Families of gauge conditions in BV formalism

Journal of High Energy Physics, Jul 2017

In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some group H and Λ is an H-invariant family of Lagrangian submanifold then under certain conditions we construct a form on Λ that descends to a closed form on Λ/H. Integrating the latter form over a cycle in Λ/H we obtain numbers that can have interesting physical meaning. We show that one can get string amplitudes this way. Applying this construction to topological quantum field theories one obtains topological invariants.

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Families of gauge conditions in BV formalism

Revised: May Families of gauge conditions in BV formalism S~ao Paulo Brazil Davis U.S.A. Andrei Mikhailov 0 3 Albert Schwarz 2 0 On leave from Institute for Theoretical and Experimental Physics , Moscow , Russia 1 sica Teorica, Universidade Estadual Paulista 2 Department of Mathematics, University of California , USA 3 Instituto de Fi In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some BRST Quantization; Gauge Symmetry; Topological Strings - group H and is an H-invariant family of Lagrangian submanifold then under certain conditions we construct a form on that descends to a closed form on =H. Integrating the latter form over a cycle in =H we obtain numbers that can have interesting physical meaning. We show that one can get string amplitudes this way. Applying this construction to topological quantum eld theories one obtains topological invariants. 1 Introduction 4 Gauge symmetries 5 From BRST to BV 2 Families of equivalent action functionals 3 Families of Lagrangian submanifolds in BV phase space 6 Topological quantum eld theories. Bosonic strings 7 String amplitudes 7.1 String amplitudes for critical string 8 Pure spinor superstring A Some useful formulas B De nition of using marked points C Central extension of the group of canonical transformations C.1 De nition of b C.2 How to build a form on LAG starting from b family of physically equivalent action functionals. As was noticed in [1] this is wrong. The consideration of a family of equivalent action functionals or family of gauge conditions labeled by points of (super) manifold leads to a construction of a closed di erential form { 1 { (a closed pseudodi erential form if is a supermanifold). If our action functionals are invariant with respect to some group H then the form is H-invariant, but it does not necessarily descend to =H. Under some conditions we construct a closed H-equivariant form H and show that this equivariant form is homologous to a form descending to =H. This allows us to obtain interesting physical quantities integrating over cycles in =H. For example, we can start with topological quantum eld theory on some manifold . One can apply our results to the family of equivalent action functionals labeled by metrics on . We obtain topological invariants of this way; it would be interesting to calculate them and compare with known invariants. This machinery can be applied to string amplitudes. The worldsheet of bosonic string HJEP07(21)63 can be considered as two-dimensional topological quantum eld theory. Considering as a space of metrics and H as a group generated by di eomorphisms and Weyl transformations we get formulas for string amplitudes; for appropriate choice of Lagrangian submanifolds these formulas coincide with the standard ones. Similar constructions work for other types of strings. Some remarks about terminology and notations. We are saying \manifold" instead of \supermanifold", \group" instead of \supergroup", etc. We understand an element of super Lie algebra as a linear combination P ATA where TA are even or odd generators of Z2 -graded Lie algebra and A are even or odd elements of some Grassmann algebra; hence in our understanding an element of super Lie algebra is always an even object (see [ 2, 3 ] for the de nitions of supermanifold, super Lie algebra, etc. that we are using). We work in BV-formalism assuming that the BV action functionals are de ned on odd symplectic manifold M equipped with volume element (SP-manifold in terminology of [4, 5]). In this situation the odd Laplacian is de ned on the space of functions on M . It was noticed in [6] that in the absence of the volume element the odd Laplacian is de ned on semidensities; this allows the reformulation of BV-formalism for any odd symplectic manifold. In appendix C we show how to prove our main results in this more general setting. Some basic formulas of BV-formalism are listed in appendix A. The space of (smooth) functions on a supermanifold M is denoted Fun(M ).This space is Z2-graded: Fun(M ) = Fun0(M ) + Fun1(M ). Functions on T M (on the space of tangent bundle with reversed parity of bers) are called pseudodi erential forms (PDF) on M . (Di erential forms can be considered as polynomial functions on T M .) Di stands for the group of di eomorphisms, Vect for its Lie algebra (the algebra of vector elds), Weyl for the group of Weyl transformations. As we have noticed an element of any super Lie algebra (and hence a vector eld) is considered an even object. We use the term \canonical transformation" for a transformation of (odd) symplectic manifold preserving the symplectic form (another word for this notion is \symplectomorphism"). On a simply connected manifold in nitesimal canonical transformations can be characterized as Hamiltonian vector elds. Notice that in our terminology the Hamiltonian on odd symplectic manifold is an odd function B; the rst order di erential operator corresponding to the Hamiltonian vector eld with the Hamiltonian B is expressed in terms of the odd Poisson bracket as an operator transforming a function G into fB; Gg; this { 2 { operator is even (parity preserving). The condition B = 0 means that the Hamiltonian vector eld is volume preserving (= divergence free). 