Integration over families of Lagrangian submanifolds in BV formalism

Gauge fixing is interpreted in BV formalism as a choice of Lagrangian submanifold in an odd symplectic manifold. A natural construction defines an integration procedure on families of Lagrangian submanifolds. In string perturbation theory, the moduli space integrals of higher genus amplitudes can be interpreted in this way. We discuss the role of gauge symmetries in this construction. We derive the conditions which should be imposed on gauge symmetries for the consistency of our integration procedure. We explain how these conditions behave under the deformations of the worldsheet theory. In particular, we show that integrated vertex operator is actually an inhomogeneous differential form on the space of Lagrangian submanifolds.

1 Introduction Many physical theories have BRST-like structure. This means that there is a nilpotent fermionic symmetry Q, and the space of physical states is the cohomology of Q. Such theories come with an equivalence relation. The theory with the action S is equivalent to the theory with the action S + QΨ for any operator Ψ. Physically meaningful quantities should be invariant, i.e. should be the same for any two equivalent theories. An important question is, how can we obtain such invariants? One way to obtain an invariant is to take the path integral: where O j are some BRST-closed operators. But actually there are other possibilities [1,2], which we will now describe.
Consider the whole equivalence class of theories. It is infinite-dimensional, because S S + QΨ where Ψ could be more or less arbitrary functional.
Let us choose a basis {Ψ a }; the space of BRST trivial deformations is parametrized by the coordinates x a : Let us call this equivalence class M. It turns out that the following pseudodifferential form on M is closed: We want to obtain BRST invariants by integrating Ω over some closed cycles [1].
This construction (and related) was also used in [3,4], and on manifolds with a boundary in [4,5].
This procedure was used in [1] to define string amplitudes 1 . We have to integrate over the moduli space of metrics on a genus g Riemann surface.
When we vary the metric, the variation of the worldsheet action is Q-exact: In this case Ψ a dx a = b αβ dg αβ and Eq. (3) gives the standard string measure: 1 BV formalism was applied to string worldsheet theory in [6] where φ are the fields of the worldsheet sigma-model. In this approach, the choice of a metric g αβ should be understood as a choice of a representative in the class of physically equivalent theories (a choice of gauge fixing).
We would like to integrate over the moduli space of metrics modulo diffeomorphisms. However, the form Ω given by Eq. (5) is not automatically base. Although it is true that it is diffeomorphism-invariant, it is generally speaking not horizontal and therefore not a base form. Indeed, changing dg αβ to dg αβ + 2∇ (α ξ β) results in the same expression for Ω only when: Even in the case of standard bosonic string, this is only true on-shell, leading to restrictions on allowed operator insertions. For a general string theory sigma-model (such as pures spinor formalism in a curved background [7]) we would like to relax the condition (6) by allowing ∇ α b αβ to be Q of something. Moreover, for the worldsheet theories of the topological type, such as the pure spinor formalism [7], a deformation of the metric is not necessarily better than any other Q-exact deformation of the action. We want to extend the definition of Ω to the space of all BRST-trivial deformations. But with all these generalizations, we still want Ω to be a base form with respect to the worldsheet diffeomorphisms. A construction of base Ω was suggested in [2]. The purpose of this paper is to further develop this construction, and to fill in some technical details.
It appears that the right interpretation of Ω is a pseudo-differential form on the space of Lagrangian submanifolds in BV phase space, which we denote LAG. We will give the definition in Section 4. The space LAG does not have LAG topologically nontrivial closed integration cycles. One can integrate over a non-compact cycle, extending to infinity, if Ω is rapidly decreasing. But this is not the case in string theory. Therefore we have to consider the factorspace of LAG over the action of the group of worldsheet diffeomorphisms. In our approach, the action of the group of diffeomorphisms on the BV phase space should come as part of the definition of the string worldsheet theory. In more general applications (beyond string theory) we may use, in place of diffeomorphisms, some other symmetry group. We develop the formalism for general symmetry group. For our construction of base Ω to work, the action of the symmetry group on the BV phase space should satisfy certain properties, which we describe in Section 6. One special case is when BV formalism comes from BRST formalism; we discuss this in Section 7. Another special case is topologically twisted N = 2 superconformal theory (Section 10).
In string theory, one considers the worldsheet theory along with its nontrivial deformations. These deformations are called "integrated vertex operators" 2 . In Section 8 we study the corresponding deformation of the string measure. We show that this deformation will generally speaking involve mixing between the components of Ω of different degrees. A closely related concept is unintegrated vertex operators; we will consider them in Section 12. In Section 11 we discuss the case of worldsheet with boundary. This could be useful for the off-shell formulation of string theory [8].

Brief review of the BV formalism
The "BV phase space" is a supermanifold M with a non-degenerate odd 2form ω satisfying dω = 0. It is not true that ω −1 defines an antisymmetric bivector. (The symmetry properties of ω −1 are not of a bivector, see Appendix A.2.) Instead, it defines a second order odd differential operator ∆ on the so-called "half-densities" [9]. A half-density is a geometrical object which transforms under the change of variables as a square root of the volume element. A half-density defines a measure on every Lagrangian submanifold L ⊂ M .

Canonical transformations
Let M denote the BV phase space. We say that a vector field is Hamiltonian if it preserves the odd symplectic form. Let us assume that M is simplyconnected. Then Hamiltonian vector fields are of the form {H, }. Let g denote the Lie algebra of Hamiltonian vector fields. The commutator is g defined as the standard commutator of vector fields: Let G denote the Lie supergroup of canonical transformations of M . Its G Lie superalgebra is opposite to the Lie superalgebra of Hamiltonian vector fields 3 : Let C ∞ M denote the space of smooth functions on M . Notice that ΠC ∞ M is a Lie superalgebra under the Poisson bracket. It is a central extension of g: Let G denote the corresponding central extension of G. It can be realized G as the group of automorphisms of some contact manifold -see Appendix 4.6.1.

Canonical
where µ L [ρ 1 2 ] is some integral form on L (which of course depends on ρ 1 2 ). There exists some half-density on M , which we will denote ∆ can ρ 1 2 , such that: ∆ can In other words: Theorem 1: given a half-density ρ 1 2 , there exists another half-density ∆ can ρ 1 2 , such that for any H ∈ Fun(M ) and any Lagrangian L ⊂ M : Eq. (12) is the definition of ∆ can . We will give a proof in Appendix B.
3 If we consider the space of vector fields as a Lie algebra of the group of diffeomorphisms, then the usual definition of the Lie algebra bracket has opposite sign to the standard commutator of vector fields

Relation between ∆ can and Lie derivative
Let us fix two functions F ∈ Fun(M ) and Ψ ∈ Fun(M ). Let us suppose that Ψ is odd. Then: For any Lagrangian submanifold L ⊂ M , let us consider: Consider the case when the restriction of Ψ to L is zero. Then Eq. (14) implies that the restriction of L {Ψ, } ρ 1 2 on such L is equal to −∆ can (Ψρ 1 2 ). We will use F as a "test function" and assume that F has compact support, contained in a sufficiently small open superdomain U ⊂ M .
The submanifold U 0 ⊂ U given by the equation Ψ = 0 contains sufficiently many Lagrangian submanifolds, in the following sense: if the restriction of a density on any Lagrangian submanifold contained in U 0 is zero, then the density is zero everywhere 4 on U 0 . Indeed, when U is small enough, we can consider the space of trajectories of {Ψ, } on U 0 . It is an odd symplectic manifold (the odd analogue of the Hamiltonian reduction). It has sufficiently many Lagrangian submanifolds, in the above sense. They lift to Lagrangian submanifolds in U . Therefore Eq. (14) implies that on U 0 : ). To extend this formula from U 0 to the whole U , let us consider the superdomainÛ = R 0|1 × U ; the fermionic coordinate of R 0|1 will be denoted ζ. Consider the subspace ofÛ 0 ⊂Û given by the equation ζ − Ψ(x, θ) = 0. It has sufficiently many maximally isotropic submanifolds. Then the same computation as in Eq. (14) gives: by definition does not depend on ζ. Therefore X = −∆ can ρ 1 2 . This implies, for odd Ψ: If instead of odd Ψ we consider some even H, then this argument does not work, because when {H, H} = 0 there are no Lagrangian submanifolds contained in level sets of H. But, given some odd Ψ and a constant Grassmann parameter ε, we can apply the argument to the odd Hamiltonian Ψ + εH. Considering the coefficient of ε proves that for even H: The formula which works for both even and odd H is:

