Physics Letters B 320 (1994) 29-35 North-Holland PHYSICS LETTERS B

A note on gauge transformations in Batalin-Vilkovisky theory

A s h o k e Sen 1

Tara Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, lndla

a n d

B a r t o n Z w i e b a c h 2,3

Center for Theorettcal Physics, LNS and Department of Physlcs, MIT, Cambrtdge, MA 02139, USA

Received 19 October 1993 Editor. M. Dine

We give a generally covariant descriptmn, in the sense of symplectic geometry, of gauge transformations in Batalin-Vilkovlsky quantization. Gauge transformations exist not only at the classical level, but also at the quantum level, where they leave the actaon- weighted measure d#s=-d/t exp (2S/h) invariant. The quantum gauge transformations and their Lie algebra are h-deformations of the classical gauge transformation and their Lie algebra. The corresponding Lie brackets [ , ]q, and [ , ]e, are constructed in terms of the sympleetic structure and the measure d#s. We discuss closed string field theory as an application.

1. Introduction

In the ant ibracket , or Ba ta l in -Vi lkovisky (BV) formalism, the master act ion has long been known to de termine not only the BRST t ransformat ions but also the gauge t ransformat ions . Indeed, as explained in the original paper o f Batal in and Vilkovisky [ 1 ], and elaborated upon in a recent monograph (ref. [ 2 ], §17.4.2), the gauge t ransformat ions o f a field (or an- t i f ie ld) ~ ' are

J ~ ' = (o9"8, 3~S)A k, ( 1 )

where o9 is the symplect ic form, S i s the classical mas- ter action, and A k are f i e ld /an t i f i e ld independen t pa- rameters o f local gauge t ransformat ions with statis- tics ( - )k+1% This result, however, is not completely general. In add i t ion to leaving only the classical mas- ter act ion invariant , the above formula is not covari-

i E-mail address: sen@tifrvax.tifr.res.in, sen@tifrvax.bitnet. 2 E-mail address: zwiebach@irene.rmt.edu, zwiebach@mittns.

bitnet. 3 Supported in part by DOE, contract DE-AC02-76ER03069. ~1 0j and 07 stand for Ol/O~ J and 0~/0~ j respectively, where the

superscripts 1 and r denote left and right denvatwes.

ant under a change o f basis; i t requires the use o f Dar- boux coordinates which make the components o9'J o f the symplect ic form constant. More seriously, when A k is f i e ld /an t i f i e ld dependent , the above transfor- mat ions do not generally leave the symplect ic form invariant , and therefore they do not qualify as t rue gauge t ransformat ions or true invar iances (we recall that in BV quant iza t ion the physics is de te rmined by the ac t ion and the sympleet ic s t ructure) . This is in contras t to ord inary gauge theory where gauge pa- rameters can be chosen to be field dependent , in ad- d i t ion to being spacet ime dependent . I f the Ak's are f ie ld /anf i f ie ld independent we f ind

J ~ ' = o 9 % ( ( O ~ S ) A ~ ) = { ~ ', (0~S)Ak}, (2)

showing that the gauge t ransformat ions are canonical t ransformat ions , and are generated by the hamil to- n ian K = (0~S)A k. This form of the gauge transfor- mat ions has been widely used since most field theo- ries have been formulated using Darboux coordinates.

The ant ibracket formal i sm has been recently for- mula ted covar iant ly in the sense o f symplect ic ge- ometry. Whi le such covar iant descr ip t ion was known for the classical par t o f the formalism, the quan tum

0370-2693/94/$ 07.00 © 1994 Elsevier Scmnce B.V. All rights reserved.