2 Families of equivalent action functionals Let us consider a functional S de ned on an odd symplectic manifold M with volume element and satisfying the quantum master equation eSBV = 0. (Here stands for the odd Laplacian [23, BV-formalism/OddLaplace.html].) Then the physical quantities corresponding to the BV action functional SBV can be expressed as integrals R L AeSBV where L is a Lagrangian submanifold of M and the integral is taken with respect to the volume element induced on this submanifold; A stands for quantum observable (i.e. (AeSBV ) = 0 or equivalently A + fA; SBVg = 0). These integrals depend only on the homology class of the Lagrangian submanifold. Let us consider now a family of physically equivalent BV-action functionals S ; obeying fS ; S g = 0, S = 0. We can consider S as a function on M . We assume that is simply connected; then S being physically equivalent for di erent values of is equivalent to the existence of functions Ba such that: for some Ba 2 Fun1(M ), Ba = 0 (one can describe Ba as Hamiltonians of in nitesimal volume preserving canonical transformations giving equivalence of functionals S for innitesimally close ). The eq. (2.1) implies that will assume a stronger condition: Then the following PDF on is closed: Indeed using eqs. (A.13) and (A.14) we obtain dB fB; Bg = 0 where B = d aBa ( ; d ) = exp (S + B) Z L 1 2 2 (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) Z L d ( ; d ) = fB; Sg + fB; Bg eS+B = eS+B = 0 Z L More generally, let us de ne: Then: hF i( ; d ) = Z L the set of volume preserving canonical transformations transforming xed point of .) A continuous (even di erentiable) section of this bundle not necessarily exists globally, but always exists locally. It exists globally, in particular, in the case when is contractible. Di erentiating the section U we obtain and (2.2). (This is not quite correct: the operators B^ = d aB^a obey dB^ 1=2[B^; B^] = 0, but their Hamiltonians Ba speci ed via B^a = fBa; g are de ned only up to a -dependent constant and (2.2) is true only for an appropriate choice of these constants; see appendix C . Their Hamiltonians Ba obey (2.1) 3 Families of Lagrangian submanifolds in BV phase space We will show that one can construct some interesting quantities (including string amplitudes) considering families of Lagrangian submanifolds instead of families of action Let us x a connected family of simply connected Lagrangian submanifolds. In other words we assume that L depends on parameters 1; : : : ; k; : : : (these parameters can be odd, but for simplicity we assume that they are even). Let G be the group of canonical transformations of M (transformations preserving the odd symplectic structure), and g its Lie algebra. Elements of g correspond to odd functions on M (Hamiltonians). Tentative de nition of the closed form . We want to de ne a closed pseudo di erential form on the space LAG of all simply-connected Lagrangian submanifolds: (3.1) Roughly speaking, the value of at a point v 2 T LAG is computed as follows. Notice that v corresponds to a pair (L; ) where L 2 LAG and 2 Fun(L) is an odd function on L describing the tangent vector.1 The variation of L can be described by in nitesimal 1As in Classical Mechanics, a function on Lagrangian manifold L speci es a tangent vector to LAG (an in nitesimal deformation of L). In our case the symplectic form is odd, hence the correspondence is parity reversing. These functions are called \in nitesimal gauge fermions". We have assumed that L is transformation. (Notice that the canonical transformation is not unique, but the restriction of its Hamiltonian to L is well de ned up to a constant summand.) In other words, for any vector eld v inducing a tangent vector to LAG at L we have: d = ( v!)jL : The function depends on v 2 hence it can be considered as a one-form on LAG. TL LAG (on odd tangent vector to LAG at L) linearly, By de nition: More generally, for every function F on M we de ne: As a complication, the one-form is de ned only up to a constant: (L; v) = eSBV+ hF i(L; v) = Therefore the de nition of by eq. (3.3) is strictly speaking ambiguous. We will prove that it is always possible to resolve this ambiguity in such a way, that the form is closed. Moreover, 1; : : : ; n. This means that we have a family of Lagrangian submanifolds (L( )). Let us nd a family of volume preserving canonical transformations g( ) such that: not participate in any way in the de nition of ; we will use it just to compute d . Using g( ) we can construct a family of physically equivalent action functionals S obeying Z L0 e S = eS : Z L Here S is obtained from S by means of the transformation g . It is easy to check that the form introduced in present section coincides with the form constructed in the section 2 for the family S and denoted by the same symbol; hence it is closed. (The second summand in the de nition of in section 2 is a Hamiltonian H of the in nitesimal canonical transformation governing the variation of S . The Hamiltonian governing the variation of L enters the de nition of in present section. These two Hamiltonians coincide up to a constant; resolving the ambiguity in the de nition of second Hamiltonian in appropriate way we can say that the Hamiltonians coincide.) If we know the precise de nition of we can give also a precise de nition of hF i. The formula (3.6) follows from (2.6). A more formal proof of the results of this section is given in appendix C. { 5 { Gauge symmetries Form is not necessarily base with respect to gauge symmetries. We assume that the action functional S, the observable A, the volume element on M , and the family are invariant under a subgroup H G (or Lie algebra h g).2 We denote by h^ the set of Hamiltonians of elements of h; then the h-invariance of S; A and volume element means that for every h 2 h^ we have fS; hg = 0; fA; hg = 0 and a weaker requirement: h = 0. (It is enough to impose h + fSBV; hg = 0; (4.1) HJEP07(21)63 see [7].) It follows from these assumptions that the form is also H-invariant (or hinvariant). In general the form is not horizontal, and therefore does not descend to =H [23, omega/Descent To Double Coset.html]. However, in some important cases, in particular in string theory, the form does descend to =H for appropriate choice of the family of Lagrangian submanifolds. We will now construct a modi ed form which is base. Under the assumptions of previous section, let us make the following additional assumption [23, introduction/Proof of equivariance.html]. Suppose that there exists a map : h^ ! Fun(M ) such that every Hamiltonian h 2 h^ satis es: h = fSBV; (h)g + (h) + f (h); (h)g 1 2 (Notice that the Hamiltonian h is odd, but (h) is even.) We will also require that satis es the following \equivariance" property. For any two elements h 2 h^ and h~ 2 h^: (4.2) (4.3) (4.4) fh; (h~)g = (fh; h~g) form.) the form Let us suppose that the action of h on comes from a free action of the corresponding Lie group H (this Lie group is not necessarily connected). Then we can construct [23, omega/Descent To Double Coset.html#(part. .Modi ed .P.D.F)] closed form H , which descends to =H. (In other words this is a base form, i.e. H-invariant and H-horizontal Technically, we use the formalism of equivariant cohomology [23, equivariantcohomology/index.html]. The conditions we impose on the map allow us to prove that CH ( ; d ; h) = eSBV+ + (h) Z L represents a class of H-equivariant cohomology of in the Cartan model [23, equivariantcohomology/Equivariant Cohomology.html]. (We consider here as a one-form on .) Recall that in this model an equivariant cohomology class is represented by a di erential form depending on an element of h and belonging to the kernel of Cartan di erential d h where h 2 h. (The dependence of h should agree with the action of the group H.) 2Notice, that H is not necessarily the full group of automorphisms. In string worldsheet theory, typically H is the group of di eomorphisms. { 6 { which is essentially the de nition of . commutator hence fh; hg = 0; combining this with eq. (4.3) we get: In the case when H is a conventional group the Poisson bracket corresponds to usual (this also can be derived just from eqs. (4.1) and (4.2)). From Cartan to base. If the action of H on is free the H-equivariant cohomology is isomorphic to the cohomology of =H. An explicit formula for a base form belonging to the same class of equivariant cohomology as Ch can be written as follows ( [23, equivariantcohomology/Direct Computation.html#(elem. .Def.Underline.Alpha)]). We need to choose a connection on (the cohomology class of the resulting base form will not depend on the choice of ). Then we have to replace with the horizontal projection of , and substitute the curvature f = d 2 1 2 for h (see [8] for a review): base = Z L exp SBV + ( ( ) ) + d The proof of the fact that eq. (4.4) is equivariantly closed uses (3.6) and the relation [23, omega/As Intertwiner.html] where r 2 g and R stands for the corresponding Hamiltonian. This formula immediately follows from: the ber bundle with our map in eq. (4.8). ! The second term ( ) is the horizontal projection of . The third term d should be understood as follows. Consider the curvature d 2 1 2 of the connection in =H; this is an H-equivariant h-valued 2-form on . Composing it we get a two-form with values in Fun(M ), which is denoted d The considerations above are rigorous in nite-dimensional case, however, we will use them in in nite-dimensional case where they can be justi ed in the framework of perturbation theory. Notice in the case when the dimension is in nite one should impose some additional conditions. In particular, the quadratic part of the BV action functional restricted to the Lagrangian submanifold should be non-degenerate. This condition (nondegeneracy condition) is necessary to have well de ned perturbation theory. It is not needed in nite-dimensional case when the integral has a de nition independent of the perturbation theory and the integral of degenerate functional makes sense. The situation with the completeness condition is similar: it is necessary only in in nite-dimensional case. The conditions of irreducibility and completeness are discussed in more details in [16]. The odd Laplacian is ill-de ned in the in nite-dimensional case unless we are working in the framework of perturbation theory when we can apply the methods of [9] or [10]. { 7 { However the equation S = 0 does make sense; it just means that the nilpotent vector eld Q corresponding to the rst order di erential operator transforming a function f into ff; Sg is volume preserving. (There exist standard ways to check that an operator in in nite-dimensional space is volume preserving; for example a method based on the calculation of Seeley coe cients is explained in [11].) Replacing S by exp S~ we can write the quantum master equation e S~ = 0 as fS; Sg + ~ S = 0; in in nite-dimensional case we assume that both summands vanish: fS; Sg = 0 (classical classical master equation) and S = 0. Similarly, we assume that in (4.2) = 0. In in nite-dimensional case we require that a quantum observable A satis es the equations A = 0 and fA; Sg = 0. Let us suppose that we have a functional S( ) with an odd symmetry QBRST (BRST symmetry) that is nilpotent o -shell (i.e. nilpotent without using the equations of motion). Then we can construct an odd symplectic manifold adding anti elds and solution to the classical Master Equation given by the formula SBV = S( ) + (QBRST i) i? (5.1) In the case when QBRST is volume preserving (divergence-free) SBV obeys also quantum master equation SBV = 0. This statement is rigorous in nite-dimensional situation; it remains true also in the in nite-dimensional case. A special case of this construction comes from the \standard" BRST formalism. It works for gauge theories as Yang-Mills/QCD or Chern-Simons, and also for the bosonic string worldsheet theory [23, bosonic-string/index.html] and the RNS superstring [23, Heterotic-RNS/index.html]. One starts from the \classical action" Scl('), which is invariant with respect to group H, hence with respect to its Lie algebra H with generators TA ("gauge symmetry"). Then one introduces additional variables c A (\the ghosts") with the quantum numbers of the symmetry parameter, but opposite statistics. The nilpotent symmetry Q is de ned by the following formulas [23, BRSTformalism/Construction.html]: QBRST'i = T AicA ; QBRSTcA = 12 fBAC c c B C (5.2) where fBAC are structure constants of the Lie algebra H. To continue from BRST to BV [23, BRST-formalism/BV from BRST.html], we de ne an odd symplectic manifold adding to 'i; cA their anti elds 'i ; cA having opposite parity (geometrically this means that we consider cotangent bundle with reversed parity of bers). Here 'i is the collective notation for the \old elds". In such a situation, a solution of the classical Master Equation (a special case of (5.1)) can be written in the form: SBV = Scl(') + Our goal will be to solve the eq. (4.2) for BV action functional (5.3). Notice that this action functional is invariant with respect to the action of the group H and its Lie algebra H; the hamiltonian of the element = ATA 2 H has the form h = T i A'i? + [ ; c]Ac? . A A There exists a solution mapping this Hamiltonian into (h) = Ac?A; it satis es the conditions f (h); (h)g = To check (4.2) it is su cient to notice that f SBV; A ? cA g = TA i A'i? + [ ; c]Ac? A [ ; c]Ac? ; ~Ac?Ag = fBAC A B ~C c?A = [ ; ~]Ac? . A A solution of (4.2) should obey (4.3). To verify this condition we notice that fT Ai A'i? + Physically meaningful quantities should be obtained from integrating eSBV over a suitably chosen Lagrangian submanifold. However, a suitable choice of Lagrangian submanifold is not completely trivial. The most straightforward guess would be to put all anti elds to zero, i.e. '? = c ? = 0. But this is a wrong choice, because the restriction of SBV to such a Lagrangian submanifold would be just Scl, a degenerate functional. The next guess would be to consider some deformation of '? = c ? = 0, such that the restriction of SBV be nonzero. But this also does not quite work, for the following reason: we want to keep the ghost number symmetry. It turns out that in the class of Lagrangian submanifolds respecting the ghost number symmetry, the one given by '? = c? = 0 is typically rigid. It has no deformations, because there are no elds with negative ghost number. In fact there are several solutions to this problem. One solution is to consider a family [23, BRST-formalism/Family of Lagrangian submanifolds.html] of Lagrangian submanifolds. Each individual member of a family may be not ghost-number-invariant, but we require that the family as a whole be closed under the action of the ghost number symmetry. Another solution is to add a eld/anti eld quartet, in other words non-minimal elds, with some non-minimal elds carrying negative ghost number. (In nite-dimensional case this gives the same answer [23, omega/As integration over single L.html] as considering a family.) Yet another approach is to start with a Lagrangian submanifold [23, BRSTformalism/Family of Lagrangian submanifolds.html] where some anti elds and some elds are zero. Schematically: '?1 = '2 = c1? = c2 = 0 (5.5) In the particular case of bosonic string worldsheet theory in section 6, where H is the group of di eomorphisms, we will use this third method. Comment about anti elds. If is a scalar eld, we will consider ? a density (i.e. a volume form, or an area form in the two-dimensional case). This is very natural: (5.4) The odd symplectic form is given by the integral of the density ( 1) ! = R ( 1) ^ ? ^ ?, i.e. 7! A local in nitesimal eld rede nition + "V ( ) is generated by the odd Hamiltonian R V ( ) ? (in order for this integral to make sense, ? should be a density). In the same sense, we actually think of the \variational derivative" as a density; it is \generated by" ? in terms of odd Poisson bracket. { 9 { Topological quantum In BRST formalism a topological quantum eld theory is de ned by a family of action functionals depending on riemannian metric on some manifold X and satisfying the condition that the variation of the action functional by in nitesimal variation of the metric is BRST exact (topological quantum eld theories of Witten type). In BV formalism we should have solutions to the master equation fS; Sg = 0 depending on riemannian metric and obeying dS = fb; Sg where d is the de Rham di erential on the space MET of all metrics and b is a 1-form on this space. (If V is a vector eld on the space of metrics we can write dS=dV = fb(V ); Sg.) Alternatively we can assume that the solution to the master equation is xed, but the Lagrangian submanifold depends on the choice of metric. We can construct an n-form n on MET integrating b(V1) : : : b(Vn)eS over some Lagrangian submanifold L in the space of elds. Summing the forms n we can get an inhomogeneous closed form that can be obtained by integrating eS+b over L. Under certain conditions (see section 2) one can prove that this form is closed and descends to the quotient space of MET with respect to the action of the group Di of di eomorphisms of X. We obtain a closed form on the quotient MET=Di ; integrating this form over a cycle we can get new invariants. In particular, applying these ideas to Chern-Simons theory one obtains invariants constructed by Kontsevich [12]; see [1] for detail. (Another construction of these invariants was given in [13].) In the rest of this section we will outline applications of these ideas to string perturbation theory. The target of string theory can be regarded as two-dimensional topological quantum eld theory; the above considerations can be applied to this TQFT. We will show that string amplitudes are particular cases of new invariants we have mentioned. Instead of formalism of families of equivalent action functionals we will use more exible formalism of families of Lagrangian submanifolds. Bosonic string. Master action in terms of world sheet metric. The construction outlined in section 5 works for both bosonic string [23, bosonic-string/index.html] and RNS superstring [23, Heterotic-RNS/index.html]. Let us consider bosonic string. For de niteness we are writing formulas for bosonic string in at space. (To avoid anomalies we should assume that we work in the dimension 26.) We start with the action functional Smat[g; x] = 2 1 Z pgg (6.1) We integrate here over a compact surface of genus h with metric g . We always assume that h > 1. The subindex mat stands for \matter", although this action also involves the dynamical metric g . This functional is invariant with respect to di eomorphisms and Weyl transformations g0 = e g ; hence we can construct a BV action functional introducing di eomorphism ghosts c, Weyl ghosts and anti elds to g , xm and ghosts.3 3BV formalism was previously applied to bosonic string in [14]. HJEP07(21)63 SBV = Smat[g; x] + + Z Following the general scheme outlined in section 5 we obtain: We now choose the Lagrangian submanifold in the following way: g = g(0) ; x? = c? = ? = 0 where g(0) is a xed metric The family (6.3) of Lagrangian submanifolds is closed under the action of di eomorphisms. On Lagrangian submanifold (6.3) the action is quadratic and the form is equal to:4 (g(0); g(0)) = [Dx Dg? Dc D ] exp SBV + [Dx Dg? Dc D ] exp Smat + [Dx Db Dc ] exp Smat + [Dx Dg? Dc D ] d2z(L g )g? [Dx Dg? Dc D ] d2z exp SBV + g(0) g? = 0 exp SBV + Z Z g(0) g? = (Lcg) g ? + g g ? + ((c @ )xm)x?m + [c; c] c? + (Lc ) ? 1 2 (6.2) (6.3) (6.4) (6.5) (6.6) (6.7) (6.8) (6.9) HJEP07(21)63 We introduced the notation t = g g ? ; b = traceless part of g? ; i.e. g? = b + 12 tg . In the transition to the last line we integrated over and t. Non-degeneracy. The exponential in (6.5) is non-degenerate. (The restriction of SBV to the Lagrangian submanifold of eq. (6.3) is non-degenerate [23, BRSTformalism/Family of Lagrangian submanifolds.html#%28part. .Non-degeneracy%29] modulo a nite-dimensional space of zero modes of b . This nite-dimensional degeneracy is removed by the second term in the exponential of (6.5).) Symmetries. The form is invariant with respect to di eomorphisms; moreover on the family (6.3) it is a base form, because for any worldsheet vector eld : To check that the last line is zero we notice that the derivative with respect to c under the sign of two-dimensional integral can be replaced be variational derivative under the sign of in nite-dimensional integral. Let us study the behavior of this form with respect to Weyl transformations g0 = e g . The 0-th component 0 of inhomogeneous form can be regarded as a partition function of conformal eld theory. The variation of partition function by in nitesimal Weyl 4We denote the de Rham di erential on the in nite-dimensional space of metrics by instead of d. transformation is governed by trace anomaly Z= = ( c1R2 + const)Z where c stands for the central charge and R denotes the curvature of the worldsheet. In our case the central charge vanishes (we are working in critical dimension d = 26; in general the central charge is equal to d 26). We see that 0 does not change by Weyl transformations. The kth component of the form can be expressed