The canonical operator is nilpotent
Indeed, since the definition of ∆ can is geometrically natural, it automatically commutes with canonical transformations and therefore for any H ∈ Fun(M ): Comparing this with Eq. (17) we conclude that ∆ 2 can commutes with multiplication by an arbitrary function. Therefore ∆ 2 can is multiplication by a function. Morever, since ∆ 2 can commutes with any canonical transformation, this function should be a constant. Taking into account that for any halfdensity ρ 1 2 and any Lagrangian submanifold L: L ∆ 2 can ρ 1 2 = 0, we conclude that the this constant is zero, i.e.:

Odd Laplace operator on functions
Given a half-density ρ 1/2 we can define the odd Laplace operator on functions as follows: We will often abbreviate: when there is some obvious implicit choice of half-density.
We will now derive a formula for ∆ of the product of two functions. But first we have to discuss some properties of Lie derivative which are independent of its relation to ∆ can .
Some properties of the Lie derivative of a half-density Consider a vector field v on M , and the corresponding 1-parameter group of diffeomorphisms g t . Let us think of a half-density ρ 1 2 as a function of x and E, where x is a point of M and E a basis in T x M , depending on E in the following way: By definition, the Lie derivative of ρ 1 2 along v ∈ Vect(M ) is: Let us multiply v by a function f ∈ Fun(M ) such that f (x) = 0. The flux of f v preserves the point x, and we have: This implies that for any f ∈ Fun(M ) and v ∈ Vect(M ): In particular: Odd Laplace operator of product of functions Eqs. (17) and (25) imply: Notice that the odd Poisson bracket measures the deviation of ∆ ρ 1/2 from being a differentiation of Fun(M ).

Master Equation
We will always assume that ρ 1 2 satisfies the Master Equation: ∆ can ρ 1 Under this assumption, the operator ∆ ρ 1/2 is nilpotent:

Moment map
Let G denote the group of canonical transformations of M . Let us denote G L and G R the group of left and right shifts on G (both G L and G R are naturally isomorphic to G). Both left and right shifts naturally lift to ΠT G. Let g be the Lie algebra corresponding to G. We consider the right-invariant differential form on G with values in g, which is denoted dĝĝ −1 . Because of (9), we can consider it as a differential 1-form with values in ΠC ∞ M . There is some invariance under the left and right shifts, which can be summarized as follows: where the action of G L × G R on ΠT G is induced from the following action on G: The action of G L × G R on M is: The G R -invariance of d g g −1 follows immediately from the definitions. The G L -invariance, at the infinitesimal level, follows from the following formula, where it is enough to assume that {ξ B H B , } is an even vector field: The moment map satisfies the Maurer-Cartan equation: Minus signs are related to the minus sign in Eq. (8). The group of diffeomorphism is opposite to the Lie group whose Lie algebra is vector fields. Therefore the minus sign in e −t{ξ B H B , } x and the minus sign in Eq. (35).

Canonical operator is ill-defined in fieldtheoretic context
The considerations of Section 2 are valid when the BV phase space M is a finite-dimensional supermanifold. But in the field theory context, ∆ can is usually ill-defined. We use the BV formalism for the worldsheet sigma-model, i.e. in the field-theoretic context. We hope that it is possible to regularize ∆ can , maybe as in [13]. In many cases the string worldsheet theory is essentially free (quadratic Lagrangian) and the path integral can be computed exactly, allowing the explicit verification. In other cases, there is a natural small parameter and the computations can be verified order by order in perturbation theory.
In string theory models, we need to act by ∆ on the b-ghost. Therefore we need ad additional assumption; roughly speaking, the symmetry generated by the b-ghost should be nonanomalous. The validity of this assumption is model-dependent; it should be imposed in addition to the absense of BRST anomaly.
In this paper, we just use ∆ can in formulas as if it were well-defined. 4 Form Ω

Summary
We want to define some pseudo-differential form Ω which can serve as string measure. There are several closely related definitions: 1. PDF on the group of canonical transformations. In Section 4.2 we will start by constructing Ω as a PDF on G. The definition actually depends on the choice of a fixed Lagrangian submanifold L ⊂ M . Strictly speaking, we can characterize Ω as a map of the following type: (which associates a PDF on G to every Lagrangian submanifold).
2. PDF on the space of Lagrangian submanifolds. There is a natural map: coming from the action of G on M . A natural question is, does Ω descend to a PDF on LAG? The answer is essentially "yes", although some minor modifications are needed. In Section 4.5 we will define Ω as a PDF on the space of Lagrangian submanifolds.
3. PDF on the space of Legendrian submanifolds. Suppose that we can construct an R 0|1 -bundle over M , which we will call M , with a connection whose curvature is ω. Lagrangian submanifolds on M lift to Legendrian submanifolds in M . We can define Ω as a closed PDF on the space LEG of Legendrian submanifolds in M . LEG 4. PDF on an equivalence class of actions. In BV formalism the choice of a Lagrangian submanifold L ⊂ M is closely related to the choice of a quantization scheme. In other words, it is essentially the choice of a representative in a class of physically equivalent theories. Given S BV and L ∈ LAG, the restriction (S BV )| L gives a physical action functional which we use in the path integral. A different choice of L gives a BRST equivalent action functional. Therefore it would be natural to try to interpret Ω as a PDF on such an equivalence class. This, however, is not straightforward. The space of Lagrangian submanifolds is actually larger than the space of action functionals; to descend to the space of action functionals one has to take the factorspace over the symmetries of S BV . Generally speaking, Ω does not descend to this factorspace. We will study these issues in Section 6.

Definition of Ω as a PDF on G
For any function 5 E : C → C, consider the following pseudo-differential form on G: E where d g g −1 is the moment map of Section 2.4; its main property is that for any function F ∈ Fun(M ):

Ω is closed
Under the assumption that ∆ can ρ 1 2 = 0 the form Ω is closed.
Preparation for the proof of closedness Notice that for any even H ∈ We can interpret this formula using the notion of Lie derivative of half-density as follows: (where d only acts on g).

Proof of closedness
Taking H to be the moment map H = d g g −1 , we get: Let G be a Lie group. It is possible to introduce the structure of a Lie group on ΠT G. In fact: and the structure of the group is introduced by pointwise multiplication.
Consider the corresponding Lie superalgebra: g We can interpret g as the algebra of maps from R 0|1 to g; therefore the elements of g are g-valued functions f (θ) of an odd parameter θ ∈ R 0|1 . It is possible to extend g by an extra odd element ∂ θ with the following commutation relations: We will call this extended algebra g . g

Cone Lie superalgebra associated to a BV algebra
Let us consider a BV algebra G with the generator ∆. Let g be the Lie superalgebra which is obtained from G by forgetting the associative algebra structure and flipping parity, and g the corresponding cone Lie superalgebra. We need to flip parity in order to turn { , } into a Lie superalgebra operation. If the parity of b as an element of G, is |b|, then the parities of the corresponding elements of g are: |(b, 0)| = |b| +1 and |(0, b)| = |b|.

Theorem 2:
The following formulas define the representation of g on G: We have to check that: Indeed, we have:

Form Ω . . . as an intertwiner
Let us consider the particular case when G is the algebra of functions on the odd symplectic manifold M (our BV phase space). In this case, g naturally acts on the differential forms on G. Indeed, every element α ∈ g = Lie(G) determines the corresponding right-invariant vector field on G. Then (α, 0) would act as a Lie derivative along this vector field, and (0, α) acts as a contraction.
We can consider Ω as a linear map from G to the space of differential forms on G; for each a ∈ G, this map computes Ω a -the corresponding differential form. Eqs. (51) and (52) can be interpreted as saying that Ω is an intertwiner: In the language of half-densities, we have an intertwiner between: • action of g on half-densities in M : and • action of g on PDFs on G:

Deviation of Ω from being horizontal
In Section 4.2 we defined a form Ω on ΠT G × LAG. The action of G on LAG defines a natural projection: It is natural to ask if Ω is constant along the fibers of π. The answer is, strictly speaking, no, but the dependence on the fiber is rather mild. Let us present ΠT LAG as follows: where × G means factor over the symmetry: For the descent to work, Ω should be base, i.e. both invariant and horizontal, with respect to (72). Actually, it is invariant and almost horizontal. The invariance is straightforward. Let us study the question of horizontality, and understand why it is "almost horizontal" instead of "horizontal". Let ξ ∈ g denote a Hamiltonian whose flux preserves the Lagrangian submanifold L. The horizontality would be equivalent to the statement that ι Ad( g)ξ Ω is zero. But in fact, Eq. (52) implies: Notice that ξ ∈ St(L) implies Ad( g)ξ ∈ St( gL). Apriori this only implies that (Ad( g)ξ)| gL is a constant (but not necessarily zero). This is potentially a problem (we certainly do want Ω to descend on LAG). We will now outline some possible ways of resolving this problem.

Use ghost number symmetry
In string theory there is usually an action of U (1) called "ghost number symmetry". Various objects are either U (1)-invariant or have definite charge.
Let us restrict ourselves to only considering those Lagrangian submanifolds which are invariant under this U (1) (i.e. request that the orbits of the U (1) be tangent to the Lagrangian submanifold) and request that d g g −1 have ghost number −1. This eliminates the possibility of adding a constant to d g g −1 and therefore renders (d g g −1 ) L unambigously defined from the variation of L. Our form Ω is now horizontal, and descends from G to LAG.

Use transverse Lagrangian submanifold
Suppose that we can find a Lagrangian submanifold L ∨ ⊂ M which is transverse to all Lagrangian submanifolds from our family: where p( gL) is a marked point on every gL. Let Z L ∨ ⊂ Fun(M ) denote the subspace of those Hamiltonians which vanish on L ∨ . Since L ∨ is chosen to be Lagrangian, ΠZ L ∨ is a Lie subalgebra of g = ΠFun(M ). (Slightly smaller than the stabilizer of L ∨ .) It is intuitively clear (from counting the "degrees of freedom") that we can impose the following gauge condition: This eliminates the ambiguity of a constant in d g g −1 .
What happens if we change L ∨ to another transversal Lagrangian sub-manifoldL ∨ ? Let us assume that we can choose: Then we just have to change: As Ω is invariant, this would not change the result of the integration.

Upgrade LAG to LAG +
The most elegant solution is to use, instead of the space of Lagrangian submanifolds LAG, the space LAG + of Lagrangian submanifolds with marked point [2].

Quantomorphisms
Suppose that there exists an R 0|1 -bundle over M : with a connection such that the curvature is equal to ω. Then we can realize the central extension G as the group of automorphisms of this bundle.
We have the exact sequence: Suppose that we can find a "symplectic potential" α such that ω = dα. Then we can use it to construct the connection satisfying: where ∂ ϑ is the vector field arizing from the action of R 0|1 on M . (We can think of ϑ as a coordinate in the fiber; it is only defined locally, but ∂ ϑ is globally well-defined.) Explicitly: Let us consider the subalgebra g ⊂ Vect(M ) consisting of Hamiltonian vector fields. For every even (we will restrict to even vector fields for simplicity) {H, } ∈ g consider the following vector field on M : It is defined to preserve the connection. An explicit calculation shows that the Lie derivative vanishes: Notice that the vertical component ofξ H (with respect to the connection defined in Eq. (82)) is H∂ ϑ . By construction, the space of vector fields of this form is closed under commutator. We can check it directly, using the formula: As a Lie algebra this is ΠFun(M ). It integrates to the group of automorphisms of the fiber bundleM → M which preserve the connection defined in Eq. (82).

Form Ω as a form on the space of Legendrian submanifolds LEG
A Legendrian submanifold in M projects to a Lagrangian submanifold in M : This projection is typically not one-to-one; it is a cover. We can define Ω as a PDF on the space of Legendrian submanifolds by interpreting Eq. (49) as integration over the projection: This descends to a closed PDF on the space of Legendrian submanifolds LEG.

Reduction to integration over single Lagrangian submanifold
Here we will explain that integration over a family of Lagrangian submanifolds can be reduced to integration over a single Lagrangian submanifold in some larger BV phase space. A family of Lagrangian submanifolds {L(λ) ⊂ M |λ ∈ Λ} defines a single Lagrangian submanifold in L Λ ⊂ M × ΠT * (ΠT Λ). We will now describe the construction of L Λ .
As a first step, let us consider a submanifold L ⊂ M ×Λ which is defined as follows: This can be promoted to a subspace L ⊂ M × ΠT Λ: Finally, we will construct L Λ as the following section of the vector bundle where σ(m) computes for every tangent vector to Λ the value of its corresponding generating function on m ∈ L(λ). This section defines our big Lagrangian submanifold: There is a natural BV Hamiltonian d on ΠT * (ΠT Λ). It descirbes the lift to ΠT * of the natural nilpotent vector field d on ΠT Λ. We have: In the case of Yang-Mills theory, λ is calledc and [dλ] is called π (see Section 7.7).
5 Picture changing

Integration in two steps
In our approach we integrate some closed form Ω over a cycle in the moduli space of Lagrangian submanifolds modulo gauge symmetries. The form Ω itself is defined in terms of the integration of e S BV over the Lagrangian submanifold (path integral). We therefore have a double integral: first the path integral, and then a finite-dimensional integral over a cycle in the moduli space.
However, we suspect that there is actually no fundamental distinction between these two steps. In principle, one can combine them into one integration.
It is often convenient to pull one or more integration out from the integration of Ω into the path integral or vice versa. This is "picture changing".
In the case of bosonic string we usually choose a Lagrangian submanifold corresponding to a fixed complex strucuture (or metric, depending on the flavour of the formalism). In this case the path integration contains the integration over the antifield to metric/complex structure, which is identified with the b-ghost. However, in principle we could also include into the path integral some partial integration over the moduli space of complex structures, and then integrate over the rest later.
It turns out that in the BV formalism, this (at least in some cases) corresponds to changing each Lagrangian submanifolds in the family into a different Lagrangian submanifold. This is a topologically nontrivial change, essentially a change in polarization.

Baranov-Schwarz transform
Any family of Lagrangian submanifolds can locally be considered as an orbit of some Lagrangian submanifold L 0 by an abelian subgroup of G: where U ⊂ G is an abelian subgroup. For every L(u) = uL 0 u ∈∈ U L 0 , consider a submanifold K(u) ⊂ L(u) which is the common zero set of all Hamiltonians of elements of T u U restricted on L(u). Consider the union: It is a Lagrangian submanifold in M . Moreover: Indeed, let us consider the k-form component of Ω. We have to integrate it over some k-dimensional family of Lagrangian submanifolds. We parametrize the family by s 1 , . . . , s k . Let us perform the Baranov-Schwarz transform by integrating over d [ds]. This turns Ω into an integral form. We get: (Remember that we identify ∂g ∂s I g −1 with the actual Hamiltonian, i.e. for us ∂g ∂s I g −1 is a function on M , and in particular a function on L.) This means that actually we are integrating not over the whole Lagrangian submanifold L, but over a submanifold K ⊂ L of the codimension k, which is defined by the system of equations: But on the other hand, remember that we have to integrate over the family of the dimension k. In other words, we lost k integrations by inserting the δ-functions (98), but then regained them as integration over the family. This is equivalent to the "90 degree rotation" L → L. Therefore integration over a family of Lagrangian submanifolds is equivalent to the integration over a single "rotated" Lagrangian submanifold.
However, in the physical context of path integration, it is hard to make use of this equivalence.