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part required the introduction of extra geometrical structure [ 3 ]. In the covariant formalism we need not use Darboux coordinates, and should be able to give the form of the gauge transformations when the of J's are not constants. The relevant formula was given in ref. [4],

5 ' ~ ' = (a~'-'OjO~S)Ak + ½ ( O~coU)AkO~,S, (3)

giving a variation of the form ' ~ r g S = ~ (0k {S, S}) A k, which vanishes on account of the classical master equation. While (3) gives an invariance of the clas- sical master action, it does not necessarily leave the antibracket invariant. We do not have, therefore, a parametrization of the allowed gauge transforma- tions. Moreover, with second order partial deriva- tives, and partial derivatives of the components of the symplectic form, eq. (3) is noncovariant.

In this paper we shall (i) write down the classical gauge transformations and their Lie algebra (with the associated Lie bracket [ , ] c) in the general case when the gauge transformation parameters may be field dependent, and, (ii) generalize this to the full quan- tum theory, where we find a Lie algebra of quantum gauge transformations (with the associated Lie bracket [ , ]q). The proper geometrical interpreta- tion of the gauge parameters A k is seen to be that of hamiltonian vectors arising from some hamiltonian A. In our picture, the gauge parameters are taken to be the hamiltonians, which are necessarily field/an- tifield dependent functions. Standard gauge transfor- mations arise from hamiltonians linear in the fields (or antifields); by including all possible field/anti- field dependent hamiltonians we are naturally led to Lie algebras. The construction is fully covariant and involves only the symplectic structure and the action weighted measure d p s - d/t exp (2S/h) , left invar- iant by the quantum gauge transformations. We were led to consider this measure and the notion of quan- tum gauge transformations in our study of back- ground independence of quantum closed string field theory [ 5 ].

2. A path integral measure

In the covariant description of the antibracket for- malism a measure d/t in the space of field/antifield

configurations is necessary. This measure is used to define the operator Adv. I f

d/t =f(q~) 1-- [ dq~', g

then

1 AaaA- ~-f ( - 1 )'0,(fio'S0,A).

For any arbitrary measure d# the following identities hold [3,6,7]:

ApduA = AduA + l{ln p, A}, (4)

&~{A,B}={&~A,B}+(-)A+I{A,&~B}. (5)

Other identities we will use are the exchange property

{A, B} = ( - )~+a+'{B , A},

and the Jacobi identity

( __ ) (A+ 1 ) ( C + 1){A ' {B, C}} + cyc l i c = 0 .

A volume element d# is consistent ifA,]u = 0. Assume we have a consistent volume element dfl, and con- sider the measure d # s - d# exp (2S/h) . The associ- ated delta operator, making use of (4), is found to be

1 Aaus=Aas, + ~ {S, "}. (6)

Following ref. [ 3 ] one can then show that

2 1 &~ =~ {~&,s+ ½{s, s},.}

which indicates that A2azs is a linear operator, in fact, a hamiltonian vector. This equation also shows that d#s is a consistent measure (A2~s= 0) if d/* is consis- tent, and, in addition, S satisfies the quantum master equation

½ {s, s} + t~&~s=o .

The operator Ad~ s coincides with the operator 6 dis- cussed in ref. [ 2 ], here we have only pointed out its geometrical interpretation as the delta operator of the particularly relevant measure d/is. The fact that the master equation can be encoded in the consistency condition for a measure was noticed in ref. [ 8 ] and is implicit in ref. [ 3 ]. We observe now that the mea- sure d#s is a rather fundamental object in the BV for- realism. There is no need that this measure should be

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written in terms of another consistent measure d# and a nontrivial function S satisfying the quantum mas- ter equation. To argue this we must look at the defi- nition of observables.

The observables in a theory are defined by ( A ) - f r d2sA, where L denotes a Lagrangian submanifold, defined by the condition that at any point p ~ L, for any two tangent vectors e, ej, ~ TpL, we have co(e,, ej) =0. The measure dgs=-d2 exp(S/h) , can be de- freed directly in terms of d/~s using the same pre- scription that gives us d2 in terms of d# [ 3 ]. Let p e L, and (el .... , en) be a basis of TpL. One then defines

d2s(el,..., en ) - [d/zs(et .... , e , , f l ..... f n ) ]~/2, (7)

