DGLA Dg and BV formalism

Differrential Graded Lie Algebra Dg was previously introduced in the context of current algebras. We show that under some conditions, the problem of constructing equivariantly closed form from closed invariant form is reduces to construction of a representation of Dg. This includes equivariant BV formalism. In particular, an analogue of intertwiner between Weil and Cartan models allows to clarify the general relation between integrated and unintegrated vertex operators in string worldsheet theory.


Introduction
The BV approach to the string worldsheet theory was developed in [1] and [2]. It involves the construction of a pseudo-differential form on the space of Lagrangian submanifolds in the BV phase space of the worldsheet sigmamodel, base with respect to the action of the worldsheet diffeomorphisms. The construction of equivariant form in [1], [2] involves a map of some differential graded Lie algebra (DGLA) Dg into the algebra of functions on the BV phase space of the string sigma-model. To the best of our knowledge, Dg was first introduced, or at least clearly explained, in [3]. Here we will rederive some constructions of [1], [2] using an algebraic language which emphasizes the DGLA structure, and apply some results of [3] to the study of worldsheet vertex operators. In a sense, Dg is a "universal structure" in equivariant BV formalism, i.e. the "worst-case scenario" in terms of complexity. The construction of Dg is a generalization of the construction of the "cone" superalgebra Cg (which is called "supersymmetrized Lie superalgebra" in [4]). There is a projection Dg −→ Cg. At this time, we do not have concrete examples of string worldsheet theories using Dg which would not reduce to projection into Cg. It is likely that pure spinor superstring in AdS background is an example, but we only have a partial construction [5].
For every Lie superalgebra a, we can define a Differential Graded Lie superalgebra Ca (the "cone" of a) as follows. Consider a graded vector space: where a is at grade zero, and sa at grade −1. (The letter s means "suspension".) We consider vector superspace a as a graded vector space, such that the grade of all elements is zero. Then, we denote sa the vector space a with flipped statistics at degree −1. 1 The commutator is defined as follows. The commutator of two elements of a ⊂ a ⊕ sa is the commutator of a, the commutator of two elements of sa is zero, sa ⊂ a ⊕ sa is an ideal, the action of a on sa corresponds to the adjoint representation of a. The differential d Ca is zero on a and maps elements of sa to the elements of a, i.e.: d Ca (sx) = x. This construction has an important application in differential geometry. If a acts on a manifold M , then Ca acts on differential forms on M . The same applies to supermanifolds and pseudo-differential forms (PDFs) on M . The elements of a ⊂ a ⊕ sa act as Lie derivatives. For each x ∈ a we denote L x the corresponding Lie derivative. The elements of sa ⊂ a ⊕ sa act as "contractions". For x ∈ a, the contraction will be denoted ι x . (We use angular brackets f x when f is a linear function, to highlight linear dependence on x.) In BV formalism, to every half-density ρ 1/2 satisfying the Quantum Master Equation corresponds a closed PDF on the space of Lagrangian submanifolds, which we denote Ω [1], [2]. Besides being closed, it satisfies the following very special property: where x ∈ a, a is the algebra of functions on the BV phase space, and ∆ is some differential on a, which is associated to the half-density ρ 1/2 . This form Ω is inhomogeneous, i.e. does not have a definite rank. It is, generally speaking, a pseudo-differential form. Otherwise, Eq. (2) would not make sense. We rederive Ω in Section 9.2. In the BV approach to string worldsheet theory, worldsheet diffeomorphisms are symmetries of ρ 1/2 , and therefore of Ω. Let g ⊂ a be the algebra of vector fields on the worldsheet. We are interested in constructing the g-equivariant version of Ω. Generally speaking, there is no good algorithm for constructing an equivariant PDF out of an invariant PDF. But in our case, since Ω satisfies Eq. (2), we can reduce the construction of equvariant form to the construction of a an embedding Dg → a -see Section 5.
The definition of Dg is similar to the definition of Cg. Essentially, we replace the commutative ideal sg ⊂ Cg with a free Lie superalgebra. Instead of defining the commutators to be zero, we only require that some linear combinations of commutators are d Dg -exact. In Section 3 we give the details of the construction, generalizing it to Lie superalgebras. We actually use, in our constructions, the cone of Dg, i.e. CDg. We study the "combined" construction CDg in Section 6.
In Section 11 we show that the map Dg → a can be used to convert unintegrated vertex operators into integrated vertex operators on the string worldsheet.