Special canonical transformations
A canonical transformation is called special if it preserves ρ 1 2 . Let SG denote the group of special canonical transformations, and sg its algebra. We will also consider the corresponding centrally extended S G and s g. Elements of s g are those Hamiltonians which are annihilated by ∆ ρ 1/2 .
Suppose that two Lagrangian submanifolds L 1 and L 2 are connected by a special canonical transformation: Naively one would want to consider such Lagrangian submanifolds completely equivalent. However, this does not work. The form Ω is not base with respect to the action of SG on LAG. Moreover, at least in the finite-dimensional case the action of SG on LAG is usually transitive; the factor space SG\LAG would be a discrete space. Therefore it does not make much sense to consider the factorspace over SG.
On the other hand, if we consider just LAG without any identifications, then there are no integration cycles. For our closed form Ω to produce invariants, we need some integration cycles.
The solution is to consider a factorspace of LAG over a subgroup H ⊂ SG. In fact this H can not be an arbitrary subgroup of SG; it has to be choosen carefully 6 .

Straightforward descent does not work
Letĥ be a special canonical transformation, i.e.ĥ ∈ S G. It follows immediately that Ω = gL e d g g −1 is invariant under the left shift g → h g. But Ω is not horizontal; for ξ ∈ s g we have: Therefore Ω does not descend from LAG to SG\LAG. However, we will now identify a class of subalgebras h ⊂ sg for which we can construct the base form Ω base h . Being a base form, it descends to H\LAG, where H is the Lie group generated by the flows of elements of h.

Equivariant half-densities
Suppose that we have the following data: H such that: Then the following equation is a cocycle of the H-equivariant Cartan complex on LEG (Section 4.1) : This is proven using Section 4.4.3.
The compatibility of Eqs. (105) and (106): Here ξ 2 = 0 because ξ is a fermion. (But the equivariance condition (105) is stronger than just L ξ ρ C 1 2 (ξ) = 0, because it must hold for two arbitrary elements ξ and η of h.) It seems that the main ingredient in this approach is the choice of a group H. (In the case of bosonic string this is the group of diffeomorphisms.) The rest of the formalism is built around H. We suspect that ρ C (0) is already unambigously determined. Indeed, we will now see that the constraints arizing from Eqs. (105) and (106) are very tight.
6.4 Expansion of Ω(L, dL, ξ) in powers of ξ we use angular brackets to emphasize linear depnendence: Φ ξ is some linear In this Section we will use ρ (0) 1 2 to define the odd Laplace operator on functions: Therefore the Hamiltonians generating h should be all ∆-closed. Moreover, Eq. (106) implies: so they are actually all ∆-exact. Eq. (105) implies: The image of Φ, as a linear statistics-reversing map h → Fun(M ), will be called F. Notice that the inverse map to Φ :

Properties of subspace F
Therefore the existence of equivariant analogue of Ω implies the existence of a subspace F ⊂ Fun(M ) satisfying some special properties, which we will now study. Let us define the bracket:

Higher orders of ξ-expansion
We can always write, without any further assumptions: So defined a(ξ) should satisfy (we abbreviate Φ ξ to just Φ): The first two terms in the ξ-expansion (which is the same as Φ-expansion) are: The consistency of Eq. (122) requires that ∆{a, a} = 0 at every order in Φ. This automatically implies: The first term −{{a, a}, a} is automatically zero because of the Jacobi identity. The vanishing of 2{∆Φ, a} follows from Eq. (123): Therefore ∆-closedness of {a, a} is automatically satisfied order by order. But its ∆-exactness is a nontrivial additional assumption which has to be verified on case-by-case basis. Moreover, {a, a} should be ∆-exact in an equivariant manner, i.e. satisfying Eq. (123). To the first order in Φ: The terms of the order Φ 3 are: The terms of the order Φ 4 are: Notice that: This is zero by our assumption that the function Φ → ∆ −1 {Φ, ∆ −1 {Φ, Φ}} is h-invariant. This means that in Eq. (128) we are taking ∆ −1 of a ∆-closed expression. We must assume that this expression is also ∆-exact, so we could take ∆ −1 of it. This assumption has to be verified order by order.

Summary: constraints on gauge symmetry
The symmetry group H with Lie algebra h acts on the odd symplectic manifold M (the BV phase space), in such a special way that the following requirements are satisfied.
We require to exist the solution of Eqs. (105) and (106) of the form of Eq.
(121). We require that a(ξ) be smooth at ξ = 0. This implies that ρ . Moreover, we assume that every element of ξ ∈ h is actually ∆-exact, i.e. exists Φ ξ such that: The linear subspace F of Fun(M ) generated by Φ ξ , ξ ∈ h, should satisfy the properties described in Section 6.4.2. (In simplest cases all elements of F are in involution with each other.) Moreover, we must require the existence of a solution a to Eqs. (122) and (123).

Deformations of equivariant half-density
What can we say about the moduli space of solutions of Eqs. (105), (106)? Suppose that we can find another solutionρ C 1 2 (ξ): Eqs. (105), (106) imply that R should satisfy: This means that infinitesimal deformations of ρ C Let us assume that R(ξ) is a smooth function of ξ and can be expanded in series in ξ. Suppose that the leading term is of the order n in ξ (in other words, R has a zero of the order n at ξ = 0): Eq. (136) implies: Therefore we can think of R n as a map:

Base form
Let Ω C h denote the equivariant form, given by Eq. (121). We will denote the corresponding base form Ω base We will work under the assumption that the action of h on LAG does not have fixed points; LAG → H\LAG can be considered a principal H-bundle. In order to construct the base form Ω base h from the Cartan's Ω C h , we first choose on this principal bundle some connection A. (We understand the connection as a F-valued 1-form on LAG computing the "vertical component" of a vector.) Then we apply the horizontal projection i.e. replace Finally, we replace Φ with the curvature dA + A 2 of the connection A; we get:

Example: Duistermaat-Heckman integration
Consider the case when our odd symplectic submanifold is the odd cotangent bundle of some symplectic supermanifold X: Let p : M → X be the projection. Let π denote the Poisson bivector of X. It defines the following solution of the Quantum Master Equation: In this formula we define a half-density at m = (x, x ) ∈ M as a function on the space of bases ("tetrads") in T m M T x X ⊕ ΠT * x X, such that the value on two different bases differs by a multiplication by the super-determinant.
Let H be a subgroup of the group of canonical transformations of X. A canonical transformation h of X can be lifted to a BV-canonical transformation ΠT * h of ΠT * X as follows: An infinitesimal canonical transformation of X generated by a Hamiltonian F lifts to an infinitesimal BV-canonical transformation of ΠT * X generated by the BV-Hamiltonian ∆ ρ 1 2 (p * F ). Every point x ∈ X defines a Lagrangian submanifold in ΠT * X: the fiber ΠT * x X. Therefore, we can think of X as a family of Lagrangian submanifolds in ΠT * X. In this case F is the p * of the subspace of Hamiltonians generating H. The equivariant form Ω becomes: where = π −1 .

Example: an interpretation of Cartan complex
Let us consider the case: The odd tangent bundle ΠT X has a natural volume form. This induces a half-density on M . Let ρ (0) 1 2 be the half-density on M induced from the volume form on ΠT X. It satisfies the Quantum Master Equation. Now consider a different half-density, which also satisfies the Quantum Master Equation 8 : where d is the BV Hamiltonian generating the lift to ΠT * (ΠT X) of the canonical odd vector field d on ΠT X.
Suppose that a Lie group H acts on X. This action can be lifted to ΠT X and ΠT * (ΠT X). An infinitesimal action of ξ ∈ h = Lie(H) on M is generated by the BV Hamiltonian: Therefore, in this case: 8 We can replace e d with an arbitrary function of d. This will still satisfy the Quantum Master Equation.
It satisfies Eq. (106): This formula can be generalized as follows. Suppose that α is a function on h × ΠT X. For any ξ ∈ h, we think of α(ξ) as a function on ΠT X, i.e. a pseudo-differential form on X. With a slight abuse of notaions, α(ξ) will also denote the pullback of α(ξ) along the projection, from ΠT X to ΠT * (ΠT X). Then we have: Therefore α → αe d+ ι is an intertwiner from the Cartan complex to the complex of equivariant half-densities on ΠT * (ΠT X).