where the vec to r s f ' are any set of tangent vectors of the full manifold at p satisfying co(e,, f J ) = ~ . This condition fixes the vectors f J up to the transforma- tionf~-~fJ+CJ~e,. The right-hand side of (7), how- ever, is invariant under this transformation, since it corresponds to a transformation of the complete ba- sis ({e,}; {f J} ) by a matrix of unit superdeterminant. Eq. (7) gives us the path integral measure of the gauge fixed theory in terms of co and d#s. Finally, in order for ( A ) to be independent of the choice of lagran- gian submanifold, A must be a function of fields/an- tifields satisfying

hz~.A + {S, A} = 0 ~ A = 0

(by eq. (6) ) . This condition defines physical opera- tors in terms of d/zs and co.

3. Classical gauge transformations

I f the gauge transformations are to be symplectic they must be generated by a hamiltonian function. We should therefore have

~ ' = { ~ ' , K}, (8)

for some suitable odd function K. The symmetries of the BV action are generated by K's for which ~S= {S, K} = 0. Since this is the condition for K to be a clas- sical observable, to every local observable, we can as- sociate a local gauge symmetry of the classical theory. A special class of solutions are given by the trivial observables

K={S ,A} , (9)

since {S,/Q = 0, by virtue of the Jacobi identity and the classical master equation. Here A is an even func- tion of ~ . As we will see, standard gauge transfor- mations arise from trivial observables K. We will dis- cuss later, in the context of string theory, why nontriviaI observables do not lead generically to standard gauge transformations.

Let us now see how we recover the gauge transfor- mations in eq. ( 1 ). In a Darboux frame, A = qb'co,kA k, with A k constants, leads to K = (O~kS)A k. Back in eq. (8) we then recover the original form of the gauge transformations. Note that, by construction, A k is the hamiltonian vector associated to the hamiltonian A. We have therefore found that the original parameters of gauge transformations have the geometrical inter- pretation ofhamiltonian vectors. It is also instructive to understand the way the transformations given in (3) fit into this description. Such transformations are easily rewritten as

~ , ~ , = {a~ ~, (0~S)A k}

-cov(OjAk)(O~S)+½(O~co'OA~OyS: (10)

We now recall that a transformation of the type Ot~b'= (0ffS)/l U leaves S invariant in a trivial fashion if /zu= (_)u+1#1,, although it does not necessarily leave co invariant. One can verify that the second and third term in the above equation do not correspond in general to trivial transformations. However, i fA k is hamiltonian (Ak= CO kJ0//) they do. In this case we can prove that ~2

~ ' ~ ' = {a~ ', (0~S)A k} + (0~S)u U ,

with

# 'J= ½ ( - )i[oyk0kA'+ ( -- )('+1)O+ ~)co'k0kAq.

Therefore, when A k is hamiltonian the ~' transfor- mations differ from gauge transformations by trivial transformations. Note, however, that the tS' transfor- mation still cannot be regarded as a genuine symme- try of the BV theory since it does not represent in general a canonical transformation.

Let us now compute the algebra of gauge transfor- mations. Let c~,c denote the canonical transformation generated by the odd function K: ~Kf= {f, K}. Given that [~K~, ~r~ ] = 8~K~.~a~, canonical transformations

a2 We use the Jacobi identity as well as the identities in Appen- dix B of ref. [7 ].

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generated by observables form a Lie algebra, since we have {S, {K1, Kz))=0, whenever K1 and K2 are ob- servables. While any local observable in the theory generates a symmetry, not much can be said about the algebra of non-trivial observables unless we know which specific theory we are studying. Hence we shall focus our attention on the Lie subalgebra whose ele- ments are the canonical transformations generated by trivial observables K= {S, A}. Let 6~ = 8~s~ denote the classical gauge transformations induced by A, then

[ ~ , a~,] = [8~s~ , ~s,A~] = ~ , ~ s ~ •

Moreover, the Jacobi identity and the master equa- tion imply that

{(S, AI }, {S, a2})

=½{s, {A,, {S,A~}}i-½{S, {A~, {S, A~))}.