in terms of correlation functions of the same conformal theory. The behavior of correlation functions by Weyl transformations is governed by conformal dimensions i of elds i ([22], formula (13,50)). To check the Weyl invariance of we notice that the dimension of b is 2 (it coincides with conformal dimension) and the dimension of g is is Weyl invariant. Moreover, it descends not only to MET=Di , but also to MET=Di o Weyl; that can be identi ed with the moduli space of complex structures on a compact surface of genus h. (A formal proof of the fact that is a base form for the Weyl group repeats the proof of similar statement for Di .) We can get the partition function of bosonic string integrating the form over this moduli space. (Notice that we are working with inhomogeneous forms, but the integration singles out one component of this form.) We can solve eq. (4.2) using the general considerations [23, omega/Case Of Standard BRST.html] of section 5. Namely, we should take a map sending a worldsheet vector eld (z; z) plus in nitesimal Weyl transformation '(z; z) to: ( ; ') = Z c? + ' ? Then the functional fS; ( ; ')g can be considered as a Hamiltonian of in nitesimal transformation of elds corresponding to the vector eld and Weyl factor '. This means eq. (6.11) de nes a solution of eq. (4.2) for the Lie algebra of the group Di o Weyl acting on the space of elds. This allows us to construct an equivariant form Z gL C Lie(Di oWeyl)( ; ') = exp SBV + + c? + ' ? Z (6.11) (6.12) We can then construct the corresponding base form [23, omega/Base Form.html] which descends to =(Di o Weyl). On the standard family of Lagrangian submanifolds given by eq. (6.3) c? = ? = 0. Therefore C Lie(Di oWeyl)( ; ') becomes essentially of eq. (6.5). Singular metrics. Notice that in the action functional (6.1) we can allow slightly singular metrics. We say that the worldsheet metric on a surface of genus h is slightly singular if on some real curves one of the eigenvalues of the metric g vanishes and another eigenvalue remains positive. More precisely we suppose that g = det g vanishes on a family of closed real curves and in the neighborhood of one of these curves it takes the form where = 0 is the equation of the curve and is a positive function.5 It is easy to check 2 5The simplest example of this picture is a cylinder with coordinates ( ; ) and metric ds2 = d 2 + 2d 2. Here a < < a; 0 that under these conditions the action functional (6.1) is nite if we make an additional assumption that xm is constant on every closed curve where the metric is singular. The formulas for BV action (6.2) and Lagrangian submanifold (6.3) can be applied to slightly singular metrics. We obtain a family of Lagrangian manifolds labelled by these metrics. Factorizing the topological space of slightly singular metrics with respect to di eomorphisms and Weyl transformations we obtain the space =Di o Weyl. Points of this space can be identi ed with complex curves having simplest singularities (nodes). (Every closed curve where the metric is singular should be contracted to a point; the metric speci es a complex structure in the complement to these points.) A part of this space that consists of stable curves (curves having only nite number of automorphisms) can be identi ed with Deligne-Mumford compacti cation of the moduli space of algebraic curves of genus h. This is a good topological space (an orbifold). The remaining part is a \bad" (non-separable) space, but it does not play any role (a heuristic explanation of this fact is the remark that its contribution to the partition function is suppressed by the in nite volume of the automorphism group). The form of eq. (6.5) descends to Deligne-Mumford space as a form having some singularities. To obtain physical quantities we should integrate the form over a cycle in Deligne-Mumford space; to obtain the partition function we should integrate over the fundamental cycle. (Of course, this is only a formal calculation-due to the tachyon in the spectrum of bosonic string the integral is divergent.) Master equation in terms of complex structures. A worldsheet complex structure can be speci ed by a eld of linear operators I acting on tangent spaces and obeying I2 = 1. Another way to specify a complex structure is to x a complex vector eld e such that the complex conjugate vector eld e together with e speci es a basis of complexi ed tangent space. (To relate these descriptions we de ne e as the eigenvector of I having eigenvalue i.) Notice that e is only de ned up to multiplication: e ue, where u is a Due to Weyl invariance one can express the functional (6.1) in terms of complex struccomplex function on the worldsheet. tures. We obtain the following functional: Z Smat[I; x] = (6.13) where the measure on the worldsheet is speci ed by the condition the vectors e; e span a parallelogram of measure 1 in tangent space. The functional is invariant with respect to di eomorphisms. We can now follow the standard procedure by rst introducing the di eomorphism ghosts [23, BRST-formalism/Construction.html] c (BRST formalism) and then adding anti elds [23, BRST-formalism/BV from BRST.html]. The result is the Master Action of the following form: Z SBV = Smat[I; x] + (LcI) I ? + (Lcx)x? + [c; c] c? (6.14) 1 2 In the expression for the action we integrate over a worldsheet. In the h-loop contribution the worldsheet is a surface of genus h. Notice that one can introduce a notion of slightly degenerate complex structure assuming that the vectors e and e can be linearly dependent on a family of closed curves on a worldsheet. (In a neighborhood of such a curve we should have a relation e = e + f + : : : where tangent vectors e and f are linearly independent, = 0 is the equation of the curve and : : : are higher order terms with respect to .) String amplitudes for critical string To represent the string theory in BV form we have applied the general constructions of the HJEP07(21)63 section 5 to the action functional Smat[g; x]. This functional depends on the metric g on the worldsheet (on a compact surface of genus h) and a map x( ) = xm( ) of this surface to Rd. This functional is invariant with respect to di eomorphisms and Weyl transformations. We applied the standard BRST construction in this setting and used (5.1) to get the BV action. To describe string amplitudes we should add marked points (punctures) ( 1; : : : ; n) on the worldsheet to this picture. Following [14] we will consider i as dynamical variables on equal footing with the metric. extra term c ( i) i? : Using again the constructions of the section 5 we get the new BV action SB0V with an SB0V = SBV + c ( i) i? where SBV is de ned by (6.14). As was noticed in section 5 this functional obeys quantum master equation in the case when the volume is Q-invariant; this remark forces us to use the di eomorphism invariant measure pg( 1)d2 1 pg( n)d2 n on the space of marked points. Let us consider functionals Vi( i) (vertices) which are invariant under di eomorphisms. The typical examples of such vertices are tachyoinic vertex eipx( ) and graviton vertex klg SB00V = SB0V + X iVi( i) where i are in nitesimally small. To de ne string amplitudes it is convenient to work with BV-action functional that is obtained from (7.2) by means of \integrating out" Weyl ghosts.6 We obtain the new BV action S~BV given by the formula eS~BV = eSmat[g;x] +R (Lcg) g? +((c @ )xm)x?m+ 21 [c;c] c? c ( i) i?+P iVi( i) (g? g ) Denoting the traceless part of g? by b we can represent this action functional in the form S~BV = S^BV + X iVi( i) 6If a solution A of the equation A = 0 is de ned on direct product of two odd symplectic manifolds Y0 and Y00 we can obtain a solution of similar equation on Y0 integrating over Lagrangian submanifolds L 2 Y00. (See for example [10].) In our case we integrate over Lagrangian submanifold = 0 of manifold with coordinates ; . (7.1) (7.2) (7.3) where taking S^BV = Smat[g; x] + Z (Lcg) b c ( i) i? (7.4) 1 2 Now we can use the standard construction of the form starting with the action functional S~BV. However, we prefer to construct the form starting with the functional S^BV and including the factor V1 : : : Vn into de ning integral. (The form coming from the second construction can be obtained from the rst one by means of di erentiation with respect to parameters.) We consider a family of Lagrangian submanifolds parameterized by g(0) ; i(0) HJEP07(21)63 g = g(0) ; i = i(0) ; x? = c? = 0 The form , restricted to one of these Lagrangian submanifolds looks as follows: (g(0) ; i(0) g(0) ; d i(0) ) = = Z [Dx Db D q i Dc] g( 1(0))V1( 1(0)) q g( n(0))Vn( n(0)) Z exp Smat + Using this formula we can get an expression of in terms of correlation functions of conformal eld theory. This allows us to analyze the behavior of with respect to Weyl transformations. It is easy to see that in our case of critical string this form is Weyl invariant if conformal elds corresponding to vertices Vi have conformal dimension 2 (dimension (1; 1) in the language of complex geometry). In this case the form descends to the moduli space Mh;n of compact complex curves of genus h with n marked points and to its DeligneMumford compacti cation Mh;n. Integrating over the fundamental cycle of Mh;n we obtain the h-loop contribution to string amplitudes. To check this we notice that after integration over d ? (and omitting indices (0) for brevity) we get: Z [Dx Db Dc] j g( j )( d j1 + c1( j ))( d j2 + c2( j ))Vj ( j ) Z q exp Smat + This result is equivalent to the standard expression for the string amplitude [15]. To see this we notice that j (d j1 +c1( j ))(d j2 +c2( j )) consist on 2n summands; one of them gives the standard expression for string amplitudes with non-integrated vertices, another gives the standard expression with integrated vertices, and the rest correspond to the situation when some vertices are integrated and some are non-integrated. All these summands are equal, hence we obtain the standard answer up to a factor 2n. Another way to calculate the string amplitudes is to work with in nitesimal deformations of BV action functional. Such deformations can be identi ed with (classical or (7.5) (7.6) (7.7) (7.8) (7.9) quantum) observables. In string theory they can be considered as integrated vertices. Applying our approach to the deformation of BV action we obtain the standard expression of string amplitudes in terms of integrated vertices (see [16] for detail). An important method of calculation of scattering amplitudes in string theory is based on the consideration of o -shell string amplitudes. This is the best method to calcu late amplitudes when the mass gets quantum corrections.The o -shell amplitudes should be de ned in such a way that the particle masses correspond to their poles (in momentum representation) and scattering amplitudes should be expressed in terms of residues in these poles. To de ne o -shell string amplitudes for critical string one can consider surfaces with marked points and local coordinate systems in the neighborhoods of these points [17], [18]. This is equivalent to consideration of surfaces with boundary. The BV formalism on manifolds with boundary was analyzed in [10]. It should be possible to combine our approach [23, boundary/index.html] with BV-BFV formalism of [10]; these would lead to generalization of de nitions given in [17, 18]. For non-critical strings very nice de nition of o -shell amplitudes was suggested by A. Polyakov [19]; it works well in our setting. Polyakov considers maps x( ) = xm( ) of a surface with marked points 1; : : : ; k into Rd and includes the factor i Z (xi x( i))pg( i)d2 i in the functional integral that de nes the partition function. Geometrically this means that we integrate over all surfaces in Rd that contain the points x1; : : : ; xk 2 Rd(surfaces with pinned points fxig in Polyakov's terminology). Doing the functional integral we obtain a function G(x1; : : : ; xk) that can be interpreted as o -shell amplitude in coordinate representation. The o - shell amplitude in the momentum representation G(p1; : : : ; pk) can be de ned as Fourier transform of G(x1; : : : ; xk) or directly as a functional integral for partition function with insertion j Z eipjx( j)q g( j )d2 j (7.10) (7.11) Polyakov considers o -shell amplitudes only at tree level (genus zero surfaces), however they can be considered also in multi-loop case. 8 Pure spinor superstring We hope that our ideas will lead to better understanding of pure spinor formalism in superstring theory and to simpli ed expressions for amplitudes in this formalism. The worldsheet sigma-model of the pure spinor sigma-model has di erent versions which are quasiisomorphic to each other, as usual in the topological eld theory. There is a \minimal version", which (in case of Type II theory7) describes matter elds (x; L; R) and \ghost elds" L; R constrained to live on the pure spinor cone: ( L m L) = ( R m R) = 0 (8.1) 7For the heterotic string the right-moving variables are those of the heterotic RNS formalism. The at space sigma-model [24] requires introduction of the momenta pL+ and pR conjugate to L and R, and the fermionic part of the action is of the rst order in derivatives: The action for pure spinors is, schematically: Z Z where the \conjugate momenta" w+L; wR take values in the cotangent bundle of the pure HJEP07(21)63 The model is invariant under a fermionic nilpotent symmetry Q. Importantly, it splits (for Type II case) into the sum of left and right symmetries: Q = QL + QR such that the conserved currents corresponding to QL and QR are holomorphic and antiholomorphic, respectively. In the case of at target space, it is easy to obtain the corresponding BV action functional: for every eld one should add its anti eld and a term in the action having the form (Q ) see (5.1).) pure spinor string is: However, the solution of eq. (4.2) requires di erent methods. As a rst step, let us restrict ourselves to the left sector.9 The explicit form of eq. (4.2) for the left sector of the . (This is a special case of general construction described in Sec 5; (8.2) (8.3) (8.4) (8.5) (8.6) 1 2 fSBV ; a( )g + fa( ); a( )g = Hh i + ( z@z L) ?L + (L w+)w?