Equivariant effective action
When some fields are integrated out, we get an "effective action" for the remaining fields.
Mini-review of BV effective action Let (M, ω) be an odd symplectic supermanifold (the BV phase space), and ρ 1 2 be a half-density on M . Suppose that we are given a submanifold E ⊂ M which comes with the structure of a fiber bundle with fibers isotropic submanifolds over some base B. (This can be thought of as a family of isotropic submanifolds.) For every b ∈ B we have the corresponding fiber, an isotropic submanifold which we denote I(b). Moreover, we require: (i.e. the degenerate subspace of the restriction of ω to E is the tangent space of the fiber). Since dω = 0, this condition implies that the Lie derivative of ω| E along the fiber vanishes, and therefore ω defines an odd symplectic form on the base which we will denote ω B . Finally, we require certain maximality property: Let L ⊂ B be a Lagrangian submanifold of B. Under the above conditions, it can be lifted to a Lagrangian submanifold in M as π −1 L, where π is a natural projection: We then define a half-density π * ρ 1 2 on B so that for every Lagrangian submanifold L ⊂ B and every function f ∈ Fun(B): The proof of existence of such π * ρ 1 2 is similar to the proof of Theorem 1.
We define a measure µ L [ρ 1 2 ] on L as push-forward of ρ 1 2 π −1 L to L. We observe that for any canonical transformation g bringing fibers of E into fibers: Then we define a half-density σ ]. Suppose that we are given a function Ψ on M whose restriction on E is constant along I. Then it defines a function on B which we denote π * Ψ: Notice that in this case the flux {Ψ, } is tangent to E.
It is enough to prove that for any φ ∈ Fun(M ) constant on E: {Ψ, φ } = 0. From the maximality of B, as defined in Eq. (161), follows that {φ, } is tangent to I. Since Ψ is constant along I, it follows that indeed {Ψ, φ } = 0.
We have: Equivalently, for two functions Ψ 1 and Ψ 2 on M both constant along I: In order to prove Eq. (166), notice that any vector field tangent to I can be written as a f a {φ a , } where f a are some functions on M and φ a are some functions on M constant on E. The commutator with {Ψ, } of such a vector field is a {Ψ, f a }{φ a , } again tangent to I. This means that the flow of Ψ brings fibers to fibers, which is equivalent to Eq. (166).
We will now prove that for any Ψ ∈ Fun(M ) whose restriction on E is constant along I: Indeed, for any "test function" f ∈ Fun(B): Equality of Lines (169)  satisfies the QME on M , considering Eq. (172) with both Ψ and F constant along the fiber of E → B and using Eq. (168) proves that π * ρ 1 2 also satisfies the QME.
Equivariant case Now we will consider partial integration when the halfdensity satisfies the equivariant Master Equation. We show that the resulting effective theory still satisfies the equivariant Master Equation.
Suppose that a Lie group H acts on M by canonical transformations, and moreover that every Hamiltonian ξ ∈ h is constant on the fibers of E π → B: Suppose that we are given an equivariant half-density ξ → ρ C 1 2 (ξ) on M , i.e.: Then the result of the partial integration is an equivariant half-density on B: Indeed, Eq. (175) is equivalent to the statement that for any two functions Ψ and F :

BRST formalism
We do not have a complete description of symmetry groups H satisfying the properties summarized in Section 6.5. But we do understand the special case, when BV formalism comes from BRST formalism.

Brief review of BRST formalism
One starts with the "classical action" S cl which is invariant under some gauge symmetry. Let X be the "classical" space of fields, e.g. for the Yang-Mills theory the fields are A µ (x). Suppose the gauge symmetry is H, with the Lie algebra h = Lie(H). We introduce ghost fields, geometrically: fields and ghosts ∈ ΠT H × X H where the action of h 0 ∈ H is via right shift on H: We use the coordinates dh, h on ΠT H (denoting "dh" the coordinate on the fiber ΠT h H) and x on X. Notice that this commutes with Q BRST = dh ∂ ∂h . We can always find a representative with h = 1 (i.e. choose h 0 = h −1 ). Then, with the standard notation c = dh:

Functions on ΠT H×X
H satisfy:

Integration measure
We assume that X/H comes with some integration measure: This µ be understood as an integration measure, i.e. a density of weight 1, rather than a function of x. The product of this measure with the canonical measure on ΠT H gives us a measure on ΠT H×X H which we will call µ BRST . Notice that Q BRST preserves this measure. This can be proven as follows.
For any function f ∈ Fun ΠT H×X because Q BRST comes from the canonical odd vector field on ΠT H.

Lift of symmetries to BRST configuration space
Original gauge symmetries can be lifted to the BRST field space as left shifts on H: These left shifts commute with the right shifts used in Eq. (180), therefore they act consistently on the factorspace. They also commute with Q BRST . Moreover, infinitesimal left shifts L ξ are actually BRST-exact: The measure on ΠT H×X H is invariant under left shifts, because it is constructed from the canonical measure on ΠT H which is invariant.

BV from BRST
The BV phase space is: The zero section ΠT H×X H is a Lagrangian submanifold. It comes with the integration measure, which lifts to a half-density on M of the form e S BV , where: where Q BRST is as defined in Eq. (181). We can imagine a more general situation when we have a functional S cl with an odd symmetry Q BRST nilpotent off-shell. But, just to describe the "standard BRST formalism", we explicitly break the fields into c A and x i .
We have realized the gauge algebra h as symmetries of the BRST configuration space as left shifts on ΠT H. Since the BV phase space is the odd cotangent bundle, we can further lift them to the BV phase space. The symmetry corresponding to the infinitesimal left shift (186) is generated by the BV Hamiltonian: In other words: F is generated by c α (192)

Form Ω in BRST formalism
Although the "BRST formalism in the proper sense of this word" requires splitting φ into "physical fields" ϕ and "ghosts" c, in many cases such a split is not required and does not play any role. Let us forget about the split for a moment; just require that BRST operator is nilpotent off-shell: But the split into ϕ and c will come back very soon in Section 7.6.
Form Ω in BRST formalism is given by the following expression: The family Λ is a family of gauge fermions Ψ(φ). To prove dΩ = 0 we use that [dφ]Q BRST (. . .) = 0 (no BRST anomaly). Notice that we deform the action S cl → S cl + Q BRST Ψ but do not deform Q BRST . Generally speaking, we would like to treat as "gauge symmetries" those Ψ which are BRST exact plus equations of motion (here "equations of motion" are those derived from S cl + Q BRST Ψ). In other words, for any vector field ζ on the field space and any functional F the following δΨ should correspond to a symmetry: This is just a generic field redefinition plus adding a BRST-exact term 9 No horizontality The form Ω given by Eq. (194) is not horizontal: Cartan form In order to construct equivariant (and then base) form, we will need to restrict ζ and F to belong to some linear subspaces: We will need to require that these subspaces are such that for any ζ ∈ F Vect and F ∈ F Fun : • the following conditions (analogous to {Φ, Φ} = 0) hold:

nothing goes to zero)
Then the following expression: is annihilated by the Cartan differential.
To complete the proof, we notice that the last line coincides with Eq. (197).

