Local gauge and Lorentz invariance of the heterotic string theory

Volume 166B, number 3 PHYSICS LETTERS 16 January 1986

LOCAL GAUGE AND L O R E N T Z INVARIANCE OF T H E H E T E R O T I C S TR ING THEOR Y '

Ashoke SEN

Stanford Lmear Accelerator Center, Stanford Untverstty, Stanford, CA 94305, USA

Received 15 October 1985

The Hul l -Wlt ten proof of the local gauge and Lorentz invarmnce of the o-model describing the propagation of the heterotlc string m arbitrary background field is extended to higher orders m ct' The modification of the transformatmn laws of the annsymmetnc tensor field under these symmetries is discussed Finally we point out the existence of an anomaly in the nawe N = ~_, supersymmetry transformanon, and show that it is cancelled by the same counterterms which restore local Lorentz and

gauge mvarmnce of the o-model

It has recently been conjectures [1-5] that the classical equations of motion of the massless fields in string theories may be interpreted as fixed point equations of the appropriate o-model. As a result we expect the symmetries of the classical equations of motion to be reflected in the a-model, and vice-versa. For example, in the closed bosonic string theory, and the type II superstring theory, the general coordinate and local Lorentz invariance of the classical equations of motion are consequences of the invariance of the corresponding non-linear e- models under reparametrization of the internal manifold [6]. The existence of such symmetries, although expected, is hard to prove otherwise. Usually the effective action involving the massless fields is constructed from the scattering amplitude involving the massless fields, and the rule for constructing these scattering amplitudes do not exhibit any general coordinate or local Lorentz invariance.

The heterotic string theory [7] is expected to have general coordinate, local Lorentz and local gauge invariance. One would expect that these symmetries should be manifest in the o-model which describes the propagation of the string in arbitrary background field. Indeed, these symmetries are present in the classical e-model action. However, a close look at the model tells us that these symmetries are anomalous. In fact, the presence of these anomalies was shown to be responsible for the appearance of the Chern-Simons terms in the classical equations of motion [ 1 -3 ] , which apparently destroys the local gauge and Lorentz invariance. Ultimately, however, we must recover local gauge and Lorentz invariance of the equations of motion. As was shown by Hull and Witten [8], to one loop order the anomalous variation of the effective o-model action under these symmetries may be cancelled by redefining the transformation laws of the antisymmetric tensor field. The purpose of this paper is to extend their proof to higher orders in the o-model perturbation theory, and to derive the exact transformation laws of the antisymmetric tensor field under local Lorentz and gauge transformations.

The light-cone gauge action for the o-model describing the propagation of the heterotic string in arbitrary background field is given by [1-5 ]

S = 4 - ~ f d ' f / do{gi/(X)O~xi~axf + ~C~SBij(X)~axi~#x/+ igi](X){Xi~l+ X i [ ~ / ( X ) + S~X)]p~'~klox k} 0

+ ~s[i~Sst + AM(x) (TM)stP~oxi] t~ t + ¼iFM(X)~ Sp"(TM)s t ~ t~ipcX]}. (1)

Work supported by the Department of Energy, contract DE-AC03-76SF00515

300 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


where gi/(x), Bi/(x ) and AM(x) are background gravlton, antisymmetrlc tensor, and gauge fields respectively, all taken to be transverse, and assumed to depend on the transverse coordinates only. The dilaton field is taken to be constant in space - t ime , so that it may be absorbed in various coupling constants, and does not appear explicit- ly in the o-model action. The X i denote the eight scalar fields, the X i are the eight left-handed Majorana-Weyl spinors and the ff s are the 32 fight-handed Majorana-Weyl spinors. We are working in the N e v e u - S c h w a r z - Ramond representation, so that the X i transform in the vector representation of SO(8), whereas the ~k s transform in the 32 representation of SO(32) or (16,1) + (1,16) representation of the SO(16) ® SO(16) subgroup of E 8 ® E 8. Also here

= ! B - 1 Si/k 2(~i /k + O/Bki + akBi])' Pi/k - ~(~]gik + ~kg~] -- atgjk)" (2,3)

F M the field strength associated with the vector po ten t ia lA M. The action (1) has an N = ~ supersymme- and .. is 1 try ,1 l(

~X, = ie~,i, $X,= _(Or _ Oo)Xte, $~s = (_exiAM)(TM)st t~t" (4)

It is more convenknt to rewrite the action m terms of the vielbeins e a satisfying e a e ; = gd' spin connection 6o ab constructed from Fi]k, and the fields X a = et a Xt:

~r s= 4-~, f dr f do {g,e(x)a x%x/ + i{~,a~a + La[~af(X)--s~b(X)]p~;kb~)cxk}

0

+- -s [l~J6s t + A M ( X ) ( T M ) s t e a a a X ' ] t , . M - s a M (5) t~ +.41Fab(X)~ p (T )s t~t~apaxb ).

The above action has a local gauge symmetry:

A M ( x ) T M -~ A M ' ( x ) T M = U ( x ) A M ( x ) T M u - I ( x ) + U ( X ) i a i U - I ( x ) , t~ ~ ~' = U(X)~b. (6)

This symmetry, however, is anomalous [9] due to the chiral nature of the fermions ft. A similar remark holds also for the local Lorentz symmetry

e a -+Rabe b, X a ~ R a b x b, w ab ~ [R(wi+ ai)R-1]ab , (7)

where R denotes a local SO(8) rotat ion. The one-loop effective action involving only the external bosonic fields transforms under these anomalous symmetries as *~

s(one_loop) = I f d, f do M - o f , (8)

where 0 M and 0 ab are the infinitesimal gauge and Lorentz transformation parameters respectively, and

U ab = co ab - S ab . (9)

As was pointed out by Hull and Witten [8], the anomalous variation of the effective action to one-loop order may be cancelled by redefining the transformation laws of B# under local Lorentz and gauge transformations:

6Bi/ = -~' a'(~, [i"nMAMI] -- b[ioab'~;~ )' (10)

which is identical to the result found by Green and Schwarz [10], except for the replacement of co by ~ . This, however, cannot be the end of the story, since this anomalous variation of Bi/ induces an anomalous variation of

,x The transformation law of qJ given here was not needed in refs. [1,2] to prove the supersymmelxy of the action (1), since we used the equations of motion of ~0 in our proof. If we do not use the equations of motion of the ¢ fields we need to use the exphcit transformation laws of ~p given here.

,2 Since we may add any arbitrary local counterterm to the one-loop effective action, the expression for the anomaly given in (8) is not unique. We shall adopt this particular definition of anomaly in order to uniquely define the fermiomc loop integrals.

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Si]k and hence also an anomalous variation of the connection U which couples to X. This induces a further variation of the one-loop effective action of order a ' . A simple way to get rid of this extra variation is to replace S ab by/_/~b in the original o-model lagrangian where H is determined from the equation

Hi] k = Si/k + ~ a' [f23(A ) - f23(co - H)] i/k' (i I)

where

= t | A M p M • A M A N A P I23(A)i/k ~ t [i" /k] -- ~1 Tr(TMTNTP)], (12) "[ i "1 "'k]

and ~23(co - H) is given by a similar equation with A replaced by co - H. I f the transformation law of Bz] under local Lorentz and gauge transformations is taken to be

~at] - l ot'[a[IoMA]]M ", ~ab. ab = - o[iv Lco)] _/_//a]b)], (13)

then Hi~k, as defmed above, is invariant under these symmetries. As a result, the one-loop effective action involving the bosonic fields transforms as in eq. (8) with ~ replaced by co - H. This, in turn, is cancelled by the variation of Bi/given in eq. (13). Note, however, that if we define a new field,

B ' . 1 ' ab H a b q = B i l + Y ~ o ~ c o [ t / ] , ( 1 4 )

and S~] k by eq. (2) with B replaced by B', then eqs. (11) and (13) may be written as, respectively,

I-Ityk=S~]k + ~a'[~23(A)--~23(co)]i/k +cOvariantterms' 6Bi,] =-la4 .... L°(iuMAM/] --°[i° ~ .abco)] ),ab. (15,16)

which is the standard Green-Schwarz transformation law [10]. This is related to the fact that the part of the right-hand side of eq. (8) (with U replaced by co - H) which is proportional to the torsion may be removed by adding a local counterterm to the lagrangian proportional to f ea#aaXi3 a X/coa[b i H]f.

