Perturbation theory and locality in the field-antifield formalism

The Batalin-Vilkovisky formalism is studied in the framework of perturbation theory by analyzing the antibracket Becchi-Rouet-Stora-Tyutin (BRST) cohomology of the proper solution Ss. It is concluded that the recursive equations for the complete proper solution S can be solved at any order of perturbation theory. If certain conditions on the classical action and on the gauge generators are imposed the solution can be taken local.


I. INTRODUCTION
The construction of the effective action is a key ingredient in the covariant quantization of gauge theories.One of the most powerful methods to get the effective action for such types of theories is the procedure based in the Becchi-Rouet-Stora-Tyutin (BRST) symmetry,' in particular, its field-antifield formulation.*In this approach the main open questions are the unitarity of the resulting theory and, related to this point, the locality of the gauge fixed action.
An alternative procedure to construct the effective action in the framework of perturbation theory was proposed by Frolov and Slavnov.3They studied the condition imposed by the commutativity of the S-matrix operator S and the free BRST charge and found a set of recurrent equations for the effective action.In a recent paper4 we developed the field-antifield formalism in this framework and showed that it produces a gauge fixed action which coincides with the one obtained by them.The equivalence of both procedures was proved by making two assumptions: (i) there exist suitable power series expansions in the coupling constant for the structure functions and (ii) the existence of solutions of the recurrent equations for the proper solution of the master equation at any order in perturbation theory.In particular, the first assumption allowed us to solve explicitly the lowest order recurrent equations and to show that their solutions coincide with those given in Ref. 3. The problem of unitarity was also considered.Under certain assumptions on the gauge generators, we proved the absence of pairs of singlet?' and, consequently, that unitarity ultimately relies on special features of the theory, as for example, the presence or absence of classical gauge invariant degrees of freedom with negative norm.
In this paper we will analyze the antibracket BRST cohomology defined in the space of aN functionals (i.e., local and nonlocal ones) and, in particular, the cohomology associated with the free part of the proper solution Sc, using homological techniques.By introducing a nilpotent operator o, we will construct an operator that counts the number of fields and antifields and show that the antibracket BRST cohomology of S is isomorphic to the kernel of this operator.(The idea of relating the BRST cohomology with the kernel of a suitable number operator has been widely used in the literature.See, for example,  This study allows us to verify the triviality of the cohomology associated with the free part of the proper solution Se except at ghost number 0 and, as a consequence, (i) that the recurrent equations for S can be solved at any order in perturbation theory (The proof of the existence theorem of an effective action in the field-antifield formalism was given earlier in Refs.11-13) and (ii) that the structure functions have suitable power series expansions in the coupling constant.
On the other hand, the question of the locality of the gauge fixed action in the field-antifield formalism has recently been considered by Henneaux,14 studying the local cohomology of the so-called Koszul-Tate differential, and also in Ref. 15.In this paper we also address this problem in the framework of perturbation theory.Assuming the locality of the classical action SC, and of the generators of the gauge algebra Rz;-', s=O,..., L, and the local completeness of the free part of these generators, we shall show thit the structure functions defining the gauge structure of the theory and, as a result, the proper solution of the master equation, can be taken to be local.
We have organized the paper as follows: in Sec.II we analyze in detail the antibracket BRST cohomology through the introduction of a contracting homotopy (T.In Sec.III we provide an alternative proof of the existence theorem of a solution of the master equation for a generic gauge theory, in the framework of perturbation theory.The question of the locality of the proper solution is addressed in Sec.IV and in Sec.V we give some conclusions.We end with two appendices.The first one is devoted to analyzing the questions of the properness and the classical limit of the solution of the master equation in the framework of perturbation theory.Finally, in the second appendix, we justify the form of the structure functions used to derive the above results.

