Families of gauge conditions in BV formalism

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 $\Lambda$ is an $H$-invariant family of Lagrangian submanifold then under certain conditions we construct a form on $\Lambda$ that descends to a closed form on $\Lambda/H.$ Integrating the latter form over a cycle in $\Lambda/H$ 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.


Introduction
A physical theory can be represented by various equivalent action functionals. For example, in the case of degenerate action functionals we can impose different gauge conditions. In BRST-formalism infinitesimal Q-exact variation of action functional leads to equivalent action functional. In BV-formalism the role of choice of gauge condition is played by the choice of Lagrangian submanifold.
As an example one can consider topological quantum field theories of Witten type, where the action functional in BRST-formalism depends on metric, but the variation of this functional by an infinitesimal change of the metric (the energy-momentum tensor) is Q-exact.
The first impression is that it is sufficient to consider only one functional from a 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 Λ leads to a construction of a closed differential form Ω on Λ (a closed pseudodifferential form if Λ is a supermanifold). If our action functionals are invariant with respect to some group H then the form Ω is H-invariant, but it does not necessarily descend to Λ/H. Under some conditions we construct a closed H-equivariant form Ω H and show that this equivariant form is homologous to a form descending to Ω/H. This allows us to obtain interesting physical quantities integrating over cycles in Λ/H. For example, we can start with topological quantum field theory on some manifold Σ. One can apply our results to the family of equivalent action functionals labeled by metrics on Σ. We obtain topological invariants of Σ 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 can be considered as two-dimensional topological quantum field theory. Considering Λ as a space of metrics and H as a group generated by diffeomorphisms 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 ǫ A T A where T A are even or odd generators of Z 2 -graded Lie algebra and ǫ 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 definitions of supermanifold , super Lie algebra, etc. that we are using).
We work in BV-formalism assuming that the BV action functionals are defined on odd symplectic manifold M equipped with volume element (SP-manifold in terminology of [4,5]) . In this situation the odd Laplacian ∆ is defined on the space of functions on M . It was noticed in [6] that in the absence of the volume element the odd Laplacian is defined 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 Z 2 -graded: Fun(M ) = Fun0(M ) + Fun1(M ). Functions on ΠT M (on the space of tangent bundle with reversed parity of fibers) are called pseudodifferential forms (PDF) on M. (Differential forms can be considered as polynomial functions on ΠT M .) Diff stands for the group of diffeomorphisms, Vect for its Lie algebra (the algebra of vector fields), Weyl for the group of Weyl transformations. As we have noticed an element of any super Lie algebra (and hence a vector field) 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 infinitesimal canonical transformations can be characterized as Hamiltonian vector fields. Notice that in our terminology the Hamiltonian on odd symplectic manifold is an odd function B; the first order differential operator corresponding to the Hamiltonian vector field with the Hamiltonian B is expressed in terms of the odd Poisson bracket as an operator transforming a function G into {B, G}; this operator is even (parity preserving). The condition ∆B = 0 means that the Hamiltonian vector field is volume preserving (= divergence free).

Families of equivalent action functionals
Let us consider a functional S defined on an odd symplectic manifold M with volume element and satisfying the quantum master equation ∆e S BV = 0. (Here ∆ stands for the odd Laplacian.) Then the physical quantities corresponding to the BV action functional S BV can be expressed as integrals L Ae S BV 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. ∆(Ae S BV ) = 0 or equivalently ∆A + {A, S BV } = 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 λ , λ ∈ Λ obeying {S λ , S λ } = 0, ∆S λ = 0. We can consider S as a function on Λ × M . We assume that Λ is simply connected; then S λ being physically equivalent for different values of λ is equivalent to the existence of functions B a such that: for some B a ∈ Fun 1 (M ), ∆B a = 0 (one can describe B a as Hamiltonians of infinitesimal volume preserving canonical transformations giving equivalence of functionals S λ for infinitesimally close λ). The Eq. (1) implies that ∂Ba We will assume a stronger condition : Then the following PDF on Λ is closed: Indeed using Eqs. (73) and (74) we obtain More generally, let us define: Then: Eq. (6) follows from the following chain of equalities: and L ∆(. . .) = 0. Notice that Eq. (1) does not define B a unambiguously; there is a freedom to add to B a a function {S, A a } where ∆A a = 0. One can use this freedom to obtain B a satisfying (1) and (2). This is not always possible globally, but always possible locally (in small pieces of the parameter space Λ). To check this we consider a fiber bundle over Λ having as a fiber over a point λ ∈ Λ the set of volume preserving canonical transformations transforming S λ 0 in S λ . (Here λ 0 is a fixed point of Λ.) A continuous (even differentiable) section of this bundle not necessarily exists globally, but always exists locally. It exists globally, in particular, in the case when Λ is contractible. Differentiating the section U λ we obtain infinitesimal canonical transformationsB a = ∂U ∂λ a U −1 . Their Hamiltonians B a obey (1) and (2). (This is not quite correct: the operatorsB = dλ aB a obey dB − 1/2[B,B] = 0, but their Hamiltonians B a specified viaB a = {B a , } are defined only up to a λ-dependent constant and (2) is true only for an appropriate choice of these constants; see Appendix C for details.)

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 functionals.
Let us fix a connected family Λ of simply connected Lagrangian submanifolds. In other words we assume that L depends on parameters λ 1 , ..., λ 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 definition of the closed form Ω We want to define a closed pseudodifferential form Ω on the space LAG of all simply-connected Lagrangian submanifolds: Roughly speaking, the value of Ω at a point v ∈ Π T LAG is computed as follows. Notice that v corresponds to a pair (L, σ) where L ∈ LAG and σ ∈ Fun(L) is an odd function on L describing the tangent vector 1 The variation of L can be described by infinitesimal canonical transformation; one can say that σ is a restriction to L of the Hamiltonian of this transformation. (Notice that the canonical transformation is not unique, but the restriction of its Hamiltonian to L is well defined up to a constant summand.) In other words, for any vector field v inducing a tangent vector to LAG at L we have: The function σ depends on v ∈ Π T L LAG (on odd tangent vector to LAG at L) linearly, hence it can be considered as a one-form on LAG. By definition: More generally, for every function F on M we define: As a complication, the one-form σ is defined only up to a constant: Therefore the definition of Ω by Eq. (13) is strictly speaking ambiguous. We will prove that it is always possible to resolve this ambiguity in such a way, that the form Ω is closed.
It is enough to prove this formula for restriction to any finite-dimensional submanifold Λ ⊂ LAG (i.e. a family of Lagrangian submanifolds). Let us parameterize Λ by coordinates λ 1 , . . . , λ n . This means that we have a family of Lagrangian submanifolds (L(λ)). Let us find a family of volume preserving canonical transformations g(λ) such that: (locally this is always possible). The introduction of such g(λ) is essentially a trick. It does not participate in any way in the definition of Ω; we will use it just to compute dΩ. Using g(λ) we can construct a family of physically equivalent action functionals S λ obeying 1 As in Classical Mechanics, a function on Lagrangian manifold L specifies a tangent vector to LAG (an infinitesimal deformation of L). In our case the symplectic form is odd, hence the correspondence is parity reversing . These functions are called "infinitesimal gauge fermions". We have assumed that L is simply-connected, in this case the map of functions to infinitesimal deformations is surjective and its kernel consists of constant functions .
Here S λ is obtained from S by means of the transformation g λ . It is easy to check that the form Ω introduced in present Section coincides with the form constructed in the Section 2 for the family S λ and denoted by the same symbol; hence it is closed. (The second summand in the definition of Ω in section 2 is a Hamiltonian H of the infinitesimal canonical transformation governing the variation of S λ . The Hamiltonian governing the variation of L λ enters the definition of Ω in present section. These two Hamiltonians coincide up to a constant; resolving the ambiguity in the definition of second Hamiltonian in appropriate way we can say that the Hamiltonians coincide.) If we know the precise definition of Ω we can give also a precise definition of Ω F . The formula (16) follows from (6).
A more formal proof of the results of this section is given in Appendix C.

Gauge symmetries
Form Ω 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 Λ are invariant under a subgroup H ⊂ G ( or Lie algebra h ⊂ g). 2 We denote byĥ the set of Hamiltonians of elements of h; then the h-invariance of S, A and volume element means that for every h ∈ĥ we have {S, h} = 0, {A, h} = 0 and ∆h = 0. (It is enough to impose a weaker requirement: see [7].) It follows from these assumptions that the form Ω is also H-invariant (or hinvariant). In general the form Ω is not horizontal, and therefore does not descend to Λ/H. However, in some important cases, in particular in string theory, the form Ω does descend to Λ/H for appropriate choice of the family of Lagrangian submanifolds.