As a consequence we have that

c c G [6a,, ~.~, ] =6ta,/:~o, [A~,A2]~-½{A1, {S, AE}}-½{Az,{&A~}}. (11)

In contrast with the standard description of gauge transformations via generating sets of transforma- tions which do not give a Lie algebra, and close only up to trivial symmetries (see ref. [2] ), our descrip- tion of gauge transformations arising from hamilto- nian functions A gives directly a closed Lie algebra. The commutator of two gauge transformations is a gauge transformation of the same type. The usual gauge transformations, arising from A's that are only linear in fields or antifields, as expected, do not close among themselves in general. They close when sup- plemented with A's having additional field/antifield dependence. The trivial identity

[ [ ~ , ~ ] , ~.~, ] + cyclic = 0,

implies that

t~([Al , [A2 ,ha]c ]e+eyehe) = 0 ,

and, as a consequence ( [A1, [A2,A3 ] ¢ ] ~ + cyclic) must be either zero or a gauge parameter that generates no gauge transformation. The latter is true, the Jacobi identity for [ , ] ¢, calculated using ( 11 ) gives a result of the form (S, Z}, which, as a gauge parameter, gen- erates no gauge transformation.

The above observations indicate that [ , ] ~ leads to a strict Lie bracket in the space of even functions A

modulo functions of the form {S, •}. Let us denote the equivalence relation by g , that is A ~A + {S, Z}. Indeed, the definition given above implies that [A, {S, Z} ] ¢ ~ 0, and therefore the bracket depends only on the equivalence class of the gauge parameters. This bracket defines the Lie algebra of classical gauge transformations.

4. Quantum gauge transformations

The main difficulty in understanding the notion of gauge transformations at the quantum level was due to the apparent lack of a suitable invariant object. In Darboux coordinates, the gauge transformations given in (1) do not leave invariant the quantum master action, nor the measure d/t exp (S/h), as one could naively hope. This is easily verified using the result that the variation of any measure d/z under a canonical transformation generated by K is given by ~xd#=2d/t AduK (see ref. [7], eq. (3.26)). Since we are taking K={S, A} with AoJI=0 (A is linear in fields), this result, with the help of eqs. (4), (5), and the Jacobi identity, leads immediately to

8~d~ exp(S/~)

= 2 d# exp( S/h ) (AduS + --~fi {S, S),A} .

If instead of a factor -~ multiplying the {S, S} term, we would have a ½, the master equation would imply invariance. This means that the measure d#s=d/L X exp (S/h) introduced earlier is actually invariant under the gauge transformations of eq. ( 1 ).

We can now easily generalize the result to arbitrary coordinate systems, and field dependent gauge trans- formation parameters. The variation of the measure d/Ls under a canonical transformation generated by K is given by 8K d#s=2dltsAdusK, and therefore the condition of invariance is simply AdusK=0, i.e. K must be an observable in the full quantum theory. A special class of solutions, representing trivial observ- ables, is given by

K=- hAdvsA = hAduA + {S, A}, (12)

where invariance follows due to the nilpotency of Adus. These gauge transformations close under corn-

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mutation. Defining the quantum gauge transforma- tion ~ _-__ ~a~.a, we find

[~3~, ~31 ] = ~?~, .~,

[AI, A2] q - ½{A~, hA~u~A~}- ½{Az,h~u~A 1} . (13)

The bracket [ , ]q differs from its classical counter- part [ , ] ~, given in eq. ( 11 ), by terms of order h. In exact analogy to the classical case, we have that ( [A~, [A:, Aa]q]q+eyclic) is of the form AdusZ , which, as a gauge parameter, generates no gauge transformation. Therefore [ , ] q leads to a strict Lie bracket in the space of even functions A modulo functions of the form ~usZ. We can check that [A, Z~usX]q~0, and therefore the bracket depends only on the equiva- lence class A ~A + Adus Z of the gauge parameters. This bracket defines the Lie algebra of quantum gauge transformations.