The replacement of S by H in the action (1) corresponds to the addition of a new term,

(i/32rr) [I23(A ) - ~ 3 ( c o - n ) ] i/k X/PaX/~a A ' (17)

to the action. This destroys the naive N = ½ supersymmetry given in (4). But before discussing this issue, let us dis- cuss another source of local Lorentz and gauge anomaly. So far, we have considered the one-loop effective action involving only the external bosonic lines. Since, however, we have four-fermion coupling in our theory, the anomalous contribution to the effective action from a fermion loop will involve external fermion fields as well. This may be analyzed by introducmg auxiliary fields o ab R ab and replacing the four-fermion coupling term in (1) by

--(1/4rra')(aabFa~TMpa~ + iRaba ~tapaxb + aoabRaba~'~o~ -- " (18)

where Q and R are defined to transform covariantly under the local gauge and Lorentz transformations. We may t ab ab now construct an effective action involving the fields X ' , Qa and R a by integrating out the ~ and X fields. Since

the connections coupling to $ and X fields contain new terms proportional to Q and R respectively, the variation of this effective action under local gauge and Lorentz transformations now contains new terms given by

l f dr f do eaa¢ 3 o M F M r l a b ab ab , ~ ab~ij - 3 a O R# ). (19) bzrd d

This extra variation may be cancelled by adding new terms to the lagrangian given by

l_L. f dr f do ~#a yirAmpmnab . abnab, v . . ~ i ~ a b ~ - - t o i r t~ ) . ~3 7f .I ,Y (20)

Adding (20) to (18) and eliminating the auxiliary fields by their equations of motion we get the following extra terms in the action besides the four-fermion coupling

(1/321r)FaM(iAf a~xk~ap,,,xbeo~# coabafjxi'~paTM~ea# + ½a,ao~i.,a-.,k--M ab., -- 2t o a .a k col )" (21)

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Thus by adding terms in the original lagrangian given by (17) and (21), we may recover local Lorentz and gauge invariance of the one-loop effective action obtained by integrating out the fermion fields. Furthermore, if we as- sume the validity of the Adle r -Bardeen theorem [ 11 ] , we may conclude that this result is exact, and that there is no further contr ibution to the local Lorentz and gauge anomalies ,3.

As was pomted out before, the addit ion of these new terms seems to destroy the naive N = ½ supersymmetry. This symmetry, however, is anomalous [ 1,2,12], since it involves field dependent phase transformations of the chiral fermions qJ s, and also of the chiral fermions X a, since

8X a = e~](ieXJ)x i - e ~ ( a - ao)X'e, ~ ~s = (_eXia~)(TM)st~t. (22)

In ref. [2] we conjectured that the supersymmetry anomaly may be cancelled by the extra terms (17) and (21) in the lagrangian. In the rest of the paper, we shall verify this conjecture for a sI, ecific choice of the background fields, where we set&] = 6i1 and Bi/= O, but keep A M arbitrary. The contr ibution to the effective action from the ~k-loop may be expressed as ,4

(1]16rr)aM a Ma + f(a M - aM), (23)

where • MXi~ .~j, (24) a M : A M ( x ) a e x i + ¼,F;} ~ ,,

and f i s a function o f (a M - a M) only, which transforms under gauge transformation as

8 f = - (1 /8 r r ) 07 ao + eaa)aMacfl M, (25)

so that (23) transforms under a gauge transformation as given in eqs. (8) and (19). I t can be shown that under a supersymmetry transformation aMr - aMo transforms like a gauge transformation with parameter ieAMX t. Hence the variation o f f under a supersymmetry transformation is given directly by eq. (25) with 0 M replaced by leAMX i. On the other hand, using eqs. (4) and (24) we may directly evaluate the variation of the first term in (23) under the supersymmetry transformation. Ignoring terms of order ~' and higher powers of a ' , these variations may be shown to cancel the variation of the terms given in (17) and (21) under the supersymmetry transformation, up to terms proport ional to the classical equations of mot ion of the X ~ and the X i fields. These terms may be cancelled by redefining the supersymmetry transformation laws of X z and X i.

A complete proof o f the cancellation of supersymmetry anomaly at one-loop order will involve the evaluation of the full one-loop effective action. However the cancellation of the one-loop supersymmetry anomaly in the presence of background gauge fields is a strong indication that such cancellation indeed occurs even in the presence of arbitrary background fields. We should also ment ion at this point that the requirement of local gauge and Lorentz invariance still leaves us with the freedom of adding local gauge and Lorentz invariant counterterms to the lagrangian. Examples o f such counterterms are

-- ~ f dr f do(iftaba x t X apaX b + l o~'eat~xia¢ xJoo ab 4 ab + g~i O X ' ~ TMI9a~ - ½~'eaOa~ Xia , x/AM~t

(26)

where f and g are tensors which transform covariantly under local Lorentz and gauge transformatmns. The con- t r ibut ion to the anomalous variation of the effective action from the first and the third terms in (26) are cancelled

,3 In order to prove such a theorem, one has to find a gauge invariant regulanzation prescription for doing higher loop calcula- tions wath this effeclave action• One may be able to achieve this by adding gauge mvanant ingher derivative terms in the action involving the auxiliary fields Q and R.

M 4=4 Smce a M couples only to the right-handed fermions, the non-local terms in the effective actaon must be a function ofa M - a a Since the gaufle variation of such a function is always a funclaon of arM- a M, we need to add the local counterterm only. pro-

IVl .d Ot portaonal to a,~ a ~ a m order to reproduce the anomaly given in eqs (8) and (19).

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by the explicit variation of the second and the fourth terms, respectwely. The terms in (26) however are not su- persymmetric by themselves, and we expect the requirement o f N = ½ supersymmetry to uniquely determine the coefficients of all such counterterms (up to redefinitions of the background fields gi/(X), Bi/(X ) and AM(x)).

Thus we have shown that the o-model given in (1), plus the counterterms given in (17) and (21) makes the model invariant under local Lorentz and gauge transformations, with the transformation law of the antisymmetric tensor field given in (13) [or (16)] , the covariant torsion H appearing in this equation being given as a solution of eq. (11). (This equation may be solved iteratively for H.) There is also strong indication that this model retains

1 the N = ~ supersymmetry. The equivalence between eqs. (11), (13) and (15), (16) also shows us that in Witten's • t

consistency condit ion [12] f Tr(R A R - F A F ) = 0 we may take R as the ordinary curvature or as the general- ized curvature including torsion. This is related to the fact that the Pontryagin class of a manifold is invariant under the addit ion of a globally defined tensor to the connection.

Noted added. After completion of this work, we learned about some work by Nepomechie [14], which dis- cusses issues similar to that o f re f . [8] in a bosonized formulation.

The a-model approach to the string theory has also been used recently to derive information about the spec- trum of massless particles in the theory [ 15].

I wish to thank J. Att ick, A. Dhar, E. Martinec, R. Nepomechie, B. Ratra, A. Strominger, Y.S. Wu and S. Yankielowicz for useful discussions. I also wish to thank A. Dhar for pointing out a mistake in the original version of the manuscript.

References

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to be published. [3] C.G. CaUan, D. Friedan, E.J. Martinec and M.J. Perry, Princeton preprint. [4] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46. [5] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. 158B (1985) 316. [6] A. Sen, preprint FERMILAB-PUB-85/60-T;

S. Das and S.R. Wadia, preprint FERMILAB-PUB-85/90-T. [7] DJ. Gross, J. Harvey, EJ. Maxtinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B256 (1985) 253; Princeton

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[10] E. Bergshoeff, M. De Roo, B. De Wit and P. van Nieuwenhuizen, Nucl. Phys. B195 (1982) 97; G. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105; M.B. Green and J.H. Schwarz, Phys. Lett. 149B (1984) 117.

[ 11 ] S. Adler and W. Bardeen, Phys. Rev. 182 (1969) 1517. [ 12] J. Attick, A. Dhar and B. Ratra, in preparation. [13] E. Witten, Phys. Lett. 149B (1984) 351. [14] R. Nepomechie, University of Seattle, Washington, preprint No. 40048-28 P5. [15] D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. 151B (1985) 37;

C. Imbimbo and S. Mukhi, preptint No. LPTENS 85/19; R. Rohm and E. Witten, in preparation.

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Local gauge and Lorentz invariance of the heterotic string theory