II. ANTIBRACKET BRST COHOMOLOGY
In order to analyze the antibracket BRST cohomology defined in the space of all functionals (local and nonlocal ones) for a generic proper solution S, it would be useful to recall the so-called regularity conditions'2v'6 and the concept of observable.16Let I be the infinite dimensional space of all histories and Gj=~SC,/&$ the classical equations of motion.Then, the regularity assumption states that Gi provide a regular representation of the stationary surface 8, where the equations of motion hold.This means that the functions Gi can be locally split as Gi= (GA-,, G,-,> in such a way that (8 G,-, = 0 is a consequence of GA-, = 0, and (ii) the rank of dGA-, is maximal on the stationary surface 8.If for a given theory the regularity conditions are verified, any smooth function vanishing on the stationary surface can be expressed as AA-lGA-, , with AA-1 smooth functions.Throughout this paper, we will consider classical actions SC, for which the regularity conditions hold.
With respect to the dynamical observables, following Ref. 16, these are described as smooth functions defined on the stationary surface C. The definition of observable can be extended to the space of all possible histories, 1, by identifying two functions which coincide on shell, i.e., F(4) -F'(4) e F-F'=Az'Gi.
Moreover, in the case of gauge theories, one must further impose that the observables be gauge invariant, at least on the stationary surface, that is aFi v Ro%=/ZlGi, being R& the generators of the gauge transformations.
Given that functions defined on I form an algebra, C(I), and that functions which vanish on shell form an ideal N of this algebra, the algebra of functions defined on the stationary surface Z is isomorphic with the quotient C(I)/N.
In presence of gauge invariance, however, this is not the complete story.As it is well known, the gauge transformations are integrable on Z and generate the so-called gauge orbits.Since the gauge invariant functions are constant along the gauge orbits, they induce well defined functions in the quotient space Z/G, G being the gauge orbits.Therefore, the space of observables can be identified in this case with the set of smooth functions on Z/G, C(Z/G).
A cohomological interpretation of this construction was given in Ref. 16.It was found that the set of physically inequivalent observables is isomorphic with the cohomology at ghost number zero of a nilpotent operator S, Ha(S), induced for the proper solution of the master equation through the antibracket, and defined as fiF=(-l)F(F,S), H,(6)= where the Fin the exponent stands for the Grassmann parity of the function F.
In what follows, we will try to relate locally the antibracket BRST cohomology of a generic proper solution S and, in particular, the set of physically inequivalent observables, with the kernel of a certain number operator N. As a preliminary step, we will perform this construction for a regular action without gauge degrees of freedom.

A. Nongauge theory
Consider a regular action SC,(#) without gauge degrees of freedom and introduce an antifield # for each classical field @; with opposite statistics.(For simplicity, throughout this paper we will assume that the classical fields 4' obey Bose statistics.)When no gauge degrees of freedom are present, the proper solution is nothing but the classical action S,t(#') and the BRST operator 6 is defined as Introduce in the space of all histories I the change of coordinates d . (2.1) Due to the fact that the action S,, defines a regular theory without gauge degrees of freedom, the Jacobian matrix of this change (2.2) does not have null vectors around the stationary surface Gi=O (which in the regular case reduces to a point) and, as a consequence, the above change is invertible, at least in a neighborhood of the stationary point.In this neighborhood, the BRST operator S acts on the local coordinates as 6Gi=O, S$F=Gi.where (H-')'j is the inverse matrix of the Jacobian matrix (2.2).This operator results to be an antiderivation u(AB) =uA .B+ (-l)AA.oB.
It is clear that, in general, u will only be defined in a neighborhood of the stationary point, where the change (2.1) and the Jacobian matrix (2.2) are invertible.In terms of the local coordinates (Gi, #), the action of (T reads uGi=4r, u~T=O. (2.4) Now, with the operators S and (+ at hand, one can construct a new operator as the following quadratic combination N=uS+Su.
In this sense, the operator u (2.4) is a contracting homotopy.