+ | this has to be solved for the unknown a( ); notice that Hh i is linear in , but a( ) does not have to be linear in . (We have assumed that a( ) = 0; otherwise we should add an ill-de ned term a( ).) One solution can be obtained as follows. Since the worldsheet theory is conformal, a holomorphic vector eld + is symmetry; it is generated by +T++. It was shown by Berkovits that the energy-momentum tensor is BRST-trivial: T++ = QLb++ (even o -shell) where b++ is a composite b-ghost. This means that one should expect that the worldsheet action can be included into topological conformal eld theory. A rigorous proof of this statement is still unknown; the most convincing treatment of this problem was given in [20].10 Notice that +b++ is a holomorphic current and therefore also 8One way of describing the pure spinor system is to cover the cone with patches. On each patch, both L and wL take values in at 11-dimensional space. 9Applying this to the \full" sigma-model, i.e. left plus right sector, is work in progress in collaboration with R. Lipinski Jusinskas. 10That paper contains also the calculation of superstring amplitudes in the framework of BV-formalism; some ideas of this calculation can be used in our approach. corresponds to some symmetry. We can identify a( ) = h i, a BV Hamiltonian generating the in nitesimal action of that symmetry. Then the second term in (8.5) vanishes and this equation is satis ed. However we hope that there exist simpler solutions of eq. (8.5) with non-vanishing second term; we leave this question for future work. We believe, that applying the techniques described above one can not only justify the pure spinor formalism, but also simplify the formulas (hopefully we can avoid using the complicated and not very well de ned \composite b-ghost"). A Some useful formulas BV phase space [23, BV-formalism/index.html] is an odd symplectic supermanifold M with a nondegenerate closed odd 2-form !. For any F 2 Fun(M ) we can de ne its Hamiltonian vector eld. We will think of this vector eld as a rst order linear di erential operator, acting on Fun(M ): HJEP07(21)63 and denote this operator fF; g. (Here nition: AB(Z) is a matrix inverse to !AB(Z).) By de where is the operator of contraction, satisfying [ V ; d] = LV . This implies: In coordinates: (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) G 7! fF; Gg = F AB(Z) dF = ( )F +1 fF; g! fF; Gg = fF; gdG = ( )G+1 fF; g fG; g! ! = dZAdZB!AB !AB = ( 1)(A+1)(B+1)!BA V = V A F = divfF; g Locally it is possible to choose the Darboux coordinates: 0 If the manifold M is equipped with a volume element (with a density) we can de ne the odd Laplacian acing on functions by the formula In Darboux coordinates is: where div stands for the divergence of vector eld with respect to the volume element. The volume element should be chosen in such a way that 2 = 0. The relation between odd Laplace operator [23, BV-formalism/OddLaplace.html] and f ; g is: (XY ) =( X)Y + ( ) X Y + ( )X fX; Y g e = + f ; g e 1 2 One can prove that X given by this formula does not depend on the choice of Darboux coordinates if X transforms as a semidensity (recall that semi-densities transform as square roots of densities= volume elements). Hence for any odd symplectic manifold one can de ne on semi-densities (volume element is not necessary), see [6]. B De nition of using marked points Let LAG+ denote the space of Lagrangian submanifolds with marked points. A point of LAG+ is a pair (L; a) where L 2 LAG and a 2 L. This de nes the double bration: M p LAG+ ! LAG Given v 2 T(L;a)LAG+, we can consider two projections v 2 TLLAG and p v 2 TaM . We will de ne is a pseudo-di erential form, i.e. a function of L; a; v. It will depend on v only through v. We can characterize v as a section of T M jL modulo T L. We then de ne as follows: 2 Fun(L) d = (a) = 0 ( v!)jL This de nition speci es as a linear function of v, i.e. as a one-form on LAG+ In order to make sense of v! we must think of v as a section of T M ; the fact that it is only de ned up to tangent to T L does not matter because L is isotropic. Eq. (B.3) eliminates the ambiguity, and we can now safely de ne a function on T LAG+ (a pseudodi erential form on LAG+) as in eq. (3.3): More generally, for every function F on M we de ne: (L; a; v) = eSBV+ hF i(L; a; v) = We will now prove the following formula: (d p !) hF i = h F + fSBV ; F gi (A.12) (A.13) (A.14) As a straightforward generalization, we can consider a product of with the pullback under p of any di erential or pseudo-di erential form on M . It satis es: [23, omega/Descent To LAGs.html#(part. .Upgrade to )] d (p hF i) = ( )j j+1p h F + fSBV ; F gi + p (d + ! ) hF i (B.7) Notice the appearance of the nilpotent operator d + ! which was studied in [21]. Proof. We take a family of Lagrangian submanifolds with marked points (L( ); a( )) and represent it in the form where g( ) are volume preserving canonical transformations (locally this is always possible). It is su cient to analyze the restriction hF i( ; d ) of the form (B.5) to this family. As in section 3 using the canonical transformations g( ) we can construct a family of action functionals S These forms do not coincide with the forms and corresponding forms that will be denoted by ~ and ~ hF i. hF i( ; d ) constructed by means of family of Lagrangian submanifolds with marked points, but they are closely related. As we noticed in section 3 the second summand in the exponential in the formula de ning ~ hF i( ; d ) is the Hamiltonian of the in nitesimal canonical transformation governing the variation of S . The second summand in the formula de ning hF i( ; d ) is the Hamiltonian H( ; d ) of the in nitesimal canonical transformation11 governing the variation of L . They coincide up to a constant summand. This constant can be calculated from (B.2). We obtain [23, omega/Descent To LAGs.html#(part. .Upgrade to )] hF i( ; d ) = C ~ hF i( ; d ) where C = e H( ;d )(g( )a0). (One can say that C is expressed in terms of the value of the Hamiltonian of the in nitesimal canonical transformation at the marked point.) We have calculated already the di erential of ~ hF i( ; d ). But we also have to evaluate d of the prefactor C. Using eq. (A.3), appendix, and p ! = 12 ! we get: 1 2 1 2 1 2 1 2 d e H( ;d )(ga0) = = e H( ;d )(ga0) ( (d H( ; d ))(ga0) fH( ; d ) ; H( ; d )g(ga0)) = = e H( ;d )(ga0) fH( ; d ) ; H( ; d )g(ga0) = = e H( ;d )(ga0)(( fH( ;d ) ; g)2!)