We will now construct a modified form Ω which is base. Under the assumptions of previous section , let us make the following additional assumption. Suppose that there exists a map Φ :ĥ → Fun(M ) such that every Hamiltonian h ∈ĥ satisfies: (Notice that the Hamiltonian h is odd, but Φ(h) is even.) We will also require that Φ satisfies the following "equivariance" property. For any two elements h ∈ĥ andh ∈ĥ: Let us suppose that the action of h on Λ comes from a free action of the corresponding Lie group H (this Lie group is not necessarily connected). Then we can construct closed form Ω H , which descends to Λ/H. (In other words this is a base form, i.e. H-invariant and H-horizontal form.) Technically, we use the formalism of equivariant cohomology. The conditions we impose on the map Φ allow us to prove that the form represents a class of H-equivariant cohomology of Λ in the Cartan model. (We consider here σ as a one-form on Λ.) Recall that in this model an equivariant cohomology class is represented by a differential form depending on an element of h and belonging to the kernel of Cartan differential d − ι h where h ∈ h. (The dependence of h should agree with the action of the group H.) We modify the definition allowing pseudodifferential forms instead of differential forms. We do not impose the condition of polynomial dependence of h. The proof of the fact that Eq. (21) is equivariantly closed uses (16) and the relation where r ∈ g and R stands for the corresponding Hamiltonian. This formula immediately follows from: which is essentially the definition of σ.
In the case when H is a conventional group the Poisson bracket corresponds to usual commutator hence {h, h} = 0; combining this with Eq. (20) we get: (this also can be derived just from Eqs. (18) and (19)).

From Cartan to base
If the action of H on Λ is free the H-equivariant cohomology is isomorphic to the cohomology of Λ/H. An explicit formula for a base form belonging to the same class of equivariant cohomology as Ω C h can be written as follows. We need to choose a connection θ on Λ (the cohomology class of the resulting base form will not depend on the choice of θ). Then we have to replace σ with the horizontal projection of σ, and substitute the curvature f = dθ − 1 2 θ 2 for h (see [8] for a review): The second term σ − ι(θ)σ is the horizontal projection of θ. The third term dθ − 1 2 θ 2 Φ should be understood as follows. Consider the curvature dθ − 1 2 θ 2 of the connection in the fiber bundle Λ → Λ/H; this is an H-equivariant h-valued 2-form on Λ. Composing it with our map Φ we get a two-form with values in Fun(M ), which is denoted dθ − 1 2 θ 2 Φ in Eq. (25).
The considerations above are rigorous in finite-dimensional case, however, we will use them in infinite-dimensional case where they can be justified in the framework of perturbation theory. Notice in the case when the dimension is infinite 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 defined perturbation theory. It is not needed in finite-dimensional case when the integral has a definition 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 infinite-dimensional case.
The odd Laplacian ∆ is ill-defined in the infinite-dimensional case unless we are working in the framework of perturbation theory when we can apply the methods of [9] or [10]. However the equation ∆S = 0 does make sense; it just means that the nilpotent vector field Q corresponding to the first order differential operator transforming a function f into {f, S} is volume preserving. (There exist standard ways to check that an operator in infinite-dimensional space is volume preserving; for example a method based on the calculation of Seeley coefficients is explained in [11]

From BRST to BV
Let us suppose that we have a functional S(ψ) with an odd symmetry Q BRST (BRST symmetry) that is nilpotent off-shell (i.e. nilpotent without using the equations of motion). Then we can construct an odd symplectic manifold adding antifields ψ * and solution to the classical Master Equation given by the formula In the case when Q BRST is volume preserving (divergence-free) S BV obeys also quantum master equation ∆S BV = 0. This statement is rigorous in finite-dimensional situation; it remains true also in the infinite-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 th and the RNS superstring .
One starts from the "classical action" S cl (ϕ), which is invariant with respect to group H, hence with respect to its Lie algebra H with generators T A ("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 defined by the following formulas: where f A BC are structure constants of the Lie algebra H. To continue from BRST to BV, we define an odd symplectic manifold adding to ϕ i , c A their antifields ϕ * i , c * A having opposite parity (geometrically this means that we consider cotangent bundle with reversed parity of fibers). Here ϕ i is the collective notation for the "old fields". In such a situation, a solution of the classical Master Equation (a special case of (26)) can be written in the form: Our goal will be to solve the Eq. (19) for BV action functional (28). 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

There exists a solution mapping this Hamiltonian into
To check (19) it is sufficient to notice that A solution of (19) should obey (20). To verify this condition we notice that In Section 6 we will illustrate these calculations in the particular case of bosonic string worldsheet theory, where H is the group of diffeomorphisms.
Comment about antifields If φ is a scalar field, we will consider φ ⋆ a density (i.e. a volume form, or an area form in the two-dimensional case). This is very natural: • The odd symplectic form is given by the integral of the density is generated by the odd Hamiltonian V (φ)φ ⋆ (in order for this integral to make sense, φ ⋆ should be a density).
In the same sense, we actually think of the "variational derivative" δ δφ as a density; it is "generated by" φ ⋆ in terms of odd Poisson bracket.

Topological quantum field theories. Bosonic strings
In BRST formalism a topological quantum field theory is defined 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 infinitesimal variation of the metric is BRST exact (topological quantum field theories of Witten type). In BV formalism we should have solutions to the master equation {S, S} = 0 depending on riemannian metric and obeying dS = {b, S} where d is the de Rham differential on the space MET of all metrics and b is a 1-form on this space. (If V is a vector field on the space of metrics we can write dS/dV = {b(V ), S}.) Alternatively we can assume that the solution to the master equation is fixed, but the Lagrangian submanifold depends on the choice of metric.
We can construct an n-form Ω n on MET integrating b(V 1 )...b(V n )e S over some Lagrangian submanifold L in the space of fields. Summing the forms Ω n we can get an inhomogeneous closed form Ω that can be obtained by integrating e S+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 Diff of diffeomorphisms of X. We obtain a closed form on the quotient MET/Diff; 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 field 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 flexible 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 and RNS superstring.
Let us consider bosonic string. For definiteness we are writing formulas for bosonic string in flat space. (To avoid anomalies we should assume that we work in the dimension 26.) We start with the action functional We integrate here over a compact surface of genus h with metric g αβ . We always assume that h > 1. The subindex mat stands for "matter", although this action also involves the dynamical metric g αβ . This functional is invariant with respect to diffeomorphisms and Weyl transformations g ′ αβ = e φ g αβ ; hence we can construct a BV action functional introducing diffeomorphism ghosts c, Weyl ghosts ζ and antifields to g αβ , x m and ghosts. 3 Following the general scheme outlined in Section 5 we obtain: Here L c is the Lie derivative along the vector field c α ∂ α . We now choose the Lagrangian submanifold in the following way: αβ is a fixed metric The family (32) of Lagrangian submanifolds is closed under the action of diffeomorphisms. On Lagrangian submanifold (32) the action is quadratic and the form Ω is equal to: 4 We introduced the notation t = g αβ g ⋆αβ , b αβ = traceless part of g ⋆αβ , i.e. g ⋆αβ = b αβ + 1 2 tg αβ . In the transition to the last line we integrated over ζ and t.
Non-degeneracy The exponential in (34) is non-degenerate. (The restriction of S BV to the Lagrangian submanifold of Eq. (32) is non-degenerate modulo a finite-dimensional space of zero modes of b αβ . This finite-dimensional degeneracy is removed by the second term in the exponential of (34).) Symmetries The form Ω is invariant with respect to diffeomorphisms; moreover on the family (32) it is a base form, because for any worldsheet vector field ξ: To check that the last line is zero we notice that the derivative with respect to c α under the sign of two-dimensional integral can be replaced be variational derivative under the sign of infinite-dimensional integral. Let us study the behavior of this form with respect to Weyl transformations g ′ αβ = e φ g αβ . The 0-th component Ω 0 of inhomogeneous form Ω can be regarded as a partition function of conformal field theory. The variation of partition function by infinitesimal Weyl transformation is governed by trace anomaly δZ/δφ = (− cR 12 + 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 Ω 0 does not change by Weyl transformations. The kth component of the form Ω 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 fields Ψ i : ( [22], formula (13,50)). To check the Weyl invariance of Ω we notice that the dimension of b αβ is 2 (it coincides with conformal dimension) and the dimension of g αβ is −2.
We have proved that in critical theory Ω is Weyl invariant. Moreover, it descends not only to MET/Diff, but also to MET/Diff ⋊ Weyl, that can be identified with the moduli space of complex structures on a compact surface of genus h. (A formal proof of the fact that Ω is a base form for the Weyl group repeats the proof of similar statement for Diff.) 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. (19) using the general considerations of Sec 5. Namely, we should take a map sending a worldsheet vector field ξ α (z,z) plus infinitesimal Weyl transformation ϕ(z,z) to: Then the functional {S, Φ(ξ, ϕ)} can be considered as a Hamiltonian of infinitesimal transformation of fields corresponding to the vector field ξ and Weyl factor ϕ. This means Eq.
(40) defines a solution of Eq. (19) for the Lie algebra of the group Diff ⋊ Weyl acting on the space of fields. This allows us to construct an equivariant form We can then construct the corresponding base form which descends to Λ/(Diff ⋊Weyl).
Therefore Ω C Lie(Diff⋊Weyl) (ξ, ϕ) becomes essentially Ω of Eq. (34) Singular metrics. Notice that in the action functional (30) 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 αβ vanishes and another eigenvalue remains positive. More precisely we suppose that g = det g αβ vanishes on a family of closed real curves and in the neighborhood of one of these curves it takes the form ρ 2 σ where ρ = 0 is the equation of the curve and σ is a positive function. 5 It is easy to check that under these conditions the action functional (30) is finite if we make an additional assumption that x m is constant on every closed curve where the metric is singular. The formulas for BV action (31) and Lagrangian submanifold (32) can be applied to slightly singular metrics. We obtain a family of Lagrangian manifolds labelled by these metrics. Factorizing the topological space Λ of slightly singular metrics with respect to diffeomorphisms and Weyl transformations we obtain the space Λ/Diff ⋊ Weyl. Points of this space can be identified with complex curves having simplest singularities (nodes). (Every closed curve where the metric is singular should be contracted to a point; the metric specifies a complex structure in the complement to these points.) A part of this space that consists of stable curves (curves having only finite number of automorphisms) can be identified with Deligne-Mumford compactification 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 infinite volume of the automorphism group). The form of Eq. (34) 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 specified by a field of linear operators I acting on tangent spaces and obeying I 2 = −1. Another way to specify a complex structure is to fix a complex vector field e such that the complex conjugate vector fieldē together with e specifies a basis of complexified tangent space. (To relate these descriptions we define e as the eigenvector of I having eigenvalue i.) Notice that e is only defined up to multiplication: e ∼ ue, where u is a complex function on the worldsheet.
Due to Weyl invariance one can express the functional (30) in terms of complex structures. We obtain the following functional: where the measure µ on the worldsheet is specified by the condition the vectors e,ē span a parallelogram of measure 1 in tangent space. The functional is invariant with respect to dif- 5 The simplest example of this picture is a cylinder with coordinates (ρ, φ) and metric ds 2 = dρ 2 +ρ 2 dφ 2 . Here −a < ρ < a, 0 ≤ φ < 2π.
feomorphisms. We can now follow the standard procedure by first introducing the diffeomorphism ghosts c (BRST formalism) and then adding antifields. The result is the Master Action of the following form: 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 andē can be linearly dependent on a family of closed curves on a worldsheet. (In a neighborhood of such a curve we should have a relationē = λe + ρf + ... where tangent vectors e and f are linearly independent, ρ = 0 is the equation of the curve and ... are higher order terms with respect to ρ.) 7 String amplitudes 7.1 String amplitudes for critical string To represent the string theory in BV form we have applied the general constructions of the Section 5 to the action functional S mat [g, x]. This functional depends on the metric g αβ on the worldsheet (on a compact surface of genus h) and a map x(ξ) = x m (ξ) of this surface to R d . This functional is invariant with respect to diffeomorphisms and Weyl transformations. We applied the standard BRST construction in this setting and used (26) to get the BV action. To describe string amplitudes we should add marked points (punctures) (ξ 1 , ..., ξ n ) on the worldsheet to this picture. Following [14] we will consider ξ i as dynamical variables on equal footing with the metric. Using again the constructions of the section 5 we get the new BV action S ′ BV with an extra term c α (ξ i )ξ i⋆ α : where S BV is defined by (43). 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 diffeomorphism invariant measure g(ξ 1 )d 2 ξ 1 · · · g(ξ n )d 2 ξ n on the space of marked points. Let us consider functionals V i (ξ i ) (vertices) which are invariant under diffeomorphisms. The typical examples of such vertices are tachyoinic vertex e ipx(ξ) and graviton vertex ǫ kl g αβ ∂ α x k (ξ)∂ β x l (ξ)e ipx(ξ) . We can introduce a new action functional where ǫ i are infinitesimally small.
