the unwanted fields from (6.2.19) in a minimal fashion we can try to use the Y = 0 multiplets [tp,~tl- 12, vpnl+T, vA)nax and ~xa,~"- X2,vv, v a n 3 m+T~p which differ by their overall dimension. In fact these are the only possible ones to use since only the second can remove the 1~4 and the first the 1e in (6.2.19). The resulting subtraction sum is

Y 32 1~ 1+ 1~ ½2 Ol~) 1 (2~

1-2 12 o,~) 0 - - ( 03 2

0 - - (0~ ½2 - - 1 p 0 I )

huv qJ,~ Vu tuv Aam 2i, rf B,, S S 3 (6.2.22)


and concisely as

1 F 2 t2 AiJ 2 ,~i B 2 _ S 2 L I - ( L o + L o ) = LEinst(h#v)WLRs(~l#a)--g[( #v(V)] - # v - ( rn) -- q i - - ra - S i J S i j (6.2.23)

The non-linearized version of this (40 + 40) set was obtained (de Wit and van Holten 1979; Fradkin and Vasiliev 1979; de Wit, van Holten and van Proeyen 1980); we will not give it in detail here, but refer the interested reader to these references.

There are no other minimal (40 + 40) versions of N = 2 supergravity without central charges, since the Y = ½ multiplet has new spin 3 fields which can only be made auxiliary by going to a larger (non-minimal) set (Rivelles and Taylor 1982b). Nor does it appear that solutions with auxiliary fields of higher spin can be constructed easily. This situation is important to clarify if the construction of off-shell N-supergravities for N > 4 are to be achieved since then the multiplet containing the graviton has fields of spin ~> 3 which must be made auxiliary. We will reconsider this problem further after we have brought in multiplets with central charges. We will not go beyond N = 2 in our analysis, since the N = 3 barrier intervenes (see Section 6.6).

6.3 . Constra ined superspace

In order to deduce the exact nature of the constrained superspace geometry appropriate to a given set of auxiliary fields as described in the last section we proceed by expanding the torsion (6.1.16) and curvature (6.1.17) and the transformation laws (6.1.19)-(6.1.21) about the fiat SUSY background in powers of the gravitational constant x as:

to obtain

with

= eM + KeMHB, f~MA

(o) A (o) A T~BC = T~ic + Xt~c, TBc = 2i(7"C)~r

= +

rBDA B (°)E B = DtCdPD]A + TdDqbEA

(6.3.1a)

(6.3.1b)

(6.3.1c)

(6.3.1d)

(6.3.2a)

(6.3.2b)

(6.3.3a)

(6.3.3b)

where ~,4 = ~MEA and D A is the flat SUSY background covariant derivative. After a certain amount of algebra we may also obtain the closure relations

C (°)A (o) (o) (o) [ j I , J 2 ] H A ~-lz TpB + A B C A C A C A = DB(12 -~I~12A + H B A 1 2 c - A 1 2 B H c + ~ x 2 D c H B

[J1 , 62]~bBA : - - DCL~2A + ~I~2DE¢cA + q~caA12o -- A12A¢CD -- A12CCDA

(6.3.4a)

(6.3.4b)

80 J .G. Taylor

(o) B (o) A where the parameters ~2, A I 2 A and ~12 are given by

(o) A (0-3- (o) ~A (O~B-- x A ~12 = 2i~17 a ~2'(a12 = ~[2]')BblJ (6.3.4c)

(o) b 1o) (o). (0).4 b (0) (01 A 1 2 a = AElaA2]c,I_~2a = ~ [ 2 D A L 1 ] a + c b b b c ~ AxEaL2]c+LDaA2] c. (6.3.4d)

We choose the identification of the component linearized vielbein h~ and gravitino ¢~ and connection co~, as usual as

b • • b ego (x) (6.3.5a) H~ I 0=o oc h . (x ) ,H . I 0=o oc ~b. (x), ~bc. 10=o =

as well as the (gauge) conditions

n~lo=0 = H~10=o = ~ol0=o = 0. (6.3.5b)

In order to preserve the gauge conditions (6.3.5b) we find that at 0 = 0 we require, taking only pure SUSY transformations into account, with

~"= A. =0 , = ~ ,

that

1o) 1o) D t~" = - ~ Tra~ - ~? D rHa~ (6.3.6a)

Dl3~ ~ = -(°~)~D~H~- L~ (6.3.6b)

DrL~ (°~n ~.b (6.3.6c) = % x,~'~Wy a.

These latter equations allow the 0(01) component of ~.4 and L~A to bc calculated from 0 0 values, as required for the method of gauge completion to work. In order for the r.h.s, of (6.3.6a-c) to bc expressed more geometrically it is natural to reparametrize ~A as

(o) - -A ~A = p A _ ~71-17 (6.3.7)

so that (6.3.6a-c) become at 0 = 0

(o) (o). to) T ~ ) - p Tra (6.3.8a)

(o) 10). D #p*' = ~? ( T , g H J - T ~ # ) - L~ (6.3.8b)

(% (°)c b b o b D~L a ~ (R6~ a - T,~)c, ) (6.3.8c)

and involves purely geometrical quantities on the r.h.s. From (6.3.2a,b) and using (6.3.5a) and (6.3.7) we also find

6 (0) 6 Hb. = O .p b + g~ ( T~a -- H° T$~ ) + L ] (6.3.9a)

6H~, = O.p ~ + g~(- Ta~ + C~a~) (6.3.9b)

6 ~ +, -6 ~, (6.3.9c) = - c3cL . + e R&.

and we also add the transformation

6TBA = gD~,r~.4 (6.3.9d)


Also (6.3.4a-d) become

(o) [ 6 , , 8 2 ] - H ~ aap~2 + ~ b b = #a2OcHa +L12a (6.3.10a)

(o) [61, 62]-Ha ~ = 63aP~2 + ~;2OcHaa (6.3.10b)

b (0)d ['(~l, 32] - (Pc% = - - 63cL12a+ ~12 ad~bL (6.3.10c)

(0) P~z = ~ (T~ H ~ - T~) (6.3.10d)

(o) P~2 _g~2L~]B + -p-r d • = gZE1 (Tv~H a - Tv~) (6.3.10e)

= e2e 1 ( - (6.3.10f) • ~,~tVca - r ~ x # a p.

We may now use equations (6.3.9a-c) and (6.3.10a-f) and the transformation laws and commutator parameters of a given set of auxiliary field transformation laws, such as (6.2.10) and (6.2.11) for the minimal N = 1 set to determine the required torsion and curvature constraints (initially at 0 = 0 and hence for all 0).

In the case of the "new-minimal" set (6.2.16) and (6.2.17) we must include a further U(1) group in the tangent space beyond the purely local Lorentz group, and for N = 2 a local SU(2) group can be included (though this is not essential unless the SU(2) group is to be gauged). Such gauging is performed introducing a further gauge connection superfield A M (and A M N ( Z ) for the antisymmetric tensor Wvv in (6.2.17)). Associated field strengths and their transformations can now be included, as well as gauge conditions similar to (6.3.5a,b), such as at 0 = 0

Aa[o=o=Aa(x), A ~ l o = o = 0 , Aab[O=o=Wab(X), A~a[O=o=A~a[O=o=O. (6.3.11)

It is also necessary to go beyond the lowest non-trivial power of 0 to determine a suitable complete set of constraints for the non-minimal N = 1 model and the minimal N = 2 model. This is due to the presence of auxiliary spinors in these models, and gauge completion must be applied to these fields as well (since it is on spinors that the gauge algebra fails to close without auxiliary fields). The auxiliary fermions of dimension L-3/2 can be only part of the torsion T~+ = T~+#_, so that we must use the equation (6.3.9d) and the known SUSY transformation law of that auxiliary spinor to identify the missing constraints. The torsions resulting from these steps vary with the models being considered, but can be unified by means of a parameter n first introduced from a different point of view (Siegel and Gates 1979). Such a unified set of torsions, combining the minimal and non-minimal versions can be obtained. They include the conventional constraints for determining the components of the connection

T~ b = 0 (6.3.12a)

that arise from the transformation law of the graviton (in four-component spinor notation)

T]a = 0 (6.3.12b)

and from the commutator law on the graviton

T ~ = 2i(ycc),,& (6.3.12c)

82 J.G. Taylor

The commutator on the gravitino gives results depending on the auxiliary field set being considered:

T~+/~+ = 0, T~+~+ ~+ = -2~'~+,/~+,!x~+'r~ T~+~_ = ½~-T~+. (6.3.12d)

For the minimal set

T,+ = 0 (6.3.12e)

so T~a --- 0, whilst for the non-minimal set we may take

R = ( n + l ) - ( D r + T 2 / n + l \ " ( 3 n + l ) \ ~+ + ~3ff--~) TyTTr+)' (6.3.12f)

where R = ..~+/~+o~+~-+ and the auxiliary spin A~+ oc T~+ ", the minimal set corresponds to the singular limit n = -½, and the most geometrical form to n = - 1 (Bedding, Downes- Martin and Taylor 1979). We add that other versions of torsion constraints, with the same physical content, are obtained by covariant redefinitions of supercovariant derivatives (see, for example Brown and Gates 1980).

For N = 2 the torsion constraints for the (40 + 40) minimal set have also been obtained by this method (CasteUani, Gates and van Nieuwenhuizen 1980) (and by a different geometric approach by Breitenlohner and Sohnius 1980), and are

T{+__~_ r - = T ~ + ~ + = 0

a a T~_+~+ = T~_#__ = 0

T~+~_ = 2i(7~C)~+#_~}

T~b = 0

T ~ = 0

(6.3.13a)

(6.3.13b)

(6.3.13c)

(6.3.13d)

(6.3.13e)

(6.3.13f)

(6.3.13g)

t + T~_~+ = T{+~_ = 0

E P+ T a _ i ~ + j k = 0 (i,j.k)

D r - ,~i/J + ,~_/~+j+h.c. = 0. (6.3.13h)

This set of constraints has been solved at the linearized level to prove that they contain one Y = 1 and two Y = 0 irreps, together with gauge irreps (Grimm 1982).

As we have already mentioned, there are numerous other approaches to obtaining superspace constraints. The constraints (6.3.12a-e) were originally written down (Wess and Zumino 1977) where it was shown by use of the Bianchi identities that they did not imply equations of motion. It is also possible to argue on dimensional ground, where the various components of the torsion have the following dimensions:

L ° : T~/~ L- 1/2 : T~#, T~,

L- * : T~b, T~, L- 3/2. 'r~ (6.3.14) • Lab .

These torsion components at 0 = 0 are to be identified with the possible auxiliary fields of corresponding dimension, as we saw earlier from the presence of the torsions in the transformation laws of the fields in an identical fashion to the auxiliary fields in component


versions. If there are no scalar fields then there can be no contributions to the torsions of dimension L ° other than the constant (TaC)~#. If there are no auxiliary spinors, similarly all dimension L-1/2 torsions must be zero, This approach may be exploited (Howe 1982) to determine constraints for N ~< 4 conformal supergravity, and also to obtain the constraints for on-shell Poincar6 supergravity (where setting certain torsions to zero forces the theory on-shell since they are proportional to the field equations).

Another method, already described for supersymmetric gauge theories, considers the requirement of preserving certain irreps covariantly in the presence of gauge fields--the method of representation preservation (Gates 1979; Stelle and West 1979, 1980). This allows the constraints (6.3.12) or 6.3.13) to be divided into three classes: conventional, representation preserving or symmetry breaking. The conventional constraints, such as (6.3.12a) allow a superfield (in this case f~b) to be solved for algebraically in terms of the other superfields present. We have already discussed representation preserving constraints in the context of supersymmetric gauge theories, and they are very similar in the super- gravitational context. Thus for N = 1 to preserve the chiral constraint (4.2.9) we require

[D~+,D/~+]+q~ = 0 = - T~+~+DAdP. (6.3.15)

For this to be valid for any q~ requires

Tr~+t~+ = T~+~+ = 0

which are constraints contained in (6.3.12c) and (6.3.12d). The new feature of the conventional and representation preserving supergravity constraints is that they have invariances under complex dilatations and local SU(2) transformations (for N = 2) as acting on the fundamental parts E M of the vielbeins EA M. Partial gauge choices can then be made on recognition of the torsions T~+ = T~+~__ and T~+#_Uk ) as connections respectively for those transformations. Thus the constraints (6.3.13f- h) impose restrictions on the gauge parameters so allowing the former to be considered as partial gauge choices. Again this approach will have difficulties when attempting to get through the N = 3 barrier, especially since a crucial input is that as to which representations are to be preserved. We will return to this question later when we discuss local superspace in the central charge framework.

An alternative approach has been formulated using the superspace analogue of the energy-momentum tensor and called the superspace translation tensor or STT (Bedding and Lang 1982). This is defined from the invariance of a superspace Lagrangian ~ under local translations ~a(z) by the derivation of the conserved Noether current jB which, if suitably improved, can be written as

jB _ ~ATnA. (6.3.16)

Due to the conservation equations on jB and on the parameters ~a (which restrict them to belong to the super-conformal or super-Poincar6 group) large parts of T ] are not independent of other parts, and may be transformed to zero. By coupling H I , defined by (6.3.1a) where we are working at the linearized level, to T4B similar relations between parts of H I are then predicted. This leads to torsion constraints which depend on the shifts, and can be made to agree with the known sets for N = 1. However the reducibility of further proposed sets (Rivelles and Taylor 1983d) is not clear in this approach. It would be interesting also to see if this method allows one to proceed to off-shell torsion constraints beyond the N = 3 barrier.

There are also completely different approaches with the same final results. One of these is to consider the translation group as the analogue of a Yang-Mills group, so that the

84 J .G. Taylor

translation generators correspond to those of a Lie group. By analogy with the prepotential superfield V in (Section 5.2) we introduce the prepotential

H = H M ~ M = H ÷ (6.3.17)

with transformation similar to (Section 5.1):

e lit --* e iA+e iHe- iA . (6.3.18)

In order to preserve the chiral superfield ~b with ~l,_~b = 0 under the transformation 6~b = - iAq~, with A = AMdM we require a~_ A m = 0~_ A ~ + = 0. Since A u- is unconstrainted it can be used to gauge H ~'- and H ~'+ to zero, leaving the axial vector superfield H m. For example we can see the component fields h b, ~] of supergravity as entering in the 0-expansions of H a and ~b:

H a = C a ( x ) + . . . + iOTbOh'~,(X) + t~O~k a + (OO)2Aa(x)

dp = 1 + . . . + ~ O ~ B ( x )

where A a and B are the minimal auxiliary fields of (6.2.10), with B = S + iP. This is clearly satisfactory to encompass the minimal version of N = 1 presented in the last section. Constraints must be imposed in the remaining parameters to eliminate superconformal invariance: it is more convenient to introduce a compensating chiral scalar superfield ~b which can act as a compensation for this extra gauge freedom by relaxing constraints otherwise present. The non-minimal version requires the spinor superfield H ~' in addition.

In the minimal case it is then possible to construct covariantly transforming derivatives / ~ + , / ~ _ and/~a from H " and ~b; these are

ff~,,_ = d?- 1/2 D , ,_ , ff~,,+ = (1. e i H ) - 1/2 e - i n dp- 1/2 D ~ + e iH

where

(6.3.19)

1. e - / ~ = e - / In+0 "~)] 1,~/- = HM~M.

Finally the derivative/~a is defined as

/ ~ a = - 2i(CT,) ~ + ~- [ /~ + , / ~ - ] + + suitable Lorentz connections.

Multiplication o f / ~ + by E ~/2 and/~a by E, where E = (s d e t / ~ ) - 1/3 gives vector fields E M which may now be used to determine torsions and curvatures, which correspond to the minimal case (6.3.12a-e). The non-minimal case may also be considered (with the parameter n entering).

Another very similar approach starts solely from the axial vector superfield H m ( z ) and constructs a covariant differential geometry by means of an elegant use of geometry in chiral superspace, the set z+ = (x m, 0 ~+) (Ogievetsky and Sokatchev 1978). This has also been extended to N = 2SGR (Sokatchev 1981) with interesting results, though we refer the reader to the references for further details. We are brief on this because it appears difficult to make such powerful use of the notion of chirality when going past the N = 3 barrier. We must also be brief for somewhat the same reason about other interesting approaches, such as that of relating co-ordinate and field variables (Schwarz 1980) and of use of the group manifold approach (Chamseddine and West 1977; Ne'eman and Regge 1978; D'Adda, d'Auria, Fre and Regge 1980).


6.4. Superspace supergravity actions

It is now necessary to construct actions for supergravity which are integrals over the whole of superspace. This should then allow the supergraph rules and power counting described in Section 4.4. to be applied to the resulting theories, so maximally including any possible bose-fermi ultra-violet divergence cancellations. The most natural superspace action for N = 1 supergravity is the volume of superspace (Wess and Zumino 1978) since it is clearly of the correct dimension (where we have included x explicitly):

I=~fd'xd40E (6.4.1)

where E = det E~t (defined as a superdeterminant by using the supertrace rule det E~t = exp (str lnE~), where s tr2] is defined using the Chang convention). We note that this definition is unique to N = 1, though for N = 2 the chiral measure ~-2d4xd40+ is also dimensionless; for N > 2 even the chiral measure has dimension and no such simple expression as (6.4.1) can be written down.

The action I of (6.4.1) is to be considered a constrained action, being defined in terms of the independent superfields satisfying whatever constraints on the torsions or curvatures, corresponding to a given set of auxiliary fields, are being considered. That (6.4.1) is a satisfactory action may be shown by deriving the field equations as the variational equations 6I = 0, where the variations in E ~ are subject to the constraints. By solving the Bianchi identities for N = 1 it is possible to show (Grimm, Wess and Zumino 1978) that for the minimal set of torsions (6.3.12a-e) all components of the torsions and curvatures may be determined in terms of three superfields W~+#+~+, G~+~_, and R and of their complex conjugates. The variations 6H] which preserve the constraints can then be shown to be described purely in terms of the vector H b and the scalar ~b, (which latter can be gauged away) and related to those of the last section, with

= ~- 2 fd4x d 'OEtHb(r:~ + r:;-_ ) - 8Rck - 8R*~b*]. (6.4.2) 31

The resulting field equations are thus the correct ones

G~+#_ = R = 0. (6.4.3)

In order to be able to quantise the action (6.4.1) it is necessary to solve the constraints in terms of a suitable set of prepotentials, as was done for supersymmetric gauge theories in Section 5. From our discussion of the Siegel-Gates and Ogievetsky-Sokatchev approach described in the last section we expect the prepotentials to include a vector superfield H m along with a scalar or spinor superfields, ~, H ~. In both of these approaches it is possible to construct invariant Lagrangians most directly. These may then be analysed both to show that they imply the correct field equations for supergravity and also to determine the linearised superfield Lagrangian and interaction vertices prior to quantisation.

In the Siegel-Gates formalism, with inclusion of the compensating scalar ~b for conformal invariance, the resulting action is (Siegel and Gates 1979)

6 fd'xd'O(g)- I = -~-~ 1/3(1' e-il~)l/a(ck e-iHq~) (6.4.4)

where the notation of (6.3.19) is used, but instead of (6.3.19) we have

g~+ = e-inD~+eiH, ff~_ = D~_, ff~ = - i (Cya)~+~-[~+,g~_]+ (6.4.5)

86 J .G. Taylor

and /~ = s det ga s. The total is invariant under the transformation (6.3.18) with ~b transforming as a chiral density

i ~b --, e i^ - ~ (~,A" - D~A ~)q~ (6.4.6)

= D_E , e-mD2 U -, where A = iAADA with A " = -i(yaC)~+~_Dl~-U +, A ~+ 2 ~+ A ~- = all depending on the spinor gauge parameter L~. The quadratic action in H and 4, arising from (6.4.4) is

12 = fdgx&O[-6~x+2 i ( x - ~)OH - ~Hla [] H a - ~ ( a • H) 2

- I ([D~-'D~+ ]-H~-~+ )2 + D2+Ha " (6.4.7)

where q~ = (1 +x). The action (6.4.7) is invariant under the infinitesimal transformations

6H~ + t~ - = - 2(D~ _ L~ + - D~ + L~_ )

6x = -D2_U+ D~+ x-½(D2_D~+ L~+ )(1 + x ) (6.4.8)

with an unconstrained spinor gauge parameter L~. Similar constructions may also be given for the non-minimal and new minimal auxiliary fields (6.2.13) and (6.2.17) (Siegel and Gates 1979; Gates, Ro~ek and Siegel 1982). Thus there is a complete solution to the problem of constructing superspace actions in terms of unconstrained superfields for the three known sets of auxiliary fields and their corresponding torsions for N = 1 supergravity.

The situation for N = 2 is not at all as satisfactory. There have been attempts at solving the constraints (6.3.13a-h). A solution has been obtained in terms of a constrained N = 2 vector and spinor-isospinor superfields H a, H *+ of dimensions L-a, L-1/2 respectively (Sokatchev 1981), analogous to the unconstrained superfields of the non-minimal version of N = 1 supergravity. A solution has also been given_(Grimm 1982) of the linearized constraints in terms of constrained superfields Z a, S, T, R~ (where R~ is traceless, i,j = 1, 2). In the former case a superfield Lagrangian has been constructed in terms of multibeins, though in terms of the constrained basic superfields. On the other hand an unconstrained superspace action has been constructed at the linearized level in terms of the prepotential ~O ~ (Siegel and Gates 1982). This is expected to be the appropriate prepotential, since it corresponds to the component field of highest (mass) dimension present in the component version of the action (6.2.21) which is the auxiliary spinor of dimension L-5/2. A linearized superfield action can then be written down using the unique local combination of projectors (see also Rivelles and Taylor 1982a)

[] (1rl,O + - zr~,o + - 2rCo,o - ) (6.4.8)

(where 7rr,1+ projects onto N = 2SUSY irreps with superspin Y, super-isospin ! and chirality ___ ). However it is difficult to understand how it is possible to construct a multibein EA M with a world vector index m without having an unconstrained superfield with such an index.

6.5. Supergravity ultra-violet divergences

Supergravity has the gauge invariances of local Lorentz invariance, general co-ordinate invariance and local supersymmetry, so requires gauge fixing with associated Faddeev-


Popov ghost terms. If auxiliary fields are not present, so that the gauge algebra is not closed, then in general there may be ghost-ghost interactions of arbitrarily high order (Fradkin and Fradkin 1978; de Wit and van Holten 1978). These terms may all be shown to be absent except for the 4-ghost interaction by analysis of N = 1 supergravity in d = 11 (de Wit, van Nieuwenhuizen and van Proeyen 1981). This fourth order term arises, for example, on eliminating the auxiliary field A from the ghost Lagrangian (~Ac + A 2).

The first important indication that supergravity had ultra-violet divergence cancellations was the discovery that, although it was not renormalizable, S-matrix elements are finite for 1- and 2-loop effects. This was shown initially by explicit calculation at the l-loop order (Grisaru, van Nieuwenhuizen and Vermaseren 1976) due to the one-loop counter-terms being proportional to field equations. At the 2 loop level there are no supersymmetric quantities which are of the right dimension to act as 2-loop counterterms and be non-zero (Grisaru 1977). However this is not true at the 3-loop level for N = 1 supergravity, where the supersymmetric completion of the square of the Bel-Ribonson tensor is an on-shell superinvariant, at least at the linearized level (Deser, Kay and Stelle 1977). It is necessary to confirm that the 3-loop invariant survives at the non-linear level.

For N = 1 supergravity only the tensor W~+#+~+ survives on-shell, where

W~p~ ~ F~p~ + C~p~O ~ (6.5.1)

with F~r the spin -3 field strength and Capr~ the Weyl tensor (now in 2-component spinor notation). We could consider the quantities

W~,prW "pr, ( W, pr W~'~r) *. (6.5.2)

but on integration over superspace these reduce to linear combinations of the topological invariants equal to the Euler number and the Pontryagin number of the manifold (Duff and Stelle 1982). The first non-trivial counterterm is therefore, contributing at 3-loop order

( W~p~, W ~l~') ( W~p,z W ap;~ )* (6.5.3)

which is the square of the Bel-Robinson tensor described earlier. Invariants for all higher loops can be obtained by taking covariant derivatives of the above.

For extended supergravity, higher loop invariants have been constructed which might act as counterterms (Deser and Kay 1978; Howe and Lindstr6m 1981; Kallosh 1981, 1982; Howe, Stelle and Townsend 1981). In particular by use of the expression of all on-shell components of the torsion and curvature in N = 8 supergravity in terms of the spinor superfield W~ k (Brink and Howe 1978) it is possible to construct full non-linear invariants which can act as counter-terms at the 8-loop level. This invariant is

x14 .fd4x d320E . tl, r~ I/I/ijk'~2 (6.5.4) ~ r r i j k V r ~ t ! •

This can be suitably contracted to give a 4-loop counter term for N = 4 supergravity. It may be that such fully non-linear counter-terms are already present at 3-loops, as they can be constructed at the linearized level for N = 1, 2, 3, 4 and 8 (Kallosh 1981). Thus it would appear that unless a miraculous cancellation of the co-efficients of these counter-terms occurs then these N-extended supergravities cannot be finite beyond ( N - 1)-loop order for N >I 3, and beyond 2-loop order for N = i and 2.

This disappointing conclusion is not necessarily valid, however. Even if we resolve the problem of discovering a satisfactory supersymmetric regularization scheme we still need to consider the nature of the off-shell quantum properties carefully. It has been shown (Gates,

88 J .G. Taylor

Grisaru and Siegel 1982) that the radiative corrections to matter-supergravity couplings uniquely determine the auxiliary fields for N = 1 supergravity; to avoid the violation of local supersymmetric Ward identities only the minimal version of auxiliary fields is allowed. This result indicates that the question of which auxiliary fields are chosen can determine radiative corrections very crucially. It may be that the non-linear counter-terms arising from (6.5.4) have no satisfactory off-shell continuations when a suitable off-shell superspace formulation of these theories has been developed. We can argue this by analogy with the counter-term constructions described above, where symmetry arguments are used to show that certain lower loop counter-terms are not allowed since they would violate the symmetry. It is possible that some higher symmetry exists for N = 8 supergravity, when constructed in a full superspace form, which prevents the appearance of any counter-terms ?

A further possibility is that of miraculous cancellations. Such a one until recently was that of the vanishing of the fl-function for N = 4 SYM described in Sections 2 and 4. That can now be understood in terms of non-renormalization theorems preventing the construction of any counter terms. Yet before the construction of the N = 2 superfield version or the N = 4 light cone superspace version of that theory it was only by detailed calculation of the fl-function up to 3 loops that the theory was expected to be finite (Avdeev, Tarasov and Vladimirov 1980; Grisaru, Ro~ek and Siegel 1980; Caswell and Zanon 1981). The analogous construction of N = 8 SGR has not yet been completely possible, but even if it had the non-renormalization theorem (Grisaru and Siegel 1982) would only have been able to prove finiteness up to ( N - 1) loops for N-SGR due to the possible construction of the N-loop counter-term. This agrees closely with the results mentioned earlier, but may be misleading, due to the new features needed to construct N-extended supergravities past the N = 3 barrier; we will return to that question later.

We can also take some heart from the result on the vanishing of the fl-function for gauged N-extended supergravities at l- loop for N > 4. This result depends on the helicity sum rule for supermultiplets (Curtwright 1981)

( - 1) 2~ d(2)2 k = 0, N > k (6.5.5) 2

where d(2) is the number of states with helicity 2 in a supermultiplet. This may then be used to prove the vanishing of the fl-function determining the amount of renormalization of the gauge coupling constant y by means of the result that fl depends on 22 and 24 (Duff 1982); fl = 0 for N > 4 then follows by use of (6.5.5). Similarly the conformal and axial anomaly also vanish for N > 2.

We conclude that a crucial problem in the analysis of ultra-violet divergences of extended supergravities at higher loops is to construct an extended superfield version of such theories. The presence of the N = 3 barrier to such attempts must now be described.

6.6. The N -- 3 barrier in local superspace

Since the original construction of N = 8 supergravity in terms of physical fields by means of dimensional reduction from N = 1 supergravity in eleven dimensions (described in Section 3.4) a vigorous search has been made for possible auxiliary fields or superfield formulations. Yet the present progress has been very slow, with the only possibility which has been completed satisfactorily being the case of N = 1 supergravity. Partial results are available for N = 2, as we discussed in Section 6.2 though an unconstrained superfield formulation has not yet been constructed. No auxiliary field or super field formulations have


been discovered for N/> 3 supergravity in d = 4. The great optimism expressed by various researchers in the late 70's that the major problems had been solved by constructing superfield versions of N = 1 SGR described in Section 6.2 has evaporated, helped by discovery of the on-shell counter-terms presented in Section 6.5.

That there is a true barrier at N = 3 to the existence of such auxifiary field or superfield formulations was first realised by attempting to combine together representations of N = 3 and N = 4 SUSY by means of the field redefinition rules mentioned in Section 5.4 and Section 6.2 (Rivelles and Taylor 1981, Taylor 1981b). This result was then simplified and also extended to higher dimensions, along the lines of the analysis of Section 5.4 (Rivelles and Taylor 1983a). We will use the same method of considering solely the fermionic terms in a linearized Lagrangian, to see if the field redefinition rule (5.4.5), and its higher spin fermionic analogues, can achieve recombination of the fields into the physical spinors and a set of auxiliary spinors.

As in the N-SYM case we use that the fundamental irrep of the N-SUSY algebra SP N has 22N-lr/v complex degrees of freedom. We have included the factor rn to take account of the possible reality of this irrep, as we did in Section 5.4 when N = 4. r~; = ½ when N is even since then the highest spin component field is bosonic and can be required to be real, otherwise r s = 1 :

{~ N o d d (6.6.1) rN = N even"

The physical fields of N-extended supergravity are N-gravitini and M spin-~ fields with M = 0, 0, 1, 4, 11, 26 and 56 for N = 1 to 8 respectively. The linearized N-extended supergravity Lagrangian must therefore be

N M

L N = ~ LRs(~//.ui)q'- ~ I f f f l J~ j+~t r l t . (6.6.2) i = l j = l

We use the creation rule (6.2.7) to decompose the Rarita-Schwinger Lagrangian LRS = L(~)-L(½) , where L(j) is the Lagrangian for a field of pure spin j. We also use the annihilation rule (5.4.5), so that also express the auxiliary field contribution in (6.6.2) as

~ £ ¢ q t = ~ felkZt - ~ #¢lkPt (6.6.3)

where the spinor fields z¢, p¢ may have vector indices, but always occur in pairs with opposite sign s for their kinetic energies.

As in Section 5.4 we now use that the linearized Lagrangian LN can only be constructed from sums and differences of the quadratic global SUSY-invariant actions of a set of irreps of ~ v . If there are a total of m times the fundamental irrep of 5aN with positive sign and n with negative sign then we may equate degrees Of freedom with positive and negative sign separately. This results in the equations

4N + 2 M + A = m "22U-lrs

2N + A = n" 22N- lr N (6.6.4)

where A is half the total number of degrees of freedom of the auxiliary spinors. Subtracting the two equations in (6.6.4) we obtain

(m - n) = (M + N)/(22n-2rN). (6.6.5)

The values of ( m - n) are given in Table 5. For N i> 3 we see that ( m - n) is less than one, which is impossible, thus indicating the existence of a barrier at N = 3.

90 J. G. Taylor

T a b l e 5.

d N m - n d N m - n

11 1 2 -7 5 1 2 10 1 1 2

2 2 - 7 3 2 - 4 9 1 2 - 7 4 2 - 7

8 1 ~<¼ 4 1 1 2 2 - 7 2 1

7 1 ~<s ~ 3 2 - 2 2 2 - 7 4 2 - 2

6 1 3 5 2 -7 ~1 2 - 4 2 -~r 6

3 ~<s x- 7 2 -~ 4 2 - 7 8 2 - 7

We note that for N = 1, the value of m - n = 1 agrees with the cases presented in Section 6.2. Thus m = 2, n = 1 corresponds to the minimal or new-minimal sets of (12+12), whereas m = 3, n = 2 corresponds to the non-minimal (20 + 20) set. For N = 2 the minimal (40+40) set corresponds to m = 3, n = 2.

We may extend the above analysis to d > 4, for suitable N, and find that the barrier is reduced to one at N = 2. We remark that the value of ( m - n) is the same for N = 1, d = 11 and all supergravities obtained by trivial dimensional reduction without contraction to d = 4. A similar equality occurs for either values of N and d. We also remark on the unique higher dimensional case of N = 1, d = 10, for which ( m - n ) = 1. A linearized superfield version of this has indeed been constructed (Howe, Nicolai and van Proeyen 1982). The dimensional reduction of this theory is o f N = 4 supergravity plus 6N = 4 super-Yang-Mills theories. This is to be seen by noting that the component fields of N = 1, d = 10 SGR are (Chamseddine 1981) e~, ~kM~, AMN and a scalar S. On dimensional reduction there are 6 vectors e~ u as required for N = 4 SGR in d = 4 and a further 6 vectors Aip; the other fields give precisely the extra fields to give the stated theory. This is also seen to avoid the N = 3 barrier with the same value of ( m - n ) as for its N = 1, d = 10 ancestor. Undoubtedly a non-linear version of these theories exists, and a constrained superspace geometry is possible to construct, though that has not yet been obtained.

Disappointingly N = 8 SGR has m - n = 2- 7. A similar approach to that of the preceding paragraph of adding further multiplets, might be considered for N = 8 SGR in d = 4, or N = 1 SGR in d = 11. There are, however, no matter multiplets available in this case, only the fundamental preon of (3.4.7) and its higher superspin analogues. Thus to avoid the N = 3 barrier in this manner would require the addition of 27 similar preonic multiplets. The difficulty of coupling these together non-linearly looks insuperable due to the well-known problems of consistent couplings of higher spin fields, so that this avenue of escape seems unavailable.

We may scale the N = 3 barrier for N-SGR in a similar manner to that for N-SYM, as discussed in Section 5.4. In principle the basis of the method is to reduce the number of spinorial generators i (S,+, S~_i) by a factor of 2. If that is achieved then it is possible to extend the N = 3 barrier in d = 4 to N = 8 (and similar extension occurs in higher dimensions). For the fundamental irrep of 6e 4 has 2 6 (complex) fermi degrees of freedom, so reducing the denominator of (6.6.3) in the case N = 8 by a factor of 28, and giving ( m - n) = 1 (if reality of the fundamental irrep is not taken account of). Thus we have to


choose methods to reduce the number of spinor generators by a factor of 2 (higher factors are needed i fN > 8 is to be considered).

As we discussed in Section 5.4 there appear to be three methods to achieve this reduction: (i) by requiring only N/2-SUSY, and representation in terms of N/2-superfields, as in the construction of 4-SYM by means of N---2 superfields described in Section 5.4. For N = 8 SGR this corresponds to the use of N = 4 superfields; (ii) by using the light-cone approach in supergravity, in analogy to its use in SYM described in Section 5.6; (iii) by use of spin-reducing central charges.

We will not consider (i) in detail here, except to note that the N = 4 relaxed hypermultiplet can be constructed by direct analogy for N = 2. For the latter this can be considered as the combination (in a Lagrangian)

A A ~o,o + (~o,3 -~oa, a) (6.6.6)

where @oa,~ is the Y = 0 irrep of N = 2 with super-isospin I and auxiliary highest spin (vector) field; the field redefinition rules of Section 5.4 and Section 6.2 allow the physical spectrum of (6.6.6) to be shown to be (½2, 02), which is that for an on-shell multiplet of N = 2. Similarly for N = 4 the relaxed hypermultiplet can be written in the same notation as

(I)oAo -~ ((I)oA,15 -- (I)0A,15). (6.6.7)

This has the physical spectrum (34, 116, ½2s, 032). To this may be added the N = 4 SGR + 6(N = 4 SYM) mentioned above as avoiding the N = 3 barrier, which has the physical spectrum (2, 34, 112, ½2s, 03s), and resulting in the N = 8 preon. The details of this are given elsewhere (Bufton and Taylor 1983b).

The light cone approach has also been attempted (Taylor 1983c), though only the third order vertex has been constructed for N = 8 SGR in the light cone superspace. In any case the resulting theory does not appear to have enough total symmetry to prove finiteness of the resulting perturbation theory to all orders. This is also true of the N = 4 superfield form of N = 8 SGR described as (i) above, since the non-renormalization theorems discussed in Section 5.5 for supersymmetric Yang-Mills theories (Grisaru and Siegel 1982) may be used to prove finiteness of N-SGR only up to (N - 2) loops. At (N - 1)-loops a counter-term may be written down in terms of N-extended superfields as

x 2~N-1) _fd4x d4NO • E (6.6.8)

(where there may also be a further dimensional factor involving the scalar field strengths in (6.6.8)). Thus the N = 4 superfields version of N = 8 SGR can only be proved finite for 1 and 2 loops. In order to go beyond this we must extend the construction to an N = 8 superfield formulation, but now by means of enlarging the algebra by method (iii), by addition of spin-reducing central charges. It is possible that they may modify the structure of the theory drastically and, in particular, may change the ultra-violet divergence properties.

6.7. Central charges

We wish to develop a framework for N-SYM and N-SGR beyond the N = 3 barrier which is geometric but involves the central charges on the r.h.s, of (1.3.13) in a non-trivial fashion. This was briefly discussed in Section 5.4, where the spin reducing "degeneracy" con-

PPNP-D

92 J .G. Taylor

dition (5.4.9), and its complex conjugate, was noted as essential. In fact the possibility of using the algebra (1.3.11) and (1.3.13) to construct N-SGR has been considered by various workers. In all cases the degeneracy condition (5.4.9) is necessary, since by its consequence (5.4.10) the central charges vanish on-shell. They therefore do not violate the commuting nature all such central charges would need, as we discussed in Section 1.3 (Haag. Lopuszanski and Sohnius 1975) since the Z ° we are now considering do not belong to the centre of SU(N). As we shall see they cause the on-shell chiral SU(N) symmetry to be reduced (see the discussion associated with eqn. (6.7.5).

An early attempt was made to construct N --- 8 SGR, using only the fundamental irrep of 6e s with one central charge satisfying (5.4.9) (Cremmer, Ferrara, SteUe and West 1980). This followed a similar version of N = 4 SYM with a degenerate central charge on the gauge multiplet (Sohnius, Stelle and West 1980a; Taylor 1980b). The reason this is possible may be seen simply from the field redefinition rules of Section 6.2. The Y = 0 spin reducing multiplet of N = 4 SUSY with a single central charge has the same content as N = 2 SUSY, but classified by USp(4) (the antisymmetric metric of USp(4) arising from Z ~i itself). The

1 x ½4, 05 content is therefore ( P,A, P,A), where doubling up of all fields occurs by the action of Z on a component field; on shell, with 1 a ,~ Op, this reduces to the correct physical fields of N = 4SYM, (1p, ½", 0~,). A similar situation occurs for N = 8SGR, with the Y = 0 degenerate multiplet being ("~1 38 127 148, ,*2 ~'-P,A, ~-, P,A, 0e,a) (the degeneracies being the dimensions of the antisymmetric representations of USp(8)) , and the use of 2 A ~ 1p, 1A ~ 0p leads to the N = 8 preon on-shell. In both cases the components are constrained, and though in the latter case the constraint can be solved in a local fashion in the non-Abelian case the resulting theory is not explicitly Lorentz covariant (Hassoun, Restuccia, Taylor and West 1983); similar problems arise in the supergravity situation (Hassoun, Restuccia and Taylor 1983).

A fully non-linear version of N = 2 SGR was constructed with a single degenerate central charge on one of the two compensating multiplets (de Wit, van Holten and van Proeyen 1980; de Wit and van Holten 1981). Linearized versions of N --- 2 SGR were constructed with two central charges on compensating multiplets (Rivelles and Taylor 1982b) though the presence of only one gauge vector indicates that it would be impossible to construct a non-linear version of such a theory.

Candidates for linearized N = 4, 6 and 8 supergravities were suggested in which only the compensating multiplets has degenerate central charges (Taylor 1981 a) as well as candidates for N = 4 SYM (Taylor 1982b). In particular the solution

~oe.6 z z z - - (I) 1,6 - - (I)0,5 x 6 + (1)0, 0 (6.7.1)

is valid for N = 4 SYM at the linearized level, where cb z denotes a multiplet with degerarate central charges. This solution has auxiliary fields of spin no higher than 2. It may be shown by means of the field redefinition rules, that this solution has no internal symmetry at all, so that there must be at least 5 non-trivial central charges present. A similar situation arises in the case of N = 8 SGR, and also for N = 4 and 6 SGR.

Let us consider this in a little detail for N = 4 (Rands and Taylor 1983). In that case we may take the 6 real generators of S U (4) as the real 4 x 4 anti-symmetric matrices _~i~, _flij with

~--- /3 2 [~*',flm]- = O, ot~m = etm.ct., fleflm - era.ft., fie = ~" = 1 (1 <~ ~ , m , n <~ 3). We may write

Z ~j = Z " ot ~j + Z_ 1 • fl_ij. (6.7.2)

We may take representations with an increasing number of the set _Z, _Z') non-zero. The symplectic symmetry has symmetric generators anti-commuting with Z ~j and antisymmetric


generators commuting with Z iJ. This is because if M is an infinitesimal matrix the transformation S~+ ~ (1 + M)S~,+ leaves (1.3.13) unchanged provided

( I + M ) Z x ( I + M ) T = Z or M Z = - Z M 7".

Thus if M = M r then [M, Z] ÷ = 0 and if M = - M T, [M, Z] _ = 0. Along the direction Z 1 the ten USp(4) generators are

(~l,_fl;~2fl, ~3fl) (6.7.3)

where we have divided the set into antisymmetric and symmetric ones respectively. Similarly along Z 2, Z a, Z~, Z~, Z~ the generators are respectively

(~2,_~;~1~,~3~); (~3,_~;~1~,~); (/~,-~,/~-~,/~-~); (/h,~;/~-~,/~-~); (/h,-~;~l-~,~-~).

(6.7.4)

If two different irreps are present with different directions for Z ~j then the total symmetry will be generated by the intersection of the sets in (6.7.3) and (6.7.4). Thus if there are central charges Z 1 and Z 2 the common generators are (fl ;~3fl) with group USp(2) × USp(2); if there are three central charges the common generators are (~_), generating SU(2). The chain of symmetries is, for increasing numbers of central charges

SU(4) ~ USp(4) ~ USp(2) × USp(2) ~ USp(2) ~ U(1) ~ 0 (6.7.5)

where the first group in (6.7.5) is the symmetry with no central charges and the last with 5 or 6. A similar situation occurs for N = 8 and higher N. Thus the largest possible symmetry group the redefined fields allow determine the number of central charges needed to break down the original SU(N) by a chain like (6.7.5).

In the case of N = 4 SGR there is another criterion determining the number of central charges. A more careful analysis shows that in order for the resulting redefined fields to transform in a local fashion under SUSY, for example in the solution for N = 4 SUSY (Taylor 1982b)

z z Do - ~o,o - ~o,s (6.7.6)

it is necessary to have at least 2 independent central charges. It is not possible to use the method of dimensional reduction by Legendre transformation

(Sohnius, Stelle and West 1980b) to obtain multiplets with more than one central charge, since after reduction in one dimension a further dimensional reduction uses the higher dimensional field equations which set the previously obtained auxiliary fields to zero.

Multiplets have been obtained with more than one central charge (Bufton and Taylor 1983c) and the general theory of representations of SUSY developed in that case (Rands and Taylor 1983). The general representation is still highly reducible, however, and in general involves infinite sets of component fields (Sohnius, Stelle and West 1981; Gorse, Restuccia and Taylor 1983), though constraints can be determined which allow reduction to finite multiplets (Bufton and Taylor 1983c; Restuccia and Taylor 1983a).

These developments would therefore seem to allow the detailed analysis of various candidates for auxiliary field structures for N-SGR and N-SYM beyond the N = 3 barrier. However in order to make full use of the central charges we need to consider the problem from a geometrical point of view.

This is especially so since none of these analyses has allowed the construction of a superfield action. The only superfield actions that have been presented have not used a full

94 J .G. Taylor

integration over superspace but only the integration of a constrained superfield over a subspace of 0 and 0 (Sohnius 1978b; Taylor 1980b). The difficulty in using the full superspace measure for N-SUSY, d4x d2NO÷ d2NO_ is that its length dimension is ( 4 - 2 N ) , which becomes progressively more negative as N increases. In order to have N-SGR analogue of the N = 1 SGR action, which is the full superspace volume presented in Section 6.4 (Wess and Zumino 1978) it is necessary to adjoin to the space- t ime variables x~(1 ~</~ ~< 4) further bose dimensions z e with 1 ~< f ~< 2 ( N - 1). Thus for N = 2,2 extra bosonic variables are required, whilst for N = 4 and 8 there are needed a further 6 and 14 bose variables respectively. Thus the naive N-SGR extension of the N = 1 value when the required extra dimensions are to be included, is

r-2 fd4xd2(n-1)z fd4NO (6.7.8)

which might be expected to be valid for N = 1, 2, 4 and 8 if suitable constraints are to be imposed on the E ~ extending those in the case of N = 1. A similar formula

f d4xd2(N-')z f d4NOTr(F~) 2 (6.7.9)

may also be conjectured for N-SYM. The same extra variables are also required to be able to give a dimensionless form to the action for a scalar superfield ~b, of dimension - 1, of form

fd4xfd2(N-l'zfd4NO'~p+(9. (6.7.10)

Such an action might be hoped for as the appropriate form to describe the N = 2 hypermultiplet (Sohnius 1978b) or its N = 4 and 8 analogues (Rands and Taylor 1983). The formulae (6.7.8)-(6.7.10) require very careful analysis before they can be expected to be used satisfactorily. In particular we have to ask how we can interpret the extra dimensions and integration over them. They are expected to be related to the central charges Z u, but exactly how is not initially clear.

A natural explanation of the extra bosonic dimensions Z 1 . . . . . Z2(N-1) are that they are truly present in nature but only become accessible at suitably high energies such as the Planck energies. Such an interpretation is the basis of the Kaluza-Kle in ( K - K ) approach, which we discussed in Sections 3.3-3.5.

However if we wish to remain in d = 11, the N o - G o theorems of the previous section are not avoided; auxiliary fields and a superfield formulation will not exist. Moreover for N = 8 SGR there are 14 extra bose dimensions, not 7, so that there is a decided mismatch between the known N = 1 SGR in d = 11 and its extension to d = 18. Indeed N = 1 SGR in d = 18 would unavoidably have fields of spin higher than two amongst its components, so would be expected to be inconsistent. We conclude that the K - K interpretation is the wrong one for the extra dimensions.

If we are not to regard the new dimensions Z1 . . . . . Z2(N-1) as ones in which we can move freely we might turn to the opposite extreme and suppose that we can never venture off a four-dimensional submanifold M 4 embedded in 2(N + 1)-dimensional space-t ime S2(N+ 1). The question we must answer is then to explain how the integrals (6.7.8)-(6.7.10) over the whole of this 2(N + 1)-dimensional space- t ime can only describe dynamics on M 4 and not in the whole of S2(N+ 1)"

A start has been made to solving this problem (Restuccia and Taylor 1983b) where it has been shown that the actions (6.7.8)-(6.7.10) must be interpreted as constrained actions, the


constraints reducing the problem to one in boundary value control theory. The boundary in this case can be shown to be precisely the four-dimensional manifold M 4 embedded in a singular fashion in the boundary of a graded manifold ~,~N+1~4~¢ (in the notation of Section 6.1). A detailed evaluation has shown that (6.7.10), suitably constrained, gives the correct equations of motion, and also has only positive energies. Further work is needed on (6.7.8) and (6.7.9), as well as the quantization rules and properties of-the resulting Lagrangians.

7. CONCLUSIONS AND FUTURE DIRECTIONS

We have now reached the point at which we can begin to assess how far the questions asked in the introductory section have been answered. The problems we faced were:

(1) the marriage of gravity and quantum mechanics, and (2) the unification of gravity with the other forces of nature.

We argued that supersymmetry, possibly in higher dimensions, should allow us to proceed some way along solving both (1) and (2). How far has it indeed done so?

As far as (1) is concerned some progress has been made with the remarkable discovery of finite supersymmetric Yang-Mills theories in four dimensions. This has been the "holy grail" of quantum field theorists ever since the realization that such theories are usually plagued with ultra-violet divergences. However the extension of these results to gravity via supergravity has not been so successful, and the situation is still an uncertain one as to whether or not any extended supergravity is strictly finite to all orders of perturbation theory. The possibilities of using central charges and the geometric features thereof, mentioned in the previous section, indicate that there still may be hope, but a lot of subtle problems have yet to be solved if that route is taken. If this path fails then it may be necessary to follow the ideas of superstring theory (Green and Schwarz 1982) which we have not been able to devote any time to here. It may also be that the finite N = 2 and 4 SYM theories themselves give a satisfactory model of the world in which both presently observed particles and gravity itself are bound states. Such a result seems presently somewhat unlikely, seen from the viewpoint of the difficulties of fitting the standard or unified gauge models into such theories, as we discussed in Section 2.5. Moreover there is a certain degree of arbitrariness associated with the parameters of the explicit SUSY-breaking mass terms allowed by the theory and also with the gauge group to be chosen. In addition there will be a very large gauge hierarchy problem of explaining why gravity is so weak at present energies; why is me so much greater than the observed particle masses ?

For (2) it is clear that maximally extended supergravity, either in d = 4 or d = 11, does allow for curvature producing forces other than gravity. Until we can construct a satisfactory quantum field version of these theories we cannot say if the forces other than gravity, or the composites of the preon, are in any way related to presently observed forces and particles. However there are intriguing questions associated with spontaneous compactification at the classical level, especially that of stability of solutions, which we have not been able to describe.

We can see therefore that the future directions in supersymmetry and supergravity are:

(1) to pursue applications of the finite quantum field theories of non-gravitational form (as described in Sections 2 and 4) to see if they are relevant to nature, and if they can encompass gravity as a bound state theory;

96 J . G . T a y l o r

(2) to comple t e the c o n s t r u c t i o n of the fully geomet r ic forms of N = 8 supe rg rav i ty ( and N = 4 s u p e r - Y a n g - M i l l s ) u s i n g the degene ra t e cen t ra l charge r e p r e s e n t a t i o n s so as to reduce the theor ies to d e p e n d on ly o n four d i m e n s i o n s (as descr ibed in Sec t ion 6);

(3) to inves t iga te fu r the r the s p o n t a n e o u s s y m m e t r y b r e a k i n g features of N = 8 super- g rav i ty in its c o m p o n e n t field fo rm (as descr ibed in Sec t ion 3) in d = 4 or 11, to see if a n y of these co u l d be r e l evan t to phys ica l app l ica t ions .

There has c lear ly been a grea t dea l of p rogress in the last year or so in the subject , b u t there is still m u c h m o r e to be d o n e before the u l t i m a t e goa l of physics , o f " s u p e r u n i f i c a t i o n " of all of the forces of na tu re , ap p ea r s to be possible .

Acknowledgements--I would like to thank my colleagues throughout the world for their help over the years in explaining the exciting things occurring in the subject and especially my research colleagues and research students at King's College for their stimulating questions and discussions.

R E F E R E N C E S

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Documents

A review of supersymmetry and supergravity