Consider now a smooth function F(4) defined in the space of all histories I.The assumed regularity conditions allow us to write F as where F( 4,J is the value of Fat the stationary point and A' are suitable smooth coefficients.For the same reason, these coefficients can be expanded in the same way

B. Lth stage reducible gauge theory
Let us see now how we can implement the constructions of u and N to the case of a general Lth stage reducible gauge theory.To perform this program it will be useful to note the so-called abelianization theorem.' 1P16S'7 This theorem states that, if the regularity conditions are verified, there always exist locally smooth coordinates 41i = (4"-1, f-1) in which the equations of motion and the reducibility functionals Rf;-' take the simple form, s Gi=(GA-,, GaelsO), (2.8) RA"-l=O R%-~-fj=S-1 R%-I= 4 4 -As % RAS-I =O R%-'=O , s=o ,..., L-1, as us R"L'a,,= (#;-LO, R;;-l&;-l), where the indices a,-,, A, range over the same values.
The particular form of the generators of the gauge transformations in these coordinates shows that the fields 4A-1 are the independent gauge invariant degrees of freedom, while 4"-1 are "pure gauge."It should be stressed also that, in general, the separation (2.8) cannot be done globally with a single set of independent equations of motion GA-, .
All these considerations imply that in the space of fields and antifields there always exist locally smooth coordinates (2.9) s=o Therefore, the BRST operator S associated with S is expressed in terms of these variables as S4A-I=0, @Lo, ,4%-s~$3, Note that, except for the classical gauge invariant degrees of freedom 4A-' and their antifields, all the other fields are organized in cohomological trivial pairs.Now, let us construct the contracting homotopy u, in such a way that the operator N=Su+uS acts as a number operator.As in the precedent subsection, consider the coordinate change '+'-'$~~,=G,~,('+~, (2.10) which will be invertible locally due to the fact that the functions GA-, have been chosen to be independent.Therefore, the Jacobian matrix HA-l.Kl= a4k, a's,, @C=a,p-la4B-ly that is, on the stationary surface, described locally by the coordinates q-1, the gauge invariant function Fis a constant.On the other hand, two gauge invariant observables F(4), F'(4) are equivalent if their constant value on the stationary surface is the same.Therefore, we obtain a similar result as that we arrived in the regular case, namely, in each class of equivalent observables we can choose a representative which is a constant function over all the space of histories I.Moreover, taking into account the expansion (2.12) and the actuation of N, we finally conclude that the kernel of this operator, i.e., the set of constant functions, is isomorphic to the set of equivalence classes of observables, or H,(S) Therefore, we conclude that the cohomology spaces Hk(S) for k#O will be trivial if u can be defined globally, that is, if the local expressions of H in Eq. ( 2.14) can be patched together without trouble.

III. PERTURBATIVE SOLUTION OF THE MASTER EQUATION
In this section we describe an alternative proof of the existence theorem1'-13 of a solution of the master equation for a generic gauge theory in the framework of perturbation theory.
Let us consider a generic Lth stage reducible gauge theory with a classical action SC,, given by a power series expansion in some coupling constant g of the form scl=s:?[421 +&,k9 431 +s,h& 441 +***,  The form of the classical action (3.1)) together with the boundary condition (A 1) imposed by the classical limit, suggest that the proper solution of the master equation S[@, a*] should have also a similar power series expansion S[Q, @,*I = 2 snw; 91, q=C@, a*>.
(3.3) n=O Moreover, as we shall show in Appendix B, a generic term S, in this expansion could be taken to be homogeneous of degree n +2 in all the fields, i.e., of the form S,[gn; q"+'].This particular form impljes in turn that we can assume a power series expansion for a generic structure function Fi,... where Rr;-' cg", #'I stand for the free part of the generators of the gauge algebra.Note that this free part o> the generators, in the framework of perturbation theory, should have the same rank as that of the generators of the complete theory c;; '[+] in order the spectrum of the free and the interacting theory coincide, i.e., rank ky;;l [4] =rank Rz;; '[$, 4'1, s=O,...,L.This condition guarantees that the free solution of the master equation (3.7) is proper.Therefore, from now on we will always assume it.
In order to show the existence of S,, for a generic n, it will be useful to introduce the BRST transformation associated with the free part of the proper solution So SF= (-l)F(F, So).For n>2, it is not obvious that the equations (3.9) could be solved.The necessary and sufficient conditions ensuring the existence of a perturbative solution of the master equation are that Q, be S closed and the cohomology group of S at ghost number one, H,(S), be zero H1(6) =o. (3.12) Let us first prove H,(S) =0 using the general results obtained in Sec.II.To this end, consider the contracting homotopy u defined there.As we have said, if the operator (T can be defined globally, that is, if the local expressions of H in Eq. ( 2.14) can be patched together without trouble, the cohomology spaces Hk( S) for k#O will be trivial.This is what happens for the Lth stage reducible free gauge theory we are dealing with.Indeed, consider the free part of the generators of the gauge algebra Rsa as;'@, 4'1, s=O,..., L, separate the independent components R>--I Jd'd'l an d t a each level complete these sets with vectors R%:[g", &"j in such a way that the matrices be invertible, i.e., there exist matrices Qz-', such that Q>;',Rr;", = 6zy:.
Next, consider the canonical transfoAation, in the &tibrifcket sense (3.13) C;;=~tisRf;,=C$s(R~, R$ = (C$, C$J, s= -1,0 ,..., L. In terms of these new fields the free proper solution (3.7) turns out to be S,=S$'(tp>+ i ~~I,as-,s~-'~9 s=o (3.14) which is precisely of the form (2.9) presented in Sec.II.What is important to note here is that, due to the fact that the matrices Q;-", Rf4;-', do not depend on the fields, the canonical transformation (3.13) and the express& (3.114) for So are valid everywhere and cover all the space of fields and antifields.On the other hand, the coordinate change (2.10) from the fields I$~--' to the free equations of motion is also globally invertible.This is so because the Jacobian of this change (2.11) is the operator defining the free part of the classical action in the space of the gauge invariant fields @-I, which is independent of the fields and invertible.Consequently, the operators u and N introduced above have well defined global expressions and, as a result, the local S-exact pieces F=SH in (2.14) can be patched together to form a globally defined object.Therefore, we conclude that the antibracket BRST cohomology defined in the space of all functionals (local and nonlocal ones) of the free proper solution So under consideration is trivial except at ghost number 0, i.e., Hk(S) =o, Vk#O, and that, in particular, the condition (3.12), H'(S) =0, is verified.
To complete the proof, let us check by induction that Q, are S closed for all n.Consider the first nonvanishing Q, , i.e., Qz.Its BRST variation is which is zero by virtue of the BRST invariance of S'.Taking into account that (3.12) is verified, this allows us to solve the equation for Sz as SS~+Q~=SS,--sF2=0 j S2=F2+S&.which is zero by virtue of the Jacobi identity for the antibracket.
In summary, we have shown that the consistency conditions SQ,=O and (3.12) are verified.This fact implies that the recursive relations (3.9) can be solved order by order in perturbation theory and ensures the existence of a perturbative solution of the master equation.

IV. LOCALITY OF THE PROPER SOLUTION
In Sec.III we have seen that it is possible to construct a perturbative solution of the master equation.Moreover, the arguments presented in Appendix A show that this solution can be taken to be proper and verify the classical limit.The question of the locality, however, has remained open until now.
In this section we will see that, if the following conditions are verified, namely, where the square brackets mean suitable symetrization of the enclosed free indices.
Let us analyze these equations and the form of the terms S, in more detail.As we argue in Appendix A, the lowest order terms in the antifields of a solution of the master equation verifying the boundary conditions (A3) and (A4) read, Eq (A5), Sat@, @*I =&lb?, V+21+ $ ~-,,a,-,R;;; 'W, 4"l~+W~)21.
Therefore, to guarantee the locality of S, we must impose, at least, that SC1 and the generators of the gauge algebra to be local, hence, conditions (a) and (b).These conditions ensure automatically that So is local and, as a consequence, that the operator 6 (3.where usjk @I9 f',,k &I are the structure constants appearing in Eq. (3.11).