Notations
When a function f (x) depends on x linearly, we will write: to stress linearity. The cone of the Lie superalgebra g is: where → ⊕ stands for semidirect sum of Lie superalgebras, with arrow pointing towards the ideal, and s is suspension, of degree −1. (We consider g as a graded vector space, all having degree zero, then sg all has degree −1.) 2 Throughout the paper we will follow the notations of [15]. From a vector space V over a field K (for us K = R) we construct an algebra A, which consists of tensors of V modulo some quadratic relations R ⊂ V ⊗ V . The coalgebra A · | consists of tensors of sV , such that the tensor product of any One motivation for using the formalism of quadratic algebras is technical, as it automates keeping track of signs. We will translate into "more elementary" language of [3] in Section 3.3.2. PDF = pseudo-differential form.
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Definition of Dg
As far as we know, Dg was introduced in [3], in the context of current algebras. We will now present it in the language of quadratic algebras.

Definition of Dg
Let us consider a larger algebra: Here → ⊕ stands for semidirect sum of Lie superalgebras, with arrow pointing towards the ideal. The embedding of g into Dg as the first summand will be denoted l: Commutator of Dg • The commutator of two elements of FreeLie s −1 A · | is the commutator of free Lie algebra.
• The commutator of two elements of g is the commutator of g.
• The commutator of elements of g and elements of FreeLie s −1 A · | corresponds to the natural action of g on A · | .
When considering a commutator of an element of g and an element of FreeLie s −1 A · | , the following description is useful. Consider the U Cg -the universal enveloping algebra of Cg, and its dual coalgebra U Cg · | : Consider the projector p: which is identity on sg⊕A · | ⊂ T c (s(g ⊕ sg)) and zero on all tensors of rank ≥ 2 containing at least one sx ∈ sg. This induces a map from Ω 1 U Cg · | to Dg which we also denote p: For any Lie superalgebra a let γ denote the commutator map: In case of a = Dg, we can consider Dg ⊗ Dg as a subspace in Ω 2 (U Cg · | ) using the projector p of Eq. (14): Then, the commutator on Dg satisfies: There is a natural projection: annihilating all tensors with rank ≥ 1 (i.e. kerπ = A · | ≥1 ). We define a differential d ′ on g → ⊕ FreeLie s −1 A · | , in the following way.
• We postulate that the action of d ′ on g be zero: • Since FreeLie is a left adjoint to the forgetful functor, we just need to We put: We then extend Eqs. (17) and (18) to the differential d ′ of g We will now prove that d ′ anticommutes with d Ω : , by definition d ′ s −1 b = 0. We must therefore check that

Representation as vector fields
Consider the cone of our free Lie algebra: and its universal enveloping algebra U C. Eq. (8) implies: This is an equation in the completion of Hom A · | , U C , and exp(x) is understood as 1 + x + 1 2 x * x + . . .. Let M be some supermanifold, and Vect(M ) the algebra of vector fields on it. Suppose that we are given a map of linear spaces: Such a map defines a representation of C in the space of pseudo-differential forms (PDFs) on M . We want to project Eq. (22) on the space of PDFs on M . It is not possible to do directly, because we do not require that d Ω act on PDFs. Instead, consider the following version of Eq. (22):

Ghost fields
Remember that we start with A -the free super-commutative algebra generated by the linear superspace V = sg. Elements of A are symmetric tensors in V ⊗· · ·⊗V . (The relations of A, on the contrary, are antisymmetric Then, elements of A · | are symmetric tensors in sV ⊗ · · · ⊗ sV .

Example
Let a ∈ V, b ∈ V be even and ψ ∈ V, η ∈ V be odd. The following tensors belong to A: The bar construction is: The subspace A · | ⊂ BA is annihilated by d B because of relations of A. In particular, the following are elements of A · | : These are symmetric tensors in sV .
To summarize, if A is the algebra of symmetric tensors in V , then A · | is the coalgebra of symmetric tensors in sV .

Standard notations
For us V = sg, therefore A · | is the coalgebra of symmetric tensors in s 2 g.
Therefore, the space Hom A where ≃ means that we are not being rigorous. We ignore the question of which functions are allowed, i.e. do not explain the precise meaning of Fun(. . .).
Let {e a } denote some basis in s 2 g, and {F a } the dual basis in the space s −2 g * of linear functions on g: In this language: for T (a 1 ...an) any tensor symmetric in a 1 . . . a n . To agree with [3], we will denote: Here the notation l F agrees with Eq. (11).

Ghost number
In our notations, if X has ghost number n then sX has ghost number n − 1.
In other words, s lowers ghost number. In particular, the cone of the Lie superalgebra g is: The generator i a 1 ...an has ghost number −2n+1.

D ′ g
We need to also consider an extension of Dg, which we will call D ′ g, which is obtained by replacing FreeLie In the language of Section 3.3.2, we allow i(0) = 0, but require:

Ansatz for equivariant form
Suppose that the Lie superalgebra g is realized as vector fields on some supermanifold M . This means that pseudo-differential forms on M are a gdifferential module (= representation of (Cg, d Cg )). Consider the equation: We can write such an equations in any g-differential module W , not necessarily PDFs on M .
Now suppose that W is a D ′ g-differential module. Since g is embedded into D ′ g, we can still write the Cartan Eq. (33): where we use the notations of Eqs. (11), (30). Then, given any d-closed form ω, consider the following anstaz for a solution of Eq. (33): where i(F ) is from Eq. (29). We will show that under certain conditions this substitution solves Eq. (33).

