Volume 65B, number 1 PHYSICS LETTERS 25 October 1976

S U P E R G R A V I T Y A N D G A U G E S U P E R S Y M M E T R Y *

P. NATH Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA

and

R. ARNOWITT*'t Lyman Laboratory o f Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Received 3 September 1976

It is shown that the recent model of supergravity considered by Freedman, van Nieuwenhuizen and Ferrara, and by Deser and Zumino, is a limiting case of gauge supersymmetry.

It has been suggested for some time [1 ] that super- symmetry may have a bearing on gravitational theory. Thus Volkov and Soroka proposed supersymmetric

a structures involving a spin 7 field interacting with a spin 2 (gravitational) field. That such a combination of

• 3 spins is relevant follows from the fact that a spin 7 and spin 2 particle forms a massless supersymmetric multiplet [2]. When the conventional (or global) super- symmetry [1 ] is generalized to a local gauge super- symmetry [3], a detailed theory of gravitational inter- actions emerges within the framework of a unified gauge theory. Gauge supersymmetry is based on a single tensor superfield gAB(Z) where z A - (x u , 0 '~) and 0 ~ are the anti-commuting fermi coordinates of the fermi-bose supersymmetry space. Invariance under the general coordinate supersymmetry space group

z A = z A' + ~ A ( z ) , ( 1 )

leads to unique field equations of the form RAB = XgAB, X = const. Here RAB is the contracted curva- ture tensor constructed from the "metric" gAB [3]. The Einstein gravitational field g~v(x) arises from the 0 ~ independent term in the superfield expansion of g~v(z). Other physical fields come from other sectors, e.g., the Maxwell (or Yang-Mills) fields appear in the linear 0 a terms of the fermi-bose component gu~(z),

Research supported in part by the National Science Foundation.

* On sabbatical from Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA.

]" John Simon Guggenbeim Memorial Foundat ion Fellow.

and the ' 1 spin ~ ("quark") fields arise in the quadratic 0 a parts ofg~a, etc. The vacuum state supports a spon- taneous symmetry breaking of the local supersymme- try to global supersymmetry. For the simplest case of X :/= 0 (which we will consider in this note) the spon- taneous breaking occurs only for Dirac 0 ~. Simultane- ously, scale invariance is spontaneously broken allow- ing mass growth in certain sectors with M 2 ~ X. Unifi- cation of gravity with other interactions then occurs (with Einstein constant G E ~ e2/X) obeying the usual gravitational threshold theorems.

Recently Freedman et al. and also Deser and Zumino [4] have proposed a model of supergravity

• 3 based on interacting spin 2 and spin ~- fields• Since, as we will explicitly exhibit below, the local gauge trans- formations of suporgravity postulated by these authors is a subset of the general gauge supersymmetry group

• 3 of eq. (1), it would be surprising if this spin 2-spin supergravity were not in some sense a "special" case of the above described gauge supersymmetry. More

3 precisely, we will show here that the spin 2 - 7 super- gravity is a limiting case of gauge supersymmetry to the singular supersymmetric space limits of Zumino and Woo [5] when one neglects the additional "source" fields (e.g., electromagnetic and quark) which auto- matically arise in gauge supersymmetry.

In sect. 1 below we discuss the form ofgAB(Z ) 3 which is covariant under the spin 2 - 7 local gauge

transformations. The field equations that arise in the limiting case are discussed in sect. 2 and some addi- tional comments made in sect. 3.

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Volume 65B, number 1 PHYSICS LETTERS 25 October 1976

1. Gauge complete metric. Under an arbitrary trans- formation eq. (1), the gauge change of gAB is given by +1

~gAB(Z) = gAc~C,B + (--1)a+ac~C,AgCB + gAB,C~ C. (2)

The global supersymmetry transformation is generated by ~r = i~TrO, ~a = 3.% where X a is an infinitesimal constant anti-commuting c-number. The metric that is invariant under this global transformation has the form

grv = %v; g ~ = - - i ( 0 ~ ) ~ r , (3)

g , ~ = k,~,~ e + (~)~(0~.)~, where k is an arbitrary constant. As mentioned above, eq. (3) represents the vacuum state of gauge supersym- metry, (OIgABIO), for Dirac 0 a coordinates. The pa- rameter k plays a crucial role in the following. For an inverse metric gAB to exist (and hence for supersym- metry space to be Riemannian) k must be non-zero. However, as we will see, the limit k ~ 0 does indeed exist in gauge supersymmetry for Dirac 0 a precisely because the spontaneous symmetry breaking from local to global supersymmetry occurs. It is this limit- ing situation that is related to the spin 2 - -~ supergrav- ity of ref. [4].

One may at tempt to extend eq. (3) to include gravi- tation by introducing the vierbein field .2 ear(x ) and associated m e t r i c g r u ( x ) = eauea v into gAB" These fields must enter gAB in such a fashion that eq. (2) generates the correct gauge changes 8ear and 8grv for the Einstein gauge transformations [~r(z) = ~U(x), ~ = 0] and local vierbein rotations [~U(z) = 0, ~a = (i/4)COab(X)(oabO)a ] . A metric that obeys this con- dition will be called "gauge complete" with respect to these gauge subgroups. (Gauge completeness obviously

*1 Notat ion is as in ref. [3] . The comma denotes right deriva- tive with respect to z A and a = (0, 1) depending on whether A = 0z, a). Our Dirac matrices obey (~°)l" = ,r ° , (3,s) t = ~s and are pure imaginary in the Majorana repre- sentation. We take e 0123 = +1, diag r~rv = ( -1 ,1 ,1 ,1 ) and

=- ~+,'r O. For Majorana fields, 7~ a = qA~r//~ where ( r / )~ = - ( C-I ).a# and C is the charge conjugation matrix and

n°¢ -= ( n-1 )~O" ,2 We usea , b . . . . to indicate vierbein indices, ~a are constant

matrices, and oab =- (i/2) [,,t a , 3,b]. Henceforth % ( x ) = ,flear(X ).

guarantees invariance of the field equations RAB = XgAB with respect to the given gauge subgroup.) For the Einstein and local vierbein transformations, the gauge complete metric is

guy(z) = guy(x) -i07(ur~)(o) 0 - k 0 r r (o)r.(o)o +

gra(z) = -i(~Ta)aear(X) - k(0rr(0)) + . . ,

and gap is(~ven as in eq. (3). Here Pr(0)=(1/4)Wab (0) o ab where ¢Oabta is the vierbein affinity*3 and + ... means cubic and higher terms in 0 a which will not contribute to any considerations here. Eq. (4) is constructed as follows. One may expand gAB(Z) in a power series in Oa : gAB(Z) = £ gAB (n) where gAB (n) is o f n t h order in 0% The lowest non-vanishing terms independent of k are taken to be gr (0) = guy(x), g(1,) = _i(OTa)eau and the lowest k-dependent term is ga~U, = k~?at3" Under the combined Einstein and vierbein transformations one requires that 8gAB on the 1.h.s. of eq. (2) be ob- tained from 8g (x) = ~ - v" (in a geodesic frame) and /zv t r , ) 6 e a r = e a h ~ h , # -- 6Oabeb r . Inserting the 0 a expansion into eq. (2), one obtains a set of differential equations in 0 a which may be integrated to successively con- struct the higher order 0 a terms of eq. (4).

It is interesting to note how the global vacuum state metric of eq. (3) generalizes to include gravity. Aside from the additional k-dependent terms, the ~uv is replaced by grv(x) and r~ar by ear(x). Thus at spa- tial infinity (classically a vacuum) global invariance is recovered. The vierbein affinity Fu(°) also enters the

m e t r i c explicitly. However, while eq. (4) is Einstein and vierbein invariant it is no longer supersymmetric covariant. To achieve the latter one must include a

3 Majorana spin 2 field ~r(x) in grv(z) to complete the supersymmetry multiplet [4] : grv(z) = guy(x) + i~oa~/v)O + .... Further, the global supersymmetry group parameter k must be replaced by a local gauge function k(x). In constructing a metric which is gauge complete under this local supersymmetry transforma- tion, it is first convenient to calculate the k-indepen- dent parts: ~A and gAB" For these, the local super- symmetry transformation corresponds to t4

=- eahebP(eco ,h - ee?,,O) and tOab ~ J = Oaab ~c°)eC W , 4 See nex t page.

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Volume 65B, number 1 PHYSICS LETTERS 25 October 1976

%a = X@(x) - (i/2) J/,“(X(x)YQ,

where qn E $,,e,fi. Using eq. (2), one may easily cal- culate the k-independent parts of the metric which is gauge complete with respect to the transformations of eq. (5). To quadratic order in 19” these are

- ~GJCvw~vjw), (6)

&;,<z> = -iW),e,, + ($,7,wWa3

and zaP is given as in eq. (3). Here r,, = (i/4) &‘wabP where ‘&bP = aabr (O) + K,,clab and

K wb =-(i/4)[$py[0tib] ’ $a7p$bl. (7)

Fq. (2) now implies that transformation (5) corre- sponds to the field transformation law

(8)

These are precisely the transformations postulated in the supergravity of ref. [4]. The first terms in eq. (5) are, of course, the conventional supersymmetry trans- formation generalized from global to local invariance. The additional second terms (as well as the second term gMo, in eq. (6)) are required so that 6gpv be con- sistent with 6e,, (since gPV = e,,@J. Eq. (2) then automatically forces the form for 6 J/,, where, as pointed out by Deser and Zumino [4] , it is the total affinity rP which enters. The significance of the com- bination wlllz + K,,b which enters in r,,, is that the

gradient h,P terms cancel when one explicitly calcu-

lates 6 rr . To complete the analysis, one may also calculate

the k-dependent parts AtA and &A,. In the follow-

* As in [3] we use a doublet of Majorana coordinates, 8oq,

4 = 1, 2, to characterize the Qirac spinors (O$kac =OOri

- i002). The Majorana spin 5 field $y has only one com-

ponent, i.e., we chose the gauge convention $J~~(x) = 0,

and similarly ;\or2(x) = 0. When no ambiguity arises, the 4

index will be surpressed, e.q., in eq. (8) one has 6$g1

= 2D&o r , 6 $;l”’ = 0 (which also exhibits explicitly the

fact that the choice $9” = 0 is maintained by the local

supergravity gauge transformations (5)).

ing we will need only terms linear in k for the quadratic order in 8”. These may again be obtained by integrat- ing eq. (2) order by order requiring that 6gAB on the 1.h.s. be evaluated by use of eq. (8). We find

A& = ; (X?e)($,$,l) + ;(~~b~)(~~PDl,$b, ),

+ ‘k,abcd 5 32 (Y ~d@“l~%.~l&bl~

and*5

ik2 -16 E”bcd(77r5~d)orpeYcD[a~b].

Note that the total gAB is also automatically Einstein

and vierbein gauge complete. 2. Field equations. The field equations for GP(x)

and gPy(x) may be obtained by inserting the total gAB = &B + AgA, into RA, = hg~~. The wave equations

for $, and g,,,(x) arise from the BoL independent parts

of the R,, and Rp,, equations respectively. Since the

field equations are second differential order in BoL, one needs gAB only through quadratic order in fP. The in- verse metric gAB also enters in RA,. gAB contains 1 /k structures, e.g., g’@ = (1 /k)@ + . . . pointing out the singular nature of the k + 0 limit. Thus one needs gAB through 0(k2) to correctly include all (k)O terms

*’ In integrating Agnv to quadratic order in ho, it is necessary

to make a Fierz rearrangement. It is here that the

Majorana nature of $I: enters.

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Volume 65B, number 1 PHYSICS LETTERS 25 October 1976

We discuss now how the k ~ 0 limit proceeds for the Or equations obtained from R a t = Xgau. In calcu- lating RAB one finds structures that are k n, n = -2 , - 1, 0 . . . . . Thus the k ~ 0 limit will exist only if the k - 2 and k -1 terms of the field equations cancel. The k -2 terms arise from those parts Of RAB containing two 0 a derivatives and are independent of flu. Thus in a geodesic frame, these terms are identical to those calculated using the vacuum supersymmetry metric of eq. (3). The fact that for Dirac 0 a, a spontaneous sym- metry breaking of gauge supersymmetry to global supersymmetry occurs for X = 8/k 2 [3], implies that eq. (3) is a solution OfRAB = ~gAB and that the k -2 terms cancel.

The cancellation of the k -1 terms occur for a dif- ferent reason. Since our metric is gauge complete with respect to the total ~1 = ~A + A~A of eqs. (5) and (9) (or alternately with respect to eq. (8)), the field equa- tions R A B = ~'gAB must be gauge covariant with re- spect to this subset of the total gauge supersyrmnetry group eq. ( I ) . In the ~u sector of Rau = ~'gau' the 1/k terms are characteristically mass structures, e.g., (1/k)~ua, but since these are not covariant with re- spect to eq. (8) they must cancel and in fact explicit calculation shows that they do. (A similar result holds in the Einstein sector.) Thus both the k -2 and k -1 structures cancel and one may proceed smoothly to the k ~ 0 limit. In this limit then, the field equations are just the (k) 0 terms OfRAB = ~gAB" These also must be gauge invariant with respect to eq. (8) (as well as invariant with respect to the Einstein and vierbein gauges). The 0 a independent part of Ra~

3 -- - Xga~ then yields -~M~a + ~(3~tbTbT~) a where

M~ = e~bcd757~D, ~ (11) - - a l c b ] "

Hence R~u = ~g~, implies the Rarita-Schwinger equa- tion [4] M~' = 0. Similarly the 0 a independent part ofRuv(z ) = ~ g ~ yields the Einstein equations for g~ with Rarita-Schwinger source.

3. Comments. The metric given in eqs. (6) and (10) contains only two dynamical fields, g~v(x) and ~u(x). The general superfield expansion of gAB of course contains many more dynamical fields in other 0 a sec- tors. Thus the Maxwell fieldAu_(x ) enters in the linear 0 ~ sector ofg~c ~ according to (Oe)~A~(x), where e is the charge matrix [3] (and the second component ~ 2 enters in the linear 0 a2 sector ofg~v(z)). These addi- tional fields produce sources of the Einstein equa-

tions ¢6. In gauge supersymmetry these source fields must be present as gAB is "gauge complete" with re- spect to the full gauge group of eq. (1), not just the subset eqs. (5) and (9) of local supergravity. These questions are closely related to the distinctions be- tween a theory with k q: 0 and one with k = 0. In the former case, the existence of an inverse metric forces the spontaneous breaking of gauge supersymmetry to occur only for Dirac 0 a (for X ¢ 0), allowing the theory to determine its own internal symmetry group to be U(1). The resulting Maxwell-Einstein system is then unified with gravitational constant G E ~ e2/X. For k ¢ 0, one may of course keep the higher terms in the field equations proportional to k n, n = 1,2 .... as "perturbat ions" on the "leading" (k) 0 term. Since spontaneous symmetry breaking implies X = 8/k 2, the neglect of these higher terms, i.e., taking the k ~ 0 limit corresponds to G E ~ 0. The hadronic and elec- tromagnetic fields would still interact in this approxi- mation, but the gravitational field would be decoupled from its source (experimentally a very good approxi- mation!). The above discussion may also explain why the conventional global supersymmetry [1 ] which is based on k ~ 0 [5] may be sufficient in discussing "low energy" non-gravitational interactions.

We should like to thank B. Zumino for several inter- esting discussions.

¢6 The dynamical equations for these source fields arise in sec- tors other than (0a) ° OfRAB = hgAB. Thus if one wishes to examine these higher sectors, one must include the source fields for consistency.

R e f e r e n c e s

[1] D.V.Volkov, and V.A. Soroka, JETP Lett. 18 (1973) 529; J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39.

[2] A. Salam and J. Strathdee, Nucl. Phys. B80 (1974) 499; B84 (1975) 127; S. Ferrara, to be published Rivista del Nuovo Cim. (LNF- 75/46(P)).

[3] P. Nath and R. Arnowitt, Phys. Lett. 56B (1975) 177; R. Arnowitt and P. Nath, Gem Rel. Grav. 7 (1976) 89; P. Nath and R. Arnowitt, J. de Phys. 37 (1976) C2-85; P. Nath, Proc. of Conf. on Gauge theories and modern field theory, Boston, 1975 (MIT Press, Cambridge, Mass., 1976); R. Arnowitt and P. Nath, Phys. Rev. Lett. 36 (1976) 1526.

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Volume 65B, number 1 PHYSICS LETTERS 25 October 1976

[4] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; D.Z. Freedman and P. van Nieuwenhuizen, Stony Brook ITP-SB-76-25 ; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335

[5] B. Zumino, Proc. of Conf. on Gauge theories and modern field theory, Boston, 1975 (MIT Press, Cambridge, Mass. 1976) p. 255; G. Woo, Nuovo Cim. Lett. 13 (1975) 546.

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