Consider now the equations (4.4) which determine the structure constants Uo&, @I from the knowledge of the lowest order terms of the generators Rb%.By hypothesis (b), the left-hand side of these equations are local expressions verifying Moreover, by local completeness of R&$g", 47, this expression can be expressed as a combination of them with local coefficients.Hence, the structure functions U,[g, I$~, which are nothing but these coefficients, can be taken to be local.
The group of equations (4.5) determines U,[g, @'j once UoJ-i[g, $J~ are known.Suppose, thus, that UO,-J~, @I are local (which is valid for U&g, $7).Then, the left-hand side of these relations is a local expression which verifies Therefore, by local completeness, the quantity inside the bracket can be expressed as a combination of Rz;' p, @ with local coefficients.As a consequence, the structure functions U&k, $7 can also be chosen to be local.
Finally, the equations (4.6) allow us to determine recursively the remaining U,, ,,&, &'j at lowest order in perturbation theory from the knowledge of the U-structure functions with lowest or equal index Uj+I,s-lk, &, UJg, &.Wh a is important to note here is that the t unkmwn quantity uj+ I,&, PI a pp ears in the equation in the form R,, j+ ,[g', $7 U,, &, 49, while the remaining expression is a local quantity, by recursive hypothesis, which contracted with R,, jk", $7 vanishes.So, applying the same arguments as above, we conclude that U,+l,&, & can also be taken local.Proceeding in much the same way one can see from Pqs. (4.7) and (4.8) that V,[g, @j at this order can be chosen local.Therefore, with such selections S1 turns out to be local.
From the above analysis, it should be clear how to proceed in the general case.Suppose that, up to order n -1, all the terms Sk, k=O,...,n-1, have been constructed local.Being all of them local, Q,, and the corresponding coefficients Q* * * appearing in the expansion (4.2) are also local.On the other hand, assume that the coefficients have been determined as local terms.In such conditions, the right-hand side of Eq. (4.3) is a local expression which vanishes upon application of Rsa as; '[go,4q. By hypothesis (c) this expression can be written as a combination of these generators with local coefficients.Therefore, the coefficients I +I...& f;...a, Wl~ I with index higher than or equal to (4.9), can be constructed locally.Applying this argument recursively, we conclude that all the coefficients appearing in the nth term S, of the proper solution of the master equation can be taken to be local.This completes the proof of the locality of the proper solution S in the framework of perturbation theory.

V. CONCLUSIONS
In this article we have analyzed the antibracket BRST cohomology defined in the space of all functionals, especially the cohomology of the free part of the proper solution of the master equation, So, using homological techniques.The main result, namely, that the cohomological spaces Hk( 6) for k# are trivial in the free case, has allowed us to find a perturbative solution of the master equation, as well as to obtain the form of the power series expansions in the coupling constant for the structure functions, i.e., The question of the locality of the solution obtained has also been considered.Our study has shown that provided (i) the classical action is local, (ii) the generators of the gauge algebra (2.3)    Define locally in the space of fields and antifields an operator u which decreases the ghost number by one unit a#= (H-l)Uqf ,, o+:=o, A'(4) =A'(4,) +2i(4)Gj, and applying this procedure recursively we arrive at F($)=F(c$,)+ ~r@+~GjI--Gin, n(2.5)where we must understand that this expansion is valid in a neighborhood of the stationary point.If instead of functions F(4), we consider functions F(4,4*), defined in the space of fields and antifields F(4,4*) =F(4) + ~~~Fi1"'i.(4)4~...4~, with F(4), F'l""n(4) smooth functions of the fields 4i, the result (2.5) leads us to the expansion around the stationary point F(4,4*)=F(40)+ c Gi,...GinI;il"'i~~jI"'im4j*l...4~~.now how to realize the inequivalent observables in a regular theory.From the definition of observable it is clear that all the elements belonging to a given equivalence class are characterized for having the same value in the stationary point.Then, the most natural representative in each equivalence class is a constant function over all the space of histories I. Therefore, C(I)/M-{F(4) /F(4) =F(4,) =ct.)-{observables}.(2.7)On the other hand, consider the action of the operator N on a generic function F(4,4*) of the form (2.6) NF(4,4*)=0+ c [(n+m)Gi1...GinFil"'in,jl"'jm4j*l...4i*m].nm It is clear that the kernel of this operator is precisely kerN=CF(4,4*) / F=F(4,) =ct.}, which is nothing but the set of inequivalent observables (2.7).In summary, by constructing an operator N which counts the number of equations of motion and of antifields, we have characterized locally the set of physically inequivalent observables as the kernel of this operator.

F
(2.11) turns out to be invertible locally.In such conditions, it is easy to see that the contracting homotopy u which realizes the above program acts on the fields as u+qJlp-', up*=o, uca,= ( -1) "SZf-' c"L"-;' , uc"/z; = 0, and on the antifields d$,=o' u4:-,=c*,,,@ -I' : uC*L, L =Q ~c*,-l,aL-*=CZaLq-*9 being as in the regular case nilpotent, $=O, and decreasing the ghost number by one unit.With these definitions of S and u one can see that the operator N acts on the above coordinates as NCLak= CL'", NCck= Cc,, and counts the number of independent equations of motion GA-, , fields 4=-l, ghosts, . . . .and the corresponding antifields.Here and from now on we define Cy";' zz GA , Cc,,, ~42 .Now, consider a smooth function of the classical fields F(4).Assuming the rkgula$y conditions we can expand it in a given local chart as F(4) =%A-, , q-l> =F(O, 0)+ 1 GA-,***GD-, ~I:::::J' f-'***@-'.Applying the same argument, a smooth function of the fields and antifields F( C,, c> of the type admits an expansion in terms of the coordinates (C: , Cl*) of the following form: F( C,, c> =hC;, C:*) =i;;(O,O)+ 1 ~~...~~~n~~~~~~~c;:B,...c~~ nis the value of F(4) at the point GA-, = q-1 = 0. Let us consider the gauge invariant functions.We have aF Ho(S)3{observables}2:kerN={F(C,, q> /F=g(O,O)=ct.}.The construction of the contracting homotopy u and of the operator N allows us to gain a better understanding of the antibracket BRST cohomology of a generic proper solution S. As it is well known, given a proper solution S, one defines the nilpotent BRST operator S and its corresponding cohomology spaces as sF=(-l>F(F,S), H/As)= being k the grading of the corresponding space, i.e., the ghost number.Until now we have been studying the cohomology space Ho(S), which gives the physically inequivalent observables.To study the case k=#O consider a smooth function F( C,, c) such that gh F= k#-O and SF=O.Given that gh F#O, we have an expansion of the type (2.12) in terms of the local coordinates (Ci , Ci*), with no constant @-m F(O, 0).If we apply now the operator N to a homogeneous term of the expansion of F, F,, , which also verifies SF,, = 0, we have, on the one hand N~~,~=N(~PI'I...~~~~::~~~c~:B~,...c~~~ >++,any S-closed function with ghost number different from zero is locally S exact.This statement by no means signifies that this can occur globally.Indeed, due to the fact that the contracting homotopy u is only defined locally, there may appear global obstructions which do not allow us to patch together the results obtained in the local charts.As a consequence, a global expression for H will not exist in general.This is precisely what happens in the ordinary de Rham cohomology, for which the cohomology spaces are intimately related with the topology of the manifold.In this sense, the operators S, a, and N can be viewed as some sort of exterior derivative d, its adjoint fl and the corresponding Laplacian dd*+d*d.This result can be equivalently expressed by means of the following decomposition theorem, analogous to the Hodge decomposition in the ordinary de Rham cohomology.Given a function F( C', CT), it can be locally written as F=Fo+SA+uB, with Foeker N. To see this, consider the expansion (2.12)-and apply the operator N to it.Taking into account the relation (2.13) we have, for every F)B=uSB =+ B=uii 3 uB=O, and, as a result, F=SA, locally.