To define string amplitudes it is convenient to work with BV-action functional that is obtained from (45) by means of "integrating out" Weyl ghosts. 6 We obtain the new BV actionS BV given by the formula Denoting the traceless part of g ⋆αβ by b αβ we can represent this action functional in the formS Now we can use the standard construction of the form Ω starting with the action functional S BV . However, we prefer to construct the form Ω starting with the functionalŜ BV and including the factor V 1 ...V n into defining integral. (The form coming from the second construction can be obtained from the first one by means of differentiation with respect to parameters.) We consider a family of Lagrangian submanifolds parameterized by g The form Ω , restricted to one of these Lagrangian submanifolds looks as follows: Using this formula we can get an expression of Ω in terms of correlation functions of conformal field theory. This allows us to analyze the behavior of Ω with respect to Weyl transformations. It is easy to see that in our case of critical string this form is Weyl invariant if conformal fields corresponding to vertices V i have conformal dimension 2 (dimension (1,1) in the language of complex geometry). In this case the form descends to the moduli space 6 If a solution A of the equation ∆A = 0 is defined on direct product of two odd symplectic manifolds Y ′ and Y ′′ we can obtain a solution of similar equation on Y ′ integrating over Lagrangian submanifolds L ∈ Y ′′ . (See for example [10].) In our case we integrate over Lagrangian submanifold ζ * = 0 of manifold with coordinates ζ, ζ * .
M h,n of compact complex curves of genus h with n marked points and to its Deligne-Mumford compactificationM h,n . Integrating over the fundamental cycle ofM h,n we obtain the h-loop contribution to string amplitudes. To check this we notice that after integration over dξ ⋆ (and omitting indices (0) for brevity) we get: This result is equivalent to the standard expression for the string amplitude [15]. To see this we notice that Π j (dξ 1 j +c 1 (ξ j ))(dξ 2 j +c 2 (ξ j )) consist on 2 n 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 2 n .
Another way to calculate the string amplitudes is to work with infinitesimal deformations of BV action functional. Such deformations can be identified with (classical or 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 off-shell string amplitudes. This is the best method to calculate amplitudes when the mass gets quantum corrections.The off-shell amplitudes should be defined 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 define off-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 with BV-BFV formalism of [10]; these would lead to generalization of definitions given in [17], [18].
For non-critical strings very nice definition of off-shell amplitudes was suggested by A. Polyakov [19]; it works well in our setting. Polyakov considers maps x(ξ) = x m (ξ) of a surface with marked points ξ 1 , ..., ξ k into R d and includes the factor in the functional integral that defines the partition function. Geometrically this means that we integrate over all surfaces in R d that contain the points x 1 , ..., x k ∈ R d (surfaces with pinned points {x i } in Polyakov's terminology). Doing the functional integral we obtain a function G(x 1 , ..., x k ) that can be interpreted as off-shell amplitude in coordinate representation. The off-shell amplitude in the momentum representation G(p 1 , ..., p k ) can be defined as Fourier transform of G(x 1 , ..., x k ) or directly as a functional integral for partition function with insertion Polyakov considers off-shell amplitudes only at tree level (genus zero surfaces), however they can be considered also in multi-loop case.

Pure spinor superstring
We hope that our ideas will lead to better understanding of pure spinor formalism in superstring theory and to simplified expressions for amplitudes in this formalism.
The worldsheet sigma-model of the pure spinor sigma-model has different versions which are quasiisomorphic to each other, as usual in the topological field theory. There is a "minimal version", which (in case of Type II theory 7 ) describes matter fields (x, θ L , θ R ) and "ghost fields" λ L , λ R constrained to live on the pure spinor cone: The flat space sigma-model requires introduction of the momenta p L + and p R − conjugate to θ L and θ R , and the fermionic part of the action is of the first order in derivatives: The action for pure spinors is, schematically: where the "conjugate momenta" w L + , w R − take values in the cotangent bundle of the pure spinor cone. The bosonic part of the action is the usual 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: such that the conserved currents corresponding to Q L and Q R are holomorphic and antiholomorphic, respectively.
In the case of flat target space, it is easy to obtain the corresponding BV action functional: for every field Φ one should add its antifield Φ * and a term in the action having the form (QΦ) · Φ * . (This is a special case of general construction described in Sec 5; see (26).) However, the solution of Eq. (19 ) requires different methods. As a first step, let us restrict ourselves to the left sector 8 . The explicit form of Eq. (19) for the left sector of the pure spinor string is: -this has to be solved for the unknown a(ξ); notice that H ξ is linear in ξ, but a(ξ) does not have to be linear in ξ. (We have assumed that ∆a(ξ) = 0 ; otherwise we should add an ill-defined term ∆a(ξ).) One solution can be obtained as follows. Since the worldsheet theory is conformal, a holomorphic vector field ξ + is a symmetry; it is generated by ξ + T ++ . It was shown by Berkovits that the energy-momentum tensor is BRST-trivial: T ++ = Q L b ++ (even off-shell) where b ++ is a composite b-ghost. This means that one should expect that the worldsheet action can be included into topological conformal field theory. A rigorous proof of this statement is still unknown; the most convincing treatment of this problem was given in [20] 9 Notice that ξ + b ++ is a holomorphic current and therefore also corresponds to some symmetry. We can identify a(ξ) = Φ ξ , a BV Hamiltonian generating the infinitesimal action of that symmetry. Then the second term in (59) vanishes and this equation is satisfied. However we hope that there exist simpler solutions of Eq. (59) 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 defined "composite b-ghost").

A Some useful formulas
BV phase space is an odd symplectic supermanifold M with a nondegenerate closed odd 2-form ω. For any F ∈ Fun(M ) we can define its Hamiltonian vector field. We will think of this vector field as a first order linear differential operator, acting on Fun(M ): and denote this operator {F, }. (Here π AB (Z) is a matrix inverse to ω AB (Z).) By definition: where ι is the operator of contraction, satisfying [ι V , d] = L V . This implies: In coordinates: Locally it is possible to choose the Darboux coordinates: If the manifold M is equipped with a volume element (with a density) we can define the odd Laplacian acing on functions by the formula where div stands for the divergence of vector field 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 and { , } is: In Darboux coordinates ∆ is: 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 define ∆ on semi-densities (volume element is not necessary), see [6].

B Definition of Ω 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 ∈ LAG and a ∈ L. This defines the double fibration: Given v ∈ ΠT (L,a) LAG + , we can consider two projections π * v ∈ ΠT L LAG and p * v ∈ ΠT a M . We will define Ω is a pseudo-differential form, i.e. a function of L, a, v. It will depend on v only through π * v. We can characterize π * v as a section of ΠT M | L modulo ΠT L. We then define σ as follows: This definition specifies σ as a linear function of v , i.e. as a one-form on LAG + In order to make sense of ι π * v ω we must think of π * v as a section of ΠT M ; the fact that it is only defined up to tangent to T L does not matter because L is isotropic. Eq. (77) eliminates the ambiguity, and we can now safely define a function Ω on ΠT LAG + (a pseudodifferential form on LAG + ) as in Eq. (13): More generally, for every function F on M we define: We will now prove the following formula: Comment As a straightforward generalization, we can consider a product of Ω with the pullback under p of any differential or pseudo-differential form ν on M . It satisfies: Notice the appearance of the nilpotent operator d + ω which was studied in [21].
Proof We take a family of Lagrangian submanifolds with marked points (L(λ), a(λ)) and represent it in the form where g(λ) are volume preserving canonical transformations (locally this is always possible). It is sufficient to analyze the restriction Ω F (λ, dλ) of the form (79) to this family. As in Section 3 using the canonical transformations g(λ) we can construct a family of action functionals S λ and corresponding forms that will be denoted byΩ andΩ F . These forms do not coincide with the forms Ω F (λ, dλ) 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 definingΩ F (λ, dλ) is the Hamiltonian of the infinitesimal canonical transformation governing the variation of S λ . The second summand in the formula defining Ω F (λ, dλ) is the Hamiltonian H(λ, dλ) of the infinitesimal canonical transformation 10 governing the variation of L λ . They coincide up to a constant summand. This constant can be calculated from (76). We obtain where C = e −H(λ,dλ)(g(λ)a 0 ) . (One can say that C is expressed in terms of the value of the Hamiltonian of the infinitesimal canonical transformation at the marked point.) We have calculated already the differential ofΩ F (λ, dλ). But we also have to evaluate This concludes the proof.
Given a "symplectic potential" α satisfying dα = ω we can construct a closed form as follows: We will choose the following ansatz for the equivariantly closed analogue of Ω: where ν is of the same formal type as a Cartan cochain: The expression defined in Eq. (92) is a cocycle of the Cartan complex of equivariant cohomology of LAG + if in addition to (19) we have Even though ν lives in the same space as cochains of the Cartan complex, the differential defined by Eq. (94) is different. (The Cartan differential would be d − ι {h, } .) Comment In particular, when we can choose an H-invariant "symplectic potential" α such that dα = ω, Eq. (94) has a simple solution: Proof of Ω C + being equivariantly closed We have to prove that: where d is the de Rham differential on LAG + . The action of d is given by Eq. (81). The action of ι {h, } on σ is essentially as in Eq. (23), but we have to remember to subtract the compensating constant to make sure that σ vanishes at the marked point; therefore: The vanishing of (d − ι {h, } )Ω C + when Eqs. (19) and (94) are satisfied follows from direct computation.

C Central extension of the group of canonical transformations
In this Section we will give a precise definition of Ω using a well-defined closed PDF Ω on a central extension G of the group of canonical transformations. 11 This group is infinitedimensional, however, in this section we will keep the notation d for the de Rham differential on the group and on the space of Lagrangian submanifolds LAG.

C.1 Definition of Ω
Let us consider the Lie superalgebra ΠFun(M ) with the commutator given by the odd Poisson bracket. It is a central extension of the Lie superalgebra of Hamiltonian vector fields which we denote g; therefore we denote it g: We consider the central extension of the group of canonical transformations G, whose Lie algebra is g.
As a variation on our theme, we will now construct a map from LAG to the space of closed PDFs on G, which we will call Ω: Ω(L, g, d g) = gL exp S BV + d g g −1 (100) Here following [6] we consider exp(S BV ) as a semidensity, d g g −1 is the right-invariant form on G taking values in the Lie algebra (Maurer-Cartan form),and g stands for an element of G corresponding to g ∈ G. In Eq. (100) we consider d g g −1 as a function on M , using the fact that the Lie algebra of G is ΠFun(M ). This form satisfies the Maurer-Cartan equation: d(d g g −1 ) + 1 2 {d g g −1 , d g g −1 } = 0 This Ω is closed as a PDF on G, i.e.: d Ω = 0 (102) The proof of Eq. (102) is a straightforward computation very similar to the computations in Section 2.
We must stress that this Ω is well-defined (does not contain any ambiguities).
C.2 How to build a form on LAG starting from Ω Since G (and therefore G) acts on LAG, there is a natural projection: However, it is not true that Ω is constant along the fibers of π. Indeed, for a ξ ∈ Lie(St(L 0 )), where St(L 0 ) stands for the stable subgroup of L 0 ∈ LAG in G one can check that the restriction of ξ on L 0 is a constant c. Using g ξ g −1 = ξ • g −1 we get: Ω(L 0 , g, d g + g ξ) = k Ω(L 0 , g, d g) where k is some number. Therefore Ω does not automatically provide a PDF on LAG. We could impose some additional restrictions, such as ghost number symmetry 12 , which would guarantee that k = 0.
Let us suppose now that a subset of LAG is represented in the form g(λ)L 0 where g(λ) ∈ G, λ ∈ Λ. Assume that we can find a "lift" g(λ) of g(λ) to G. Then we can define a closed form Ω(L, dL) = Ω(L 0 , g(λ), d( g(λ))) This coincides with the "tentative" definition of Section 3, because the restriction of dĝĝ −1 to gL 0 gives σ. 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 (specified by a generating function usually called S) under the Hamiltonian flow. It says that ∂S ∂t equals the restriction of H on L plus a constant (which can depend on t).
Notice that by the variation of g(λ) the form Ω(L, dL) obviously remains in the same cohomology class.