Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

QUANTUM INEQUIVALENCE OF DIFFERENT FIELD REPRESENTATIONS

M.J. DUFF Physics Department, Imperial College, London SW7 2BZ, UK

and

P. van NIEUWENHUIZEN Institute for Theoretical Physics, State University of New York at Stony Brook, Long lsland, NY 11794, USA

Received 2 June 1980

The gauge theories of antisymmetric tensor potentials A~v and Atavp describe 1 and 0 degrees of freedom, respectively. Yet we show the gravitational trace anomalies of A~v and a scalar field A to be different, and that ofA#v p to be non-vanish- ing. Corresponding inequivalences also occur in their one-loop counterterms when the spacetime has non-trivial topology. Furthermore, the coupling of A~zvo to gravity provides a gauge principle derivation of the cosmological constant. Possible applications to supergravity are also discussed.

One might expect that different field representa- tions for particles with a given spin and number of de- grees o f freedom would not lead to different physical effects. For example, a scalar field A is naively equiva- lent to an antisymmetric rank-two gauge potential Auv = -A~,~ because the corresponding field strength Fro, o obeys the equation of motion OUFuv o = 0, and with Fur o =- euvoaf °, this implies fu = OuA. In fact, as we intend to show, this expectation is incorrect. M- ready at the classical levelf u is not equal to OuA (with A regular) when the spacetime has non-trivial topology and, as we shall see, this inequivalence mani- fests itself at the quantum level. In particular, the dif- ference between the gravitational trace anomaly for the field Auv and that of the field A is equal to the Gauss-Bonnet invariant

(327r2)-l*Ruvpa * RUVO ~

= (32rr2)-l(RuvoRt~VP~ - 4RuvRUV +R2) .

Similarly, the gauge theory of a rank-three antisym- metric potential Auvo, which has no degrees of free- dom at all, nevertheless yields a non-vanishing trace anomaly equal to - 2 times the above invariant. Of perhaps even greater interest, moreover, the coupling

of gravity to the field Auv o leads to a cosmological constant in the Einstein equations. Hence the cosmo- logical constant may be derived from a gauge principle!

In a previous analysis [1 ] of the one-loop counter- terms of Auv and Auv o coupled to gravity, the above Gauss-Bonnet invariants were dropped on the grbunds that they were total derivatives, and the naive equiva- lences were obtained. However, this is no longer per- mitted when the topology is non-trivial.

We expect that similar inequivalences at the quan- tum level will appear for different representations of fermions. For example, rewriting a spin-1/2 Majorana spinor ?~ as an antisymmetric tensor spinor ?~uv [2] may lead not only to different trace anomalies but also to different axial anomalies. (This could conceiv- ably have far-reaching consequences if a change of field representation also changed the Yang-Mills axial anomaly and hence the GIM cancellation mechanism. However, the coupling of such representations to gauge fields encounters certain consistency problems which shall not be discussed here.)

First, we shall derive the above mentioned trace anomalies and counterterms and then comment on possible applications to supergravity. Two derivations are presented: one based on topological considerations

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Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

and one based on a Feynman graph analysis. In both cases we follow the gauge-fixing procedure of ref. [1 ]. Thus one A [uuo] field requires two real anticommut- ing Fadeev-Popov ghosts B[uvl, three real commuting ghosts C u (namely two complex commuting Fadeev- Popov ghosts and one real anticommuting Nielsen- Kallosh ghost) and four anticommuting scalar ghosts E. The degrees of freedom count is thus 1 × 4 - 2 X 6 + 3 X 4 - 4 X l = 0 .

The topological derivation relies upon the observa- tion that the functional integration over the physical and ghost fields given above yields

(det D3)-1/2 (det D2)(det D 1 )-3/2 (det D0)2 , (1)

where Dp is the laplacian acting on p-forms. The gen- eral results of ref. [3] now allow us to write down immediately the one-loop counterterm

A 2 2 = ~ - ( b 3 -- 2b 2 + 3b 1 - 4bo) (2)

and trace anomaly

T u = b 3 - 2b 2 + 3b 1

where

- 4b0, ( 3 )

Bp = f d4x x/g bp (4)

is the t-independent term in the asymptotic expansion of Tr exp ( -Dp t). Explicitly,

180(47r)2b 0 = RuvpoRUVO° - RuuRUV + S R2 + 615]R, (s)

180(47r)2bl = 1 1RtauooRUVP° (6)

+ 86R R uv - 20R 2 - 6E1R, Uv

and

b 2 = b~ + b ~ , (7)

where

R u v o ° - 9 3 R R "u+ ~SR2 180(4r02b2 = 33Ruvoo uu (8)

- 12DR +- 30R *R uvpa, Uvpo

(the + and - refer to self-dual and antiself-dual 2- forms, respectively), while b 3 and b 4 are given by the duality relation

bp = b(4_p).

Note the special combinations

4

C (-I )PBp p=0

(9)

_ 1 j v~, #uv +R 2) 327r 2

1 f d 4x x/g *Ruvpa*RUV°° = X, (10)

32~ 2

and

B~ - B 2 - 1 f d4x x/g *R R uv°° = 7", 48~2 uvpo

(11)

where X is the Euler number and r the Hirzeburch sig- nature: topological invariants which take on integer values in spaces with non-trivial topology. Eqs. (10) and (11) are just the Gauss-Bonnet theorem and Hirzeburch signature theorem, respectively.

X counts the alternating sum of the number of (normalizable) zero eigenvalue modes of the differen- tial operators Dp i.e. the number of harmonic p-forms; while z counts the difference between the number of self-dual and anti-self dual harmonic 2-forms.

The crucial observation, using eq. (9), is that it is precisely the Gauss-Bonnet combination of eq. (10) (times - 2 ) which enters eqs. (2) and (3). Thus the counterterm and trace anomaly are determined only by these zero-modes:

fd4x V~ r " = - 2 x . (12)

The Euler number is known to contribute to part of the gravitational trace anomaly [4], but for general fields there are non-topological R 2 and RuvRUV terms as well. In the case of Auu o, however, these cancel iden- tically. In other words, after projecting out the zero modes, the ratio of determinants appearing in (1) is equal to unity. [Eq. (11) is the analogue of the axial anomaly for antisymmetric tensors and might also prove to have some physical significance.]

Similar remarks apply to Auv which requires two real anticommuting vector ghosts and three real com- muting scalar ghosts [1 ]. The degrees of freedom count is now 1 × 6 - 2 X 4 + 3 X 1 = 1, and functional integration yields

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Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

(det D2)- l /2(det D 1)(det D0)-3/2 , (13)

A ~ = n ~ -(b 2 - 2b 1 + 3b0), (14)

4

T ~ = b 2 - 2 b 1 + 3 b O = p ~ O ( - 1 ) P b p + b 0 . = (15)

Since for a scalar fieldA alone Tu u = bo, the differ- ence between the trace anomalies (and counterterms) for the fields Auv and A is again given by the Gauss- Bonnet invariant.

It is now easy to see why the naive equivalence argument breaks down in spaces with non-trivial topol- ogy. The Hodge decomposition splits any p-form uniquely into a harmonic part, an exact part (curl of a p - 1 form) and a coexact part (divergence of a p + 1 form). Thus although all exact forms are closed (have vanishing curl), not all closed forms are exact. In the present context this means that a vector with vanish- ing curl is no longer simply the gradient of a scalar, and a scalar with vanishing gradient is no longer zero. It is the non-vanishing harmonic parts which are re- sponsible for the inequivalence.

The Feynman graph analysis may be read off from ref. [1], but this time retaining the Gauss-Bonnet in- variants. Let a be the coefficient of R~voaRUV°a ap- pearing in

(o~RuvpoRU~O~ + ...). (16) 8n2(n - 4)

The contribution to a from a pure Auv o loop was - ½ X ~ from the Yu2~ term of eq. (2.20), while the Buy loop gave 2 X ~ from the Y2uv term and - ½ from the X 2 term of eq. (2.23), the C loop give ~ from the y~2 term and the E loop gave zero. Before discard- ing the Gauss-Bonnet terms, there is also a contribu- tion to a coming from the trace o f the unit matrix in eq. (2.17), but since the combination of fields has zero degrees of freedom, this cancels. Since 2x~did in fact vanish when the Gauss-Bonnet terms were dis- carded, it follows that

zS j2(Auvo)=(_2)n 1 x/g (17) - 4 32rt2 *R~vp°*RUVPo'

in agreement with the topological analysis. Similarly, 1 for the field A , v there was a contribution to a of 5-

X 1 X ( - 2 ) form the y2 term while the vector ghosts

1 1 yielded - g and the scalar ghosts - g . Hence

A.t2(Auv ) - ,fl.C(A) = 1 v7 ~ t~ ~ v p a 32n2 --~vpo -- ,

(18) which is once again in agreement.

We now turn to another interesting application of antisymmetric tensor fields, namely the derivation of the cosmological constant from a gauge principle. Con- sider the lagrangian of ref. [1 ] describing the coupling ofAuv o to gravity with, a priori, no cosmological con- stant:

.12 = - V ~ R - 2 %/~ F.uoo FUVO°, (19)

where Fuvoa is the curl ofAuv p . As mentioned pre- viously, the matter field Auv o has zero degrees of free- dom. Next consider the field equations

D"G~oo -- 0, (20)

Gt~ v + guvF 2 = 0, (21)

x 2 where we have used Fu~oaFUvpX -= ¼6aF . From eq. (20) it follows that Fuvpa = euvoaAl/2 , where A is a constant, and substituting into eq. (21) we find the Einstein equations for pure gravity with a cosmologi- cal constant A. Thus pure gravity with a cosmological constant may be reformulated as gravity coupled to the gauge field Auv o.

A final application concerns supergravity. Although it is customary to write the auxiliary fields of N -- I

supergravity as S, P, A u [5,6], and alternative formula- tion already discussed in refs. [6,7] is to replace S by OuSU + ~uo~v@ and similarly for P. In this formula- tion, the term (OuSU)2 in the lagrangian is of the kind discussed above, namely FuvooFUVOa where AUvp -~ euvo°S a. The important point to note is that replac- ing S by Auv o in the lagrangian leads to inequivalent counterterms and anomalies in the way we have de- scribed. The field Auv p (or equivalently SU) is now dy- namical and can no longer be eliminated from the lagrangian because its field equation is no longer alge- braic. This is also discussed in ref. [6]. In the standard formulation, it is a term S - ~ouv~v which produces the cosmological constant. In the A ~ p (or S t ) formula. tion, however, this is a total derivative and it is the term (OuSU + ~uouvO) 2 which yields the cosmological constant in the field equations. A superfield version of this phenomenon may be found in ref. [7].

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We are very grateful to K,S. Stelle for discussions. References

Notes added. The one-loop counterterms derived in ref. [1] from the lagrangian (19) yield on-shell the same results (except for the above-mentioned top~og i - cal terms) as those for pure gravity with a cosmological constant derived by S.M. Christensen and M.J. Duff (Nucl. Phys. B, to be published). W. Siegel has pointed out that in the version of N = 8 supergravity with anti- symmetric tensors, even the topological counterterm (and hence the trace anomaly) vanishes when the result without antisyrrunetric tensors (S.M. Christensen, M.J. Duff, G.W. Gibbons and M. Ro~ek, Phys. Rev. Lett, to be published) is amended by our (17) and (18).

[1] E. Sezgin and P. van Nieuwenhuizen, Stonybrook preprint ITP-SB-80-3 (revised version); this paper employs the correct ghost assignments given by W. Siegel, Institute for Advanced Study preprint; see also J. Thierry-Mieg, Harvard preprint.

[2] P.K. Townsend, CERN preprint TG.2790. [3] S.M. Christensen and M.J. Duff, Nucl. Phys. B154 (1979)

301. [4] M.J. Duff, Nucl. Phys. B125 (1977) 334. [5] S..Ferrara and P. van Nieuwenhuizen, Phys. Lett. 74B

(1978) 333. [6] K.S. SteUe and P.C. West, Phys. Lett. 74B (1978) 330. [7] V. Ogievetsky and E. Sokatchev, JINR preprint, Dubna,

E2-80-I 39.

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