4.1. Example L Scalar fieM theory

We first illustrate our ideas with the help of the simplest theory, namely the theory of a free scalar field O in D dimensions described by the action

S= J dDx[0u¢ 0u0 - V(~) ] .

In this case the classical theory does not possess any gauge invariance in the usual sense. The BV formu- lation Of the theory involves the field ¢ and its anti- field 0% and the BV master action coincides with the classical action S. As a result, any local function K(¢) , independent of antifields, corresponds to a local ob- servable (z~usK (0) = 0 ), and hence generates a gauge symmetry ~3. The resulting transformations are

~O =~{0, K(¢)} =0,

50*= {0% K(¢)} = 5K(0) fiO (14)

This can easily be seen to be a symmetry of the theory leaving both the action S (which is independent of O*), and the measure dO d¢* separately invariant. This, of course, need not be the case for general K(O, O*) =~A~A (0, 0").

a3 Note that a general K of this form cannot be written as Adus// for a IocalA.

PHYSICS LETTERS B 6 January 1994

4.2. Example II. Gauge transformations in closed string fieM theory

The closed string field theory master action is given by [9]

g•o ~ 1 g (e~v) S= fig Z -~.1 N ( [ ~ / ~ 1 . . . [ ~ ' / ) N . N = 2 for e = 0 N = I for e ~ i

(15)

The corresponding measure is d # = ]~,dV' (for nota- tion, see refs. [ 9, I 0,5 ]. ) Let us study gauge transfor- mations generated by observables of the form K = h ~ A . The most general form of A is given by

g ~ 0 (16)

Separating out the contribution of the terms involv- ing (V(°'2) [ = (Oh21Q (2), we get

g,N>~O

g>~O, N~ 1 • z= 1

N--1 fo rg l = g g N+I fore1 <e 1

-- ~ he ~, E ( N - m + l ) ! ( m - 1 ) ! e,N~O gl = 0 r a~ 1

X 1. (v (g- -g l 'N- -m+2) I @ 1'.. ( A (gl ,m) [ ~ 1 1 , )

×1~)2-- .I ~tr)Ar--m+2 I 7t)U .... I~)m,

e l e ~ 1, N>~0

x I ~>3... I ~>N+2, (17)

where 15e) denotes the sewing ket [ 10]. This gener- ates the gauge transformation

g,N~ 0

(18)

In particular, choosing a A for which only (A(°,l)[ --- (A I is non-zero, we get

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Volume 320, number 1,2 PHYSICS LETTERS B 6 January 1994

g ~ I ~ > e = - o < A I Q ( ° ) I ~ ¢ >

1

g = 0 N~> l ( g = 0 ) N~>0(g>0)

× I ~e>3,.-I ~'>N+2

= QIA>o

1

g=O N ~ l (g=O) N ~ 0 ( g > 0 )

× IA>2 ~r)3 ... I ~>N+2, (19)

where I A > e--- 0 (A I Seo¢ >- Note that the I ~> indepen- dent term of the gauge transformation receives a con- tribution f rom two-punctured surfaces o f all genera. This means that the notion o f an unbroken gauge symmetry at the classical level differs f rom the cor- responding notion at the quantum level. I f we trun- cate to g = 0 we recover the standard gauge transfor- mations o f classical closed string field theory.

Are there gauge transformations generated by non- trivial observables? To answer this we need to see i f there are local $4 observables in string field theory with are not of the form Ad,sA. It might perhaps be possi- ble to construct nontrivial local observables that are quadratic and of higher orders in the fields. They would generate transformations g l ~g> that are linear and of higher orders in [ ~>, but do not contain any I~g> independent piece. Although these would give rise to local symmetries o f the theory, they would not be gauge symmetries in the conventional sense o f the term. I f we want a symmetry transformation that contains a field independent piece, we need a K that is linear in the string field 1 7t). We shall now argue that in string field theory there is no local observable that is linear in I ~ and is not of the form Aa~A. The intuition is simple, such transformations would cor- respond at the linearized level to shifts o f the type

~4 Here by local expressions, we mean terms that when ex- pressed in momentum space, the integrands have well defined Taylor series expansion m momenta about the point where all momenta vanish. In position space these terms will be repre- sented by a series contaming higher derivative terms. In this limited sense the string field theory lagrangaan and the gauge transformations are local, since they contain integration over subspaces of the moduli space which do not include any de- generation point.

I ~> ~ [ ~> + I HQ >, where I HQ > is an element of the cohomology of Q. Such transformations leave the ki- netic term o f the string field action invariant, but cer- tainly do not qualify as gauge transformations.

To prove this it is enough to take K = < KI ~> linear in 1~>, and analyze the ~ independen t terms in the equation Adps/r(= 0. This gives Q (e) 0 < KI 6eo~ > = 0, and shows that o<KI ~e> must be a BRST invariant op- erator. Furthermore, since we want to exclude solu- tions o f the form Ad~A, we must also require that 0 < KI 6eo~ > be a non-trivial member o f the BRST co- homology. But in string theory we know that non- trivial members of the BRST cohomology are found only for certain specific values o f momentum k u, sat- isfying mass-shell constraints o f the form k 2 = m 2, where m is the mass of one of the particles in the spectrum of string theory. Thus, when expressed as a momen tum space integral, <KI ~F> will contain a fac- tor o f g (k 2 - m 2) in the integrand, and hence it does not correspond to a local observable in the theory.

In this example we have found the quantum gauge transformations of closed string field theory. An ob- vious question is whether our formalism can help us understand the gauge structure o f string theory. Since the present approach deals with Lie algebras, it may provide an alternative or complementary approach to current studies based on homotopy Lie algebras [ 11 ]. It remains to be seen if the Lie brackets [ , ] ¢ and [ , ]q define manageable structures in string the- ory. Another interesting project would be to isolate from the string algebra the Lie subalgebra represent- ing coordinate transformations in string theory [ 12 ].

References

[1] I.A. Batalm and G.A Vilkovisky, Phys Rev. D 28 (1983) 2567.

[2] M. Henneaux and C. Teltelbolm, Quanttzation of gauge systems (Princeton U.P, Princeton, NJ, 1992).

[ 3 ] A. Schwarz, Geometry ofBatahn-Vilkovlsky quantlzation, UC Davis preprint, hep-th/9205088 (July 1992).

[4] E. Wltten, Phys. Rev. D 46 (1992) 5467. [5]A. Sen and B. Zwiebach, Quantum background

independence of closed string field theory, MIT preprmt CTP ~2244, to appear.

[6] E. Witten, Mod Phys. Lett. A 5 (1990) 487. [ 7 ] H. Hata and B. Zwiebach, Developing the covariant Batalin-

Vilkoxqsky approach to string theory, MIT preprint CTP $2184, hep-th/9301097, Ann. Phys. (NY), to appear.

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Volume 320, number 1,2 PHYSICS LETTERS B 6 January 1994

[ 8 ] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, MIT Math preprint, hep-th/ 9212043.

[9 ] B. Zwiebaeh, Nucl. Phys. B 390 (1993) 33. [ 10 ] A. Sen and B. Zwlebach, Local background mdepelldence

of classical closed string field theory, MIT prepnnt CTP #2222, hep-th/9307088, submitted to Nucl. Phys. B.

[ 11 ] E. Wltten and B Zwiebach, Nucl. Phys. B 377 (1992) 55. E. Verlinde, The master equation of 2D string theory, Princeton prepnm IASSNS-HEP-92/5, hep-th/9202021, Nucl. Phys. B, to appear; T. Kimura, J. Stasbeffand A. Voronov, On operad structures ofmoduli spaces and string theory, hep-th/9307114.

[ 12 ] D. Ghoshal and A. Sen, Nucl. Phys. B 380 (1992) 103.

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