(ga0) = = e H( ;d )(ga0)( { 20 { (B.10) (B.11) (B.12) (B.13) (B.14) (B.15) (B.16) + = (p e ) Z L e C+ = (p ) eS+ + (h) 2 Fun (( T M ) h) d + ! fh; g + h = 0 d fh; g C+ = 0 fh; g = h h(a) Proof of C+ being equivariantly closed. We have to prove that: where d is the de Rham di erential on LAG+. The action of d is given by eq. (B.7). The is essentially as in eq. (4.6), but we have to remember to subtract the action of fh; g on The vanishing of (d computation. compensating constant to make sure that vanishes at the marked point; therefore: fh; g) C+ when eqs. (4.2) and (B.20) are satis ed follows from direct This concludes the proof. follows: Given a \symplectic potential" satisfying d = ! we can construct a closed form as We will choose the following ansatz for the equivariantly closed analogue of : The expression de ned in eq. (B.18) is a cocycle of the Cartan complex of equivariant cohomology of LAG+ if in addition to (4.2) we have Even though lives in the same space as cochains of the Cartan complex, the di erential de ned by eq. (B.20) is di erent. (The Cartan di erential would be d fh; g.) Comment. In particular, when we can choose an H-invariant \symplectic potential" such that d = !, eq. (B.20) has a simple solution: where is of the same formal type as a Cartan cochain: Central extension of the group of canonical transformations In this section we will give a precise de nition of using a well-de ned closed PDF b on a central extension Gb of the group of canonical transformations.12 This group is in nitedimensional, however, in this section we will keep the notation d for the de Rham di erential on the group and on the space of Lagrangian submanifolds LAG. 12The existence of a central extension of the group of canonical transformations (symplectomorphisms) of odd symplectic manifold M can be proven in the same way as for an even symplectic manifold. Namely, as in the even case one constructs a bundle with connection over M , the ber of this bundle is a one-dimensional odd vector space. The group Gb can be de ned as a group of transformations of the total space of the bundle that are compatible with the bration (transform bers into bers), induce canonical transformation on the base and are compatible with connection. Z L = e (B.17) (B.18) (B.19) (B.20) (B.21) (B.22) (B.23) the fact that the Lie algebra of Gb is equation: Here following [6] we consider exp(SBV) as a semidensity, dgg 1 is the right-invariant form on Gb taking values in the Lie algebra (Maurer-Cartan form),and g stands for an element of G corresponding to gb 2 Gb. In eq. (C.3) we consider dgbgb 1 as a function on M , using bb Fun(M ). This form satis es the Maurer-Cartan d(dgbgb 1) + 2 fdgbgb 1 ; dgbgb 1g = 0 This b is closed as a PDF on Gb, i.e.: b 2 Fun(LAG T Gb) b(L; gb; dgb) = exp SBV + dgbgb 1 Z gL 1 db = 0 The proof of eq. (C.5) is a straightforward computation [23, omega/De nition.html] very similar to the computations in section 2. (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) De nition of b Let us consider the Lie superalgebra Fun(M ) with the commutator given by the odd Poisson bracket. It is a central extension of the Lie superalgebra of Hamiltonian vector elds which we denote g; therefore we denote it gb: gb = Fun(M ) We consider the central extension of the group of canonical transformations Gb, whose Lie algebra is gb. closed PDFs on Gb, which we will call b: As a variation on our theme, we will now construct a map from LAG to the space of We must stress that this b is well-de ned (does not contain any ambiguities). C.2 How to build a form on LAG starting from b Since G (and therefore Gb) acts on LAG, there is a natural projection: b : LAG T Gb ! T LAG restriction of on L0 is a constant c. Using gbbgb 1 = b g 1 we get: However, it is not true that b is constant along the bers of b. Indeed, for a where St(L0) stands for the stable subgroup of L0 2 LAG in Gb one can check that the 2 Lie(St(L0)), b(L0; gb; dgb + gbb) = k b(L0; gb; dgb) where k is some number. Therefore b does not automatically provide a PDF on LAG. would guarantee that k = 0. g( ) 2 G; a closed form 2 We could impose some additional restrictions, such as ghost number symmetry,13 which Let us suppose now that a subset of LAG is represented in the form g( )L0 where . Assume that we can nd a \lift" g( ) of g( ) to Gb. Then we can de ne (L; dL) = b(L0; gb( ); d(gb( ))) (C.9) This coincides with the \tentative" de nition of section 3, because the restriction of dg^g^ 1 to gL0 gives . This is a general fact, true both in classical mechanics and in BV formalism. In classical mechanics it is essentially the Hamilton-Jacobi equation, which describes the evolution of a Lagrangian submanifold (speci ed by a generating function cohomology class. Acknowledgments usually called S) under the Hamiltonian on L plus a constant (which can depend on t). Notice that by the variation of gb( ) the form (L; dL) obviously remains in the same We are grateful to Nathan Berkovits, Alberto Cattaneo, Alexei Kotov, Misha Movshev, John Murray, Pavel Mnev, Sasha Polyakov and Kostas Skenderis for useful discussions. The work of A.M. was partially supported by the FAPESP grant 2014/18634-9 \Dualidade Gravitac; a~o/Teoria de Gauge", and in part by the RFBR grant 15-01-99504 \String theory and integrable systems". Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Phys. 158 (1993) 373 [hep-th/9210115] [INSPIRE]. Batalin-Vilkovisky Geometry, math/9909117. [1] A. 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Nelson , Covariant insertion of general vertex operators , Phys. Rev. Lett . 62 ( 1989 ) 993 [18] A. Sen , O -shell Amplitudes in Superstring Theory, Fortsch. Phys . 63 ( 2015 ) 149 [19] A.M. Polyakov , Contemporary Concepts in Physics . Vol. 3 : Gauge Fields and Strings, CRC [7] A.S. Schwarz , Symmetry transformations in Batalin-Vilkovisky formalism , Lett. Math. Phys. [8] E. Meinrenken , Equivariant cohomology and the cartan model , [10] A.S. Cattaneo , P. Mnev and N. Reshetikhin , Perturbative quantum gauge theories on [11] A.S. Schwarz , The Partition Function of a Degenerate Functional , Commun. Math. Phys. 67 [14] B. Craps and K. Skenderis , Comments on BRST quantization of strings , JHEP 05 ( 2005 ) [15] J. Polchinski , String Theory, Cambridge University Press, Cambridge U.K. ( 2005 ).

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Andrei Mikhailov, Albert Schwarz. Families of gauge conditions in BV formalism, Journal of High Energy Physics, 2017, 63, DOI: 10.1007/JHEP07(2017)063