Lifting the gauge symmetry to BRST formalism
As we have just explained, our ζ and F are restricted to belong to some subspace F Vect ⊕ F Fun , which should satisfy certain conditions. A geometrically natural solution to these conditions can be found in the case of "traditional" BRST formalism where the fields φ are split into physical fields ϕ and ghosts c. It corresponds to the following choice of ζ and F : -constant (= field-independent) shifts of ghosts. Eq. (202) becomes:

Faddeev-Popov integration procedure
The naive (or "standard") Lagrangian submanifold is: But the restriction of S BV to this Lagrangian subnamifold coincides with S cl , and therefore is degenerate; we cannot integrate. In order to resolve the degeneracy, we have to deform to another Lagrangian submanifold. However, we face a complication. It is desirable to keep the ghost number symmetry. But if we restrict ourselves to only those Lagrangian submanifolds which are invariant under the ghost number symmetry, then the standard one of Eq. (210) is rigid. It does not admit deformations. Indeed, the deforming gauge fermion should have ghost number −1, but there are no fields with negative ghost number. Therefore, there are no deformations. One solution to this problem is to introduce non-minimal fields (the BRST quartet). Here we will describe another solution (giving the same answer), based on the consideration of families of Lagrangian submanifolds. It turns out that exist ghost number preserving families of Lagrangian submanifolds deforming the standard one. We can then use our form Ω to integrate over these families, thus obtaining a regularized theory with ghost number symmetry. Let us outline the construction of such families. Let X be the space of fields ϕ. Suppose that we have a linear space V and a map F : X → V such that the orbits of the symmetry are transversal to the level set Y = F −1 (0). This F is called "gauge fixed condition". For eachc ∈ ΠV * we can consider the following section of ΠT * X: The secion (211) defines a Lagrangian submanifold of ΠT * X. (Whenc = 0 this is X ⊂ ΠT * X.) Therefore, we have a family of Lagrangian submanifolds parametrized by elements of ΠV * . Let us integrate our PDF Ω, which in this case is equal to: over this family. The corresponding integral form (the density on ΠV * ) is: This means that we have to integrate: This is the standard Faddeev-Popov integral (see e.g. Chapter 8 of [16]). The integration is convergent for an appropriate choice of a contour.

Yang-Mills theory
The Landau gauge corresponds to the following function F : X → V , where V is the space of functions on the four-dimensional spacetime: Notice that V (and therefore our family of Lagrangian submanifolds) is now infinite-dimensional. In this case (F * ) * c is A µ = ∂ µc and: The integration contour is such thatc is complex conjugate to c. The insertion of δ(F ) is usually done by means of a Lagrange multiplier.

Conormal bundle to the constraint surface
Here we will describe Lagrangian submanifolds of the type used in the bosonic string worldsheet theory. They provide another solution to the problem discussed in the beginning of Section 7.7.
Conormal bundle Let Y be some family of submanifolds Y ⊂ X closed under the action of the gauge symmetry 10 For each Y ∈ Y, the odd conormal bundle of Y (denoted Π(T Y ) ⊥ ) is a subbundle of the odd cotangent bundle ΠT * X| Y which consists of those covectors which evaluate to zero on vectors tangent to Y . For each Y ⊂ X, the corresponding odd conormal bundle is a Lagrangian submanifold. Given such a family Y, let us define a family of Lagrangian submanifolds in the BV phase space in the following way: for every Y , the corresponding Lagrangian submanifold is the odd conormal bundle of Y , times the space of c-ghosts: Irreducibility and completeness Let us ask the following question: under what conditions the restriction of S BV to each L(Y ) is non-degenerate? Or, in case if it is degenerate, how can we characterize the degeneracy? We have: The second term T i A c A ϕ i is the evaluation of the covector ϕ on the tangent vector Qϕ ∈ T X.
Let us assume that the restriction of S BV to any L(Y ) has a critical point, and study the quadratic terms in the expansion of S BV | L around that critical point. To define the perturbation theory, we need already the quadratic terms to be non-degenerate. Assuming that the critical point is at ϕ = 0: Suppose that all degeneracies of S cl and of k are due to symmetries (the completeness). In other words: Let s be the quadratic part of S BV | L : The degeneracy is characterized by the isotropic subspace of s which we denote ker s: Let us make the following assumptions: 1. The space im τ ∩ T Y is zero, in other words Y is transverse to the orbits of H. This is a constraint on the choice of Y 2. The next term, ker τ , is also zero. This kernel being nonzero corresponds to reducible gauge symmetries.
But the last term Π (im τ + T Y ) ⊥ is essentially nonzero. It can be identified with the cotangent space to our family: where Y is the moduli space of submanifolds Y ⊂ X. Therefore the quadratic part of S BV is degenerate. However this degeneration is removed by the factor e d g g −1 . Indeed, in this case: When we integrate over Y/H, the differentials dy i span the complement of imτ in T X/T Y . Since we require that the family Y be H-closed, τ defines a map h → T X/T Y which we denote [τ ]. With these notations: In the case of bosonic string (Section 9.2) dim coker [τ ] = 3g − 3 and e d g g −1 αβ .

Integrated vertex operators 8.1 Deformations of BV action
Suppose that we infinitesimally deformed the solution of the Master Equation: In string theory U is called "integrated vertex operator". In order for this to satisfy the Master Equation to the first order in ε we require: We will also postulate that U is diffeomorphism invariant, which means in our formalism that: As we deform S BV , what happens to F? The answer is simple if U satisfies the Siegel gauge condition: In this case F remains undeformed. But if Eq. (236) is not satisfied, then F should also get deformed.

Deformations of F
Eqs. (234) and (235) imply that {U, Φ} is ∆-closed: Let us also require that it is ∆-exact: Under these assumptions we can deform: In other words: • the space F does deform, according to Eq. (241), but the action of diffeomorphisms remains the same In particular, the BV Hamiltonian of diffeomorphisms ∆Φ stays undeformed (Eq. (242)) Is it true that the deformed Φ remain in involution modulo ∆-exact? Notice that the deformed {Φ, Φ} is automatically ∆ + ε{U , }-closed under already taken assumptions: Opening the parentheses, we derive that the expression 2{Φ , Let us assume that it is also ∆-exact: (the validity of this assumption depends on the cohomology of ∆). Therefore, to the first order in the bosonic infinitesimal parameter ε: This equation can also be derived immediately from Eq. (243) under the assumption that the ghost number 3 cohomology of ∆ + ε{U , } vanishes Therefore the condition of being in involution persists, but with deformed q: Remember that the construction of equivariant form Ω C requires solving the equation: The solutions deforms: And the string measure deforms: In the base form Ω base we substitute for ξ the curvature of the connection -the 2-form. In other words, a(ξ) becomes a two-form on LAG. Therefore a vertex operator is actually an inhomogeneous PDF on LAG (and not just a function). If the theory has ghost number, then U has ghost number zero and a(ξ) has ghost number −2; the sum of the ghost number and the degree of the form is zero. Lower ghost number components of vertex operators were recently used in the context of pure spinor formalism in [17].

Exact vertex operators
Let us consider the case when U is ∆-exact: Let us also assume that W is H-invariant: In this case, we can take: We observe that the resulting deformation of the equivariant form is the Lie derivative along {W, }: This means that: • deforming with an exact integrated vertex is equivalent to an infinitesimal change of the integration cycle To construct the base form from the Cartan form, we need to choose a connection. When deforming with an exact vertex, we need to also adjust the connection by carrying it along with {W, }

Relaxing Siegel gauge
Let us deform S BV by adding an integrated vertex operator not in the Siegel gauge. In this case we have to use the full base form Ω base from Section 6.7. Suppose that there exists W such that U + ∆W is in Siegel gauge: In practice W could be complicated. But if it exists, then this implies that we can satisfy the Siegel gauge by slightly deforming the integration cycle. Usually in string theory models, having the Siegel gauge satisfied on a family of Lagrangian submanifolds implies that Φ vanishes on Lagrangian submanifolds in this family; then our form Ω base reduces to the standard string theory measure. (In the case of bosonic string this is discussed in Section 9.3.) Combined with the observation that the Siegel gauge can be satisfied by deforming the contour, this implies that the amplitudes obtained by integrating our Ω base give the same result as the standard prescription for string amplitudes.

Bosonic string
Now we will explain how this formalism can be applied to the bosonic string worldsheet theory.