D ′ g-differential modules
Suppose that W is a D ′ g-differential module. This means that W is a representation of the Lie superalgebra CD ′ g, with the differential d W which agrees with the differential d CD ′ g of CD ′ g. (The differential of D ′ g, which we denote d D ′ g , does not participate in these definitions, but will play its role.) For every x ∈ D ′ g we denote L x and ι x the corresponding elements of CD ′ g, and L W x and ι W x their action in W . With the notations of Eqs. (11) and (30), Eq. (24) implies, for all ω ∈ W : Let us consider Eq. (42) in the special case when ω satisfies: for all x ∈ D ′ g. Then: Eq. (38) is a special requirement on W and ω. It is by no means automatic. Intuitively, it may be understood as an interplay between d W and d D ′ g (and ω): We do not requite that d D ′ g act in W . Instead, we want Eqs. (37) and (38) (or, equivalently, Eqs. (37) and (40)).
We will now consider two examples of D ′ g-differential modules.
6.2 Pseudo-differential forms (PDF) Suppose that a supermanifold M comes with an infinitesimal action of D ′ g, i.e. a homomorphism: This is only a homomorphism of Lie superalgebras; we forget, for now, about the differential d D ′ g . We denote d M the deRham differential on M , and F = F a e a . Then Eq. (24) implies Let us consider Eq. (42) in the special case when ω is closed: Consider a linear subspace X ω ⊂ Vect(M ) consisting of all vectors v such that exits some other vector d ω v ∈ X ω satisfying: Suppose that ω is "non-degenerate" in the sense that the map from VectM to PDFs on M given by v → ι v ω is injective. Then Eq. (44) defines an odd nilpotent operator: Moreover, X ω is closed under the operation of commutator of vector fields. Indeed: Therefore (X ω , d {ω} ) is a differential Lie superalgebra. Suppose that: Then Eq. (42) implies that: In other words, ω C defined by the equation: is a cocycle in the Cartan's model of equivariant cohomology. Notice that apriori there is no action of d D ′ g on M , and we have never used it.

Special cocycles
Similarly, suppose that D ′ g is mapped into some Lie superalgebra a: and W is a representation of a. Consider the Chevalley-Eilenberg cochain complex C(a, W ) of a with coefficients in W . The cone Ca acts on C(a, W ); for each x ∈ a we denote L x and ι x the action of the corresponding elements of Ca. Eq. (42) still holds, now ω is a cochain: We are allowing arbitrary dependence of cochains on the ghosts of a, not only polynomials. We will say that a cocycle ω is special if exists d {ω} satisfying Eq. (44) for all v ∈ im(r). Moreover, we require: 6.4 A procedure for constructing r We will now describe a procedure for constructing an embedding: This is not really an "algorithm" because, as we will see, it may fail at any step. Suppose that we can choose, for each a ∈ {1, . . . , dimg}, some φ a ∈ a so that exist φ ab ∈ a such that: where d = d ω and f ab c the structure constants of g. Then verify the existence of φ abc such that: The mutual consistency of these two equations follows from Eq. (19). Then continue this procedure order by order in the number of indices: [φ a 1 , φ a 2 ...an ] + . . . + [φ a 1 ...a n−1 , φ an ] = dφ a 1 ...an [φ a 1 ...a n−1 , dφ b ] = f a 1 b c φ ca 2 ...a n−1 + . . .

Chevalley-Eilenberg complex of a differential module
In this Section, there is no Dg nor D ′ g. We forget about them for now. As a preparation for BV formalism, we will now discuss yet another formula of Kalkman type. Consider a Lie superalgebra a and its cone Ca, which is generated by L x and ι x , where x ∈ a. We consider the quadratic-linear dual coalgebra U a · | . The dual space U a · | * = Hom U a · | , K is the algebra of functions of the "ghost variables" c A . The universal twisting morphism is: We will study the properties of the following operator: Since a is quadratic-linear, U a · | comes with the differential d is the Chevalley-Eilenberg differential d is the "internal" differential of U a · | ; it comes from U a being inhomogenous (i.e. quadratic-linear and not purely quadratic algebra). The Chevalley-Eilenberg complex C • (a, W ) with coefficients in W can be defined for any representation a-module W . Consider the special case when W is a a-differential module W , i.e. a representation of (Ca, d Ca ). We will denote L W x , ι W x and d W Ca the elements of End(W ) representing elements L x , ι x of Ca and d Ca . (Then W is also a representation of a, where x ