Generalized gauge fields

Volume 165B, number 4,5,6 PHYSICS LETTERS 26 December 1985

G E N E R A L I Z E D G A U G E F I E L D S ~:

Thomas C U R T R I G H T 1.2

The Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA

Received 14 March 1980

Higher rank Lorentz tensors with mixed spacetime index permutation symmetry are shown to be gauge fields. Theoretical applications involving such fields are proposed.

Both totally symmetric and totally antisymmetfic Lorentz tensors of arbitrary rank may be used as gauge fields. Both types of gauge field tensors appear quite naturally in theoretical analyses of some interesting physical problems. For example, symmetric gauge tensors (spinor-tensors) of rank s may be used to describe massless particles of spin s(s + ~). This is not only true for s ~ 3 [1], but has actually been shown for all spins [2]. Furthermore, antisymmetric rank-two gauge fields are naturally coupled to strings [3], and more recently such fields have been used in formulating dual transformations for vector gauge theories [4]. Finally, an antisymmetric gauge tensor of rank-three appears in the elegant eleven-dimensional formulation o f N = 8 supergravity, prior to reducing the model to four spacetime dimensions [5,6].

In this paper we describe how to generalize the concept of a gauge field to include higher rank Lorentz tensors which are neither totally symmetric nor totally antisymmetric under spacetime index permuta- tions. We have found that essentially all Lorentz tensors may be used as gauge fields. Here we mainly describe the theory of a rank-three tensor whose index permutation symmetry corresponds to the Young dia- gram ~ . However, the analysis is readily extended to more complicated cases. We briefly mention further

Research supported in part by the NSF: Grant No. PHY-79- 23669.

1 Robert R. McCormick Fellow. 2 Present address: University of Florida, Gainesville, FL 32611,

USA.

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generalizations near the end of this paper and plan to describe them in more complete detail elsewhere.

We also briefly discuss here some possible theoretical applications of such "mixed symmetry" gauge fields involving supersymmetric theories and models containing more than one fundamental spin-2 particle. Remarkably, such tensors appear in a model of a massive, self-coupled, spin-2 field formulated by Freund and the author [7].

Consider the simplest mixed symmetry tensor available: a rank-three tensor which we assume to have index permutation symmetry " ~ " . This simplest example illustrates almost all the features of the general case. We denote this tensor as T[ab] c with the antisymmetrized Lorentz indices in brackets. T has the symmetry properties

T[ab] c = -T[ba]c, (1)

Tiab] c + T[bct a + T[ca] b = 0. (2)

These properties coincide with those o f the linearized spin connection in Einstein's theory (wua b = ~ahub

- Obhua) but T is not to be inflexibly identified with ¢o in the present case. In fact, we shall usually assume a flat spacetime in the following.

Now we clef'me T[ab] c as a free local gauge field by postulating the most general form for the gauge transformation. This form is

6Tlab] c = i}aSbc - ~bSac + aaAbc -- ~bAac + 2~cAba, (3)

where S and A are, respectively, arbitrary symmetric

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and antisymmetric functions of spacetime. There are now six possible second derivative terms which may appear in the lagrangian for T. Requiring invariance of the action under (3) uniquely fixes the relative co- efficients o f these six terms.

The simplest way to present .6? is to note that the field strength

F[abc]d = igaT[bc]d + Ob TIcald + OcTIabld (4)

is invariant under S-type gauge transformations

8F[abcld = - -2~d(~aAbc + ;)bAca + bcAab ). (5)

A unique relative combination o f F 2 and (trace t7)2 is actionwise invariant under A-type gauge transformations as well. This is

1 32= --~(F[abc]dF[abc]d -- 3F[abx]xF[aby]y ). (6)

There is also an equivalent way to write .6? using a field strength similar in form to the Riemann tensor,

R lab] [xy] = ~xT[ab]y - (x ~+y), (7)

whose gauge variation depends only on the sum of S and A :

5R[ab][xy I = Ox[3a(S + A ) b y - (a ~ b ) ] - x ~ y . (8)

This alternative form is .1

1 32 = - a (R lab] [xy] R [ab ] [xy] -- 4 R [ax] [bx] R [ay] [by]

2 (9) + R [xy] Ix),] )"

Eqs. (6) and (9) give actions invariant under (3) in any number o f dimensions D. However, for D ~< 3, 32 - 0. Also, one recognizes (9) as being somewhat familiar for D = 4.

Z?(D = 4) = ~ e a b X Y e A B X Y R . (10) [ab ] [xy] R [AB ] [ XY]

This is almost the same form as the Euler density written in terms o f R Riemann • Nevertheless, there are important differences here. As defined in (7),

R lab] [xy] --/= R [xy] [ab] , (11)

and both derivatives in (10) are n o t contracted with

,1 Note that R ~ R [xy] [xy] is invariant under (3) if we re- strict S = 0 CA. In general, however, 6R = 2(OS - OxOySxy) and all three terms in (9) are required for gauge invariance.

the same antisymmetric symbol. Consequently, ./2 (D = 4) is n o t a total divergence as is the Euler density. Thus Z? leads to meaningful equations of motion for all D /> 4.

These equations o f motion are

E [ab] c - ~xF[xablc -- a cF[abx]x = 0, (1 2)

E a =-E[ay]y = 23xF[xay]y = 0. (13)

Similar expressions may be obtained using R. It should be noted that the actual Euler-Lagrange equations obtained by varying 32 are not (12), but rather

E[ab] c + ~(gacEb - g b c E a ) = 0. (14)

F o r D > 3, (14) has the nontrivial trace given in (13). Written in terms of T, (12) is

I-qT[ab] c + ac[3bTtax] x - (a +~b)]

+ bx [bbT[xa]c - (a ~+ b)] - 3cOxT[ab] x = 0. (15)

The obvious gauge condition which eliminates the last five terms in (15), giving the standard massless wave equation [] T = 0, is

~aT[bx]x + 2~xT[xb] a = 0. (16)

At this point, it is instructive to count the physical modes represented by T.

In D dimensions the number of transverse modes represented by a traceless tensor T[ab] c is

NO(/) ) = ~D(D - 2)(/) - 4). (17)

Similarly, the number o f transverse plus longitudinal components for such a traceless tensor is

N ( D ) = 1 2 ~(D - 1)(D - 3). (18)

The mode count in (17), or (18), gives the correct number of physical modes carried by the tensor T when it is massless, or massive, respectively. This can be seen for the massless case by first counting the number of constraints from the gauge condition given in (16), and then counting the additional degrees of freedom which may be eliminated from T through gauge transformations preserving (16).

For the massive case, the counting is actually simpler. The free massive T theory is defined by ad- ding to 32 the mass term

2 m (T[ablcT[ab] c - 2T[ax]xT[ay]y ). (19)

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The resulting massive wave equation is given by (14) with the replacement [] ~ [] + m 2 . Tracing this wave equation and/or taking its divergence leads to the standard F ie rz -Paul i form for the massive wave equations for T.

([] + m2)Ttablc = 0, (20)

axT[xb] c = 0, T[ax] x = 0. (21)

Counting the constraint conditions (21) in D dimensions shows that this massive T field describes precise- ly N(D) physical modes, as given in (18).

Significantly, N O (D = 4) = 0. The massless T gauge theory describes zero propagating physical degrees of freedom in four dimensions. This may be surprising, so we offer a detailed proof. It is clear from (3) or (14) that a consistent coupling of T to an external source S occurs if, and only if,

aaS[ab] c = 0 = ~cS[ab]c . (22)

For such "physical" sources, we add Slab] c T[ab] c to Z?, solve for T in terms of S, and obtain the action in D dimensions expressed in terms of the source:

--~ f dDx{S[ab]c[]-lS[ab]c

+ [2/(3 - D)] S[ax]x l-q-1S [ay]y}. (23)

Transforming to momentum space, we may now de- termine the number of propagating modes by counting the independent source bilinears which appear in the residue of the k 2 = 0 pole. Using (22), we find a vanishing residue for D = 4, five independent bilinears for D = 5, etc., thereby confirming the mode count in (17). Hence the massless T field carries only gauge, and not physical degrees of freedom in four dimensions ,2

Given this fact, one may wonder how theories involving T can lead to any physical content in four dimensions. There are two interesting mechanisms through which this could happen. First, i f m :/= 0, the massive T field describes 5 physical modes for D = 4.

It is trivial to see that T is just a spin-2 field in this case. This interesting possibility will be discussed ful- ly in ref. [6]. Second, the T field could naturally appear in a model with D > 4. It would then decompose into a variety of physically nontrivial fields upon reduction to four dimensions * s. The situation here would be completely analogous to that occurring for the antisymmetric rank.three tensor gauge field ap- pearing in the N = 8 supergravity theory [5,6]. Such an antisymmetric gauge field also describes zero degrees of freedom for D = 4.

A simple supersymmetric example of the second mechanism above immediately suggests itself for D = 5. In five dimensions, the massless T gauge field has the same number of physical modes, 5, as does a symmetric tensor gauge field gab" This suggests that simple supermultiplets containing T[abl c may be equivalent for D = 5 to multiplets containing gab" Upon reduction from D = 5 to D = 4, bo th fields give the same four-dimensional spin content: T[ab] c -~ s = 2, 1, 0 and gab ~ s = 2, 1 ,0 . Parities are not so easily correlated, nevertheless, it may be possible to formu- late in two ways an interacting model for D = 5 which reduces in four dimensions to N = 2 supergravity coupled to an N = 2 vector multiplet . This possibility is currently being examined. The T[ab] c formulation would be the "dual" [4,7] of the gab theory.

Not only external sources, but also self-interac- tions are subject to the constraints in (22). We have not found a source involving T itself which leads to a completely self-consistent model in an arbitrary number of dimensions. An obvious ansatz for such a self- coupling is to add quadratic terms in T to R in (7), so that R has the same form as the nonlinear Riemann tensor writ ten in terms of the spin connection. We have tried using such a nonlinear R in the lagrangian (9). Unfortunately, this self-coupled theory is no longer gauge invariant under (3), even when these gauge transformations are modified by " a " -+ "a + T" . We conclude this particular ansatz is in- consistent, but we believe the general problem de-

,2 A simplified form of the four-dimensional field equations for T may be found by taking explicit values for the indices in (12) and (13). These equations reduce to aaF[xyz] b = abF[xyz] a (D = 4) for all a, b, x, y and z. This implies F[xyz]b = abA [xyz] (D = 4).

*s Reducing a massless T from D > 4 down to D = 4 leads to the set of spins (D - 4, s = 2; (D - 4) 2 , s = 1 ; I(D - 3)(/) - 4)(D - 5) +D - 4, s = 0}. Reducing a massive T similarly gives (D - 3, s = 2; (D - 3)(D - 4), s = 1 ; ~(D - 3)(/) - 4)(D - 5), s = 0}. Note that spins higher than 2 are never obtained in these reductions.

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serves further study. Thus we propose an extension of the Gupta program [8] (i.e., finding all consistent spin-2 field theories) to include arbitrary massless and massive tensor fields in D dimensions.

A novel source for T[ab] c has appeared in a study of the dualization o f continuum field theories in four dimensions by Freund and the author [7]. Briefly, it was observed that

S[ab] c = 2eabef~eOfc + ecdefaeOfa -- ecaef~eOfb (24)

is a traceless tensor satisfying eq. (22), where Olin is a symmetric energy-momentum tensor. A remarkable feature o f (24) is that it vanishes identically for zero momentum transfer, due to the explicit derivative, and hence is a manifestly chargeless source. For the case of a massive T[abl c field, eq. (20) with (24) added to the RHS leads to a nontrivial scattering of ex-

t~ ext for nonzero momentum transfer. ternal sources Vlm It remains to be demonstrated whether T[ab] c can consistently and causally self-couple in such a way that Slab] c follows from a local lagrangian. We refer the reader to ref. [7] for a fuller discussion.

Before describing other types o f mixed symmetry gauge fields, we mention one of our original motiva- tions for considering T[ab] c. This particular tensor is the simplest gauge field which, upon reduction from D = 12 t o D = 4, gives vector fields in both the SO(8) multiplets "28 (antisymmetric)" and "35 (symmetric)". Both of these SO(8) multiplets appear as auxiliary vector fields in the N = 8 supergravity theory, where, in fact, they combine to permit a local SU(8) invariance [6]. Thus, it is stimulating to con- jecture that a twelve-dimensional supermultiplet exists which contains T and is relevant to the N = 8 theory. However, such a multiplet would also contain an SO(8) octet of spin-2 fields upon reduction to D = 4 and would require overcoming several problems for its realization *a

Next, we briefly sketch some of the properties of more general Lorentz tensor gauge fields. Increasing the number of antisymmetrized indices is trivial. For example, T[abc]x , whose index symmetry is " ~ " , has

*4 It is easy just to match Fermi and Bose degrees of freedom for D = 12. For example, {oWeyl, 3(~Weyl)} describes 384 massless fermiortic modes on-shell, while (T[a b ] c, g(ab) ,Aa} and {S(abc),g(ab),A [abc] } each separately describe 384 massless bosonic modes. This latter Bose set yields a single spin-3 field for D = 4.

the following gauge transformation and lagrangian:

8T[abclx = OaM[be] x + abM[ca] x + OcM[ab] x

+ ~aAxbc + abAax c + acAab x + 3~xAab c,

.13 = F[abcd]xF[abed] x -- 4F[abcx]xF[abcy]y ' (25)

where

F[abcd]x = OaT[bcd]x -- Ob T[cda]x + OcT[dab]x

- adT[abc]x.

The gauge parametersA and M are, respectively, totally antisymmetric and mixed-symmetric, as defined by (1) and (2). As another example, consider a "Riemann" gauge tensor with symmetry " ~ " . The gauge transformation is

8T[ab][xy ] = [~aM[xylb - (a ~ b)] + [(ab) ~ (xy)]. (26)

Tensors with several symmetric indices are also easily analysed. The gauge transformation for T[abtlb 2...b s with index symmet ry" ~ " is obviously constructed from totally symmetric tensors of rank s, and from mixed t e n s o r s M [ a b d b 2 b 1" In order to find invariant actions for gauge ten'sgrs with several symmetric indices, it is necessary to impose trace conditions on the gauge parameters exactly as was done for totally symmetric gauge fields in ref. [2]. It is also necessary to introduce auxiliary fields in formulating the massive versions of such symmetric tensor theories as was first done by Schwinger [1] for totally symmetric tensors (spinor-tensors) o f rank 3(2). Although T . describes zero physical degrees of freedom [abl]b 2 ...b~ . . ~ • for D = 4 when massless, the addition oi mass terms leads to an alternate description of massive spin s particles [7]. Finally, we remark that gauge models of spinor-tensors with mixed Lorentz index symmetry are also possible and are constructed in an obvious generalization of the totally symmetric spinor-tensors described in ref. [2], by using the above ideas. Again, trace conditions must be imposed on spinor-tensor gauge parameters to obtain invariant actions.

By generalizing gauge transformations to include arbitrary mixed symmetry Lorentz tensors, the number of local field theories available for use in physical models has increased. This may be a useful enlarge- ment, especially for constructing consistent field

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theories with spin higher than two, or for finding more general ways o f viewing spin.2 particles. One such alternate view of massive s = 2 fields is described in ref. [7]. It remains to be seen whether Nature is aware of these generalizations.

I thank Tohru Eguchi and Peter Freund for stimulating discussions. I also thank David and Susan Pietsch for their generous patronage.

Note added. Since submitting this paper for publi- cation, a few interesting developments have taken place.

First, the fundamental twelve-dimensional supermultiplet of fields has been constructed using gauge fields such as those described in this paper [9].

Second, it has been noted that in low-energy ap- proximation, superstrings can give rise to curvature- squared contributions to the effective lagrangian of gravity [10,11]. The effective action due to such "R 2 ' ' lagrangians should be unitary, at the perturba- tive level, and therefore have no quartic derivative terms which are bilinear in the graviton field. Only one "R 2 ' ' lagrangian meets these requirements (in any number o f spacetime dimensions), namely that given in eq. (9) above. Indeed, the absence of quartic derivative terms in the effective action due to (9) is nothing but the gauge invariance discussed in the present paper.

Finally, generalized gauge fields as defined above

explicitly appear as variables in the covariant descrip- t ion o f the quantized string [ 12,13 ].

References

[1 ] J. Schwinger, Particles, sottrces, and fields (Addison- Wesley, Reading, MA, 1970); F.A. Berends, J.W. van Holten, P. van Nieuwenhuizen and B. de Wit, Phys. Lett. 83B (1979) 188; Nucl. Phys. B154 (1979) 261.

[2] C. Fronsdal, Phys. Rev. D18 (1978) 3624; J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630; T. Curtright, Phys. Lett. 85B (1979) 219.

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(1980) 413. [8] S.N. Gupta, Phys. Rev. 96 (1954) 1783;

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LBL-190302. [12] W. Siegel and B. Zwiebach, UC Berkeley preprint

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Generalized gauge fields