Solution of Master Equation
The fundamental fields of the worldsheet theory are: matter fields x m , complex structure I α β and the diffeomorphism ghosts c α . The matter part of the action depends on matter fields and complex structure: Following the general scheme, we find the solution of the Master Equation: There is a constraint: I 2 = −1, or in components: I α β I β γ = −δ α γ . The antifield I can be identified with a symmetric tensor I αβ = I βα with only nonzero components I ++ and I −− (which are also denoted I zz and I zz .) The coupling We have just changed the polarization; I is now a field (called b) and I an antifield (called b ). The action can be written in the new coordinates: Family of Lagrangian submanifolds We now choose the Lagrangian submanifold in the following way: On this Lagrangian submanifold the action is quadratic. In particular: BRST structure The BRST operator Q BRST of the bosonic string can be understood as follows. We expand S BV in powers of the antifields x , c , b and consider only the linear term. The corresponding Hamiltonian vector field preserves the Lagrangian submanifold and is the symmetry of the restriction of S BV on the Lagrangian submanifold. There are also higher order terms, because the dependence on b is nonlinear; b γ α T γβ dz α ∧ dz β is just the linear approximation. In constructing the Q BRST we simply neglect those higher order terms. This leads to Q BRST being nilpotent only on-shell. The explicit formula for Q BRST can be read from Eq. (265): Ghost numbers

Integration over the family of Lagrangian submanifolds
Here we will first use our prescription to construct the equivariant analogue of Ω, depending on some motivated choice of the subspace F ⊂ Fun(M ). We will then implement the standard procedure to construct a closed form on H\LAG.
Choice of F We will here make use of the standard choice of F always applicable in the BRST case as we explained in Section 7.4. We will choose F so that Π∆F is the algebra of diffeomorphisms of the worldsheet: where ξ α = ξ α (z,z) is a vector field on the worldsheet.

Vect(Σ)⊕Weyl
The resulting equivariant form is: where the term dI , b comes from dgg −1 and c α t α from Φ of Eq. (269). The pairing dI , b is as in Eq. (258).
Recovery of the standard approach Usually in the literature, the integration cycle is chosen so that c = 0, and therefore the term c α t α vanishes. Moreover, we do not even need to do the horizontal projection of dgg −1 . This is a consequence of the following general statement. Suppose that {Φ, Φ} = 0 and the integration cycle in the moduli space of Lagrangian submanifolds is such that: In this case the construction simplifies: We can now use the Baranov-Schwarz transform (Section 5.2) and interpret the integration of Ω as the integration of e S over some new Lagrangian submanifold, which can be described as follows. Consider any 3g −3-dimensional surface in the space of metrics g ·· parametrized by (s 1 , . . . , s 3g−3 ): (no need to require any holomorphicity). Let us consider a submanifold defined as follows: it is the bundle over S whose fiber at a point s ∈ S consists of the subspace of ΠT * s MET orthogonal to the tangent space to S at that point. Notice that this submanifold is Lagrangian. We will promote N S to a Lagrangian submanifold N S in the BV phase space of the bosonic string by adding x and c · (and keeping x = 0 and c · = 0). We have: Notice that we can use Ω instead of Ω B because Eq. (271) is satisfied in this case. Indeed, Φ ξ = ξ α c α and we choose the Lagrangian submanifolds so that c * · = 0; this proves Eq. (271). When some part of N S contains a gauge-trivial directionġ αβ = 2∇ (α ξ β) , then the integral of e S BV over that part is automatically zero. Indeed, in this case all the b ·· ∈ ΠT * s MET orthogonal to the tangent space to S satisfy in particular b αβ ∇ α ξ β = 0 and therefore the integral of e S BV over c · will give zero (because of the zero mode c α ξ α ). In this sense, e S BV is a "base integral form".
Standard integration cycle We will now discuss the "usual" (in the bosonic string theory) integration cycle on the moduli space of Lagrangian submanifolds. Our choice of F is such that h = Π∆F is the algebra of diffeomorphisms of the worldsheet. This allows us to construct the base form on the space of Lagrangian submanifolds modulo diffeomorphisms.
On a Lagrangian submanifold from the standard family the metric g αβ (same thing as b αβ ) is fixed, and the path integral goes over b αβ , c α , x m . This picture explains why factorization by the action of H results in closed integration cycles: Here h is a large diffeomorphism; the integration cycle is a 3g −3-dimensional family of metrics g αβ ; it only becomes closed after we make an identification of g αβ and h * g αβ .

Topologically twisted N = 2 model
In this Section we will give an example of the contructrion of F in the case which is not covered by the scheme described in Section 7.4.

BV action
Consider the topologically B-twisted N = 2 superconformal theory [18]. Let us restrict to the case of flat target space. The fields are: bosons: complex scalars x a and x a fermions: one-form ρ a and scalars ϑ a and η a The "classical" action is: The BV action is:

Action of diffeomorphisms
Let us consider the infinitesimal diffeomorphism (= vector field) of the world- The corresponding BV generation function is The last term ι ρ a ι v ϑ a requires explanation. As ϑ is a scalar field, its antifield ϑ is a 2-form. The expression ι v ϑ is the usual contraction of a vector field v with the 2-form ϑ . Finally, ι ρ a ι v ϑ should be understood in the following way. We interpret ρ as a vector-valued two-form, i.e. a section of (T ⊗ Ω 2 )Σ. We contract its vector index with the single remaining covector index of ι v ϑ . What remains is a two-form, which is just integrated over Σ. Explicitly: Identities useful in proving this:

Closedness of F under [ , ]
We need to prove: The only nontrivial computation is the bracket of the last term in Eq. (279) with the last line of (280): and the bracket of the term −( Eqs. (287) and (290)

Worldsheet with boundary
The consideration of this Section is not rigorous for the reason explained in Section 2. We hope that more rigorous treatment could be obtained by an axiomatic approach along the lines of [4]. Consider string worldsheet theory on a flat disk D. Insertion of some operators inside the disk creates a state on the boundary: Let us consider a Riemann surface with a boundary, and insert our flat disk: Consider the variations of the Lagrangian submanifold (for example, metric), limited inside some compact region on the Riemann surface: As we vary the metric, the state on the boundary (marked red) remains the same state in the same theory. This is possible because we do not change the theory in the region of insertion.
If we limit the variations of the Lagrangian submanifold in this way, then we can construct the form Ω (and its base analogue) in the same way as we did on the closed Riemann surface. In this case the form Ω is not closed: We will derive this formula in a moment. If we restrict ourselves with the insertions of only physical operators Ψ, then the form Ω base Ψ is closed.

Variation of the wave function
Let us study the quantum theory on a region Σ\D (the complement of the disk on the Riemann surface); its boundary is −∂D. On the picture D is painted blue. Notice that there are no operator insertions inside D.
Let us bring our theory to some first order formalism, so that the Lagrangian is of the form pq − H. Fix the Lagrangian submanifold L. Fix some polarization in the restriction of the fields to ∂D, for example the standard polarization where the leaves have constant q. Let us consider the wave function: the integration is over the field configurations inside D, and q enters through boundary conditions. More generally, let Ψ O (q) denote the wave function obtained by the path integration with the insertion of some operator O with compact support, not touching the boundary: Any functional Ξ on the odd phase space, with compact support on the worldsheet, determines an operator insertion, just by its restriction on the Lagrangian submanifold. We will denote it Ξ 0 : Theorem 3: Suppose that ρ 1 2 satisfies the Master Equation.
The expansion of S BV around the Lagrangian submanifold in powers of antifiedls defines some Q BRST . Then: where Ξ 0 is the restriction of Ξ to L and L Ξ is the Lie derivative of the halfdensity ρ 1 2 along the Hamiltonian vector field {Ξ, } (see Eq. (17)). Notice: • Eq. (295) is not affected by the presence of a boundary Proof: Let us introduce some Darboux coordinates near L, so that L is given by the equation φ = 0. Let us expand Ξ in powers of φ . Let us first assume that only a constant in antifields term is present: In this case the Lie derivative is equivalent to inserting Q BRST Ξ 0 into the path integral, giving Eq. (295). If Ξ also depends on antifields, then we have to be careful restricting ourselves to such Ξ that the Lie derivative L Ξ ρ 1 2 is well-defined, because otherwise Eq. (295) does not make sense. This assumption must include the vanishing of the integration by parts: This is equivalent to the linear term in the antifield expansion of Ξ not contributing to the RHS of Eq. (295). has compact support which does not touch the boundary. Then Q BRST defined as in Theorem 4 becomes a symmetry of S BV | L sufficiently close to the boundary. Let us define Q BRST Ψ O via the insertion of the BRST current near the boundary:

Interpretation of Ω as an intertwiner in the presence of a boundary
The equivariant Ω C is a particular case of Ω e a when a is a solution to Eq. (122). We interpret the path integral in the theory on Σ\D, with a boundary state, as the path integral over the whole Σ with insertions O 1 , O 2 , . . . (inside D) determining this boundary state. Then our form Ω is defined by the path integral in the theory on the whole compact Riemann surface Σ: But we only allow the variations of L which do not change the theory inside the disk D. In other words, we restrict to d g g −1 of compact support inside Σ\D. We also assume that a also has compact support localized inside Σ\D.
Then, as we explained in Section 4.4.3, Ω intertwines d with ∆: Since the support of O 1 O 2 · · · is in D, and the support of a is in Σ\D, we have: The first term should be interpreted as a nilpotent operator Q BRST acting on the inserted state: Therefore we have: We conclude that the presence of a boundary modifies the intertwiner property of Ω. This is a particular case of the following general construction. Let ρ : g → End(V ) be a representation of of the cone Lie superalgebra. Then any complex K, d K defines a new representation of g : In our case K is the Hilbert space of states on the boundary, with d K = Q BRST .

Base form
In the presence of a boundary our construction of the base form does not give a closed form: At the same time, Ω base is still horizontal and invariant. Horizontality follows from the construction, as we obtain our base form from the equivariant form by horizontal projection. Invariance follows from the fact that dΨ base is also horizontal. For our constructions presented in this part, it is important that we restrict to only those variations of the complex structure which are zero at the boundary. Otherwise, the variation would change the Hilbert space of states. In such case it would be nontrivial to define dΨ, as we would need a connection on the bundle of Hilbert spaces.

Unintegrated vertex operators
Recall that H is the group of diffeomorphisms of the worldsheet and h its Lie algebra.

Definition of unintegrated vertex
Let us try to relax the condition of the h-invariance for the vertex, i.e. do not require that {∆Φ, U } = 0. In this case the deformed action will not be h-invariant. It does not seem to be possible to modify the definition of F so that the deformed action S + εU be invariant 11 . We will deal with this complication in the following way. Let us assume that, instead of being an integral over the worldsheet, U is actually a sum of insertions of some operators into fixed points on the worldsheet. We can then restrict the group of diffeomorphisms to a subgroup which preserves those points. Let h U be the subalgebra of h which preserves the points of insertions and F U = ∆ −1 h. In other words: Remember that we have to integrate Ω over some cycle in the moduli space of Lagrangian submanifolds. We postulate that, in case of unintegrated vertex: • the integration cycle should include the variations of the positions of the insertion points We will now outline the procedure of integration over those insertion points.

Fixing the Lagrangian submanifold
Therefore we have to study the restriction of Ω to the subgroup H ⊂ G.
We will for now assume that elements of F are in involution (i.e. q = 0 in Section 6.4.2). We will make use of the fact that H (as opposed to full G) preserves ρ 1 2 . This implies that the integration measure can be transformed to a fixed Lagrangian submanifold: where the action of h 0 ∈ H is induced by the right shift on H and the action of H ⊂ G on M ; in particular, any function of the form h, x → F (hx) is H-invariant. The differential comes from the canonical odd vector field on ΠT H; we will denote it d (h) . The integrand in Eq. (313) belongs to this space: We will show that this is the same as the Lie algebra cohomology complex of h with coefficients in Fun(M ). This is a version of the well-known theorem saying that Serre-Hochschild complex of the Lie algebra with trivial coefficients is the same as right-invariant differential forms on the Lie group. This is a general statement, true for a Lie group acting on a manifold.

Notations and useful identities
In this Section the tilde over a letter will denote the composition with h: Here are some identities that we will need: Elements of the space (Fun(M × ΠT H)) H can be obtained from letters Ψ and U by operations of multiplication and computing the odd Poisson bracket, or applying ∆.
We have: In other words, the operator of multiplication by e ∆ Ψ intertwines between ∆ and ∆ − d (h) . After we integrate over the Lagrangian submanifold, ∆ − d (h) becomes just −d (h) .

Integration
The one-form component

Cohomology of ∆ vs Lie algebra cohomology
In this Section we will show that the Modified de Rham complex of H is the same as the Lie algebra cohomology complex of h with coefficients in Fun(M ).

Definition of the Lie algebra cohomology H • (h, Fun(M ))
The space Fun(M ) is a representation of h; the action of h is slightly easier to write down at the level of the corresponding action of the Lie group H; h ∈ H acts on f ∈ Fun(M ) as follows: Therefore: We define d Lie Ψ to coincide with −d (h) Ψ of Eq. (320): To follow the Faddeev-Popov notations, we introduce the Faddeev-Popov ghost: Beware that Ψ is not just the Faddeev-Popov ghost; it is the product of the Faddeev-Popov ghost c A with ∆ −1 H ∈ Fun(M ). Eqs. (333) and (334) are equivalent to saying: where f AB C is the structure constants of h: It is straightforward to verify using Eqs. (330) and (332) that: The subgroup H ⊂ G preserves ρ 1 2 and therefore ∆: This implies: ∆d Lie + d Lie ∆ = 0 (341)

Proof that d (h) is the same as d Lie
This is similar to the statement that for any Lie group G, the de Rham subcomplex of right-invariant forms on G is the same as the Lie cohomology complex of g with coefficients in the trivial representation: (Fun(ΠT G)) G = C • (g, C) Our case is a variation on this theme:

Another intertwiner between d Lie + ∆ and ∆
One intertwiner between d Lie + ∆ and ∆ is already provided by Eq. (326), but it is nonlocal (because each ∆Ψ contains one integration). Motivated by the integration procedure of Section 12.2.4, we will now construct another intertwiner, a local one. Let us assume that elements of subspace F are all in involution, i.e. q(x, y) = 0. In this case:

Descent Procedure
Here we will show that the intertwining operator of Eq. (347) can be interpreted as the generalization of the string theory descent procedure, which relates unintegrated and integrated vertex operators. Consider an unintegrated vertex opearator U . We interpret it as an element of the cohomology of ∆ with some ghost number n: We could have used e ∆Ψ instead of e {Ψ, } . We prefer to use e {Ψ, } because it leads to the local result. Although Ψ contains integration, the odd Poisson bracket is local (i.e. involves a delta-function) and therefore removes the integral.
In string theory the use of such an inhomogeneous expression is often referred to as "the descent procedure".

Integrated vertex and Lie algebra cohomology
We have shown that the cohomology of ∆ is the same as the cohomology of ∆ + d Lie . The cohomology of ∆ + d Lie can be computed using the spectral sequence, corresponding to the filtration by the ghost number. Let F p ⊂ Fun(M ) consist of the functions with the ghost number ≥ p. At the first page, we have: Therefore, if E 1 = E ∞ , then the cohomology of ∆ is equivalent to the cohomology of h with values in Fun(M ).
Remember that dZ parametrizes a point in the fiber of ΠT M over the point Z in M . Then dZ + V is a new point in the same fiber, linearly depending on .
If V is odd, we define ι V as follows: ι V ω = ∂ ∂ ι V ω. In coordinates: The relation to Lie derivative: A.2 Symplectic structure and Poisson structure Consider a supermanifold M , with local coordinates Z A , equipped with an odd Poisson bracket of the form: The Poisson bivector A π B should be symmetric in the following sense: The odd symplectic form ω can be defined from the following equation:

A.3 Darboux coordinates
In Darboux coordinates: Solutions can always be found, order by order in φ , to any order n. For example, when n = 1: where χ i (φ) should satisfy: This equation always has a solution in a sufficiently small neighborhood U of x.
What we have to prove is that: Let us consider Eq. (371) in the special case when g ∈ G is such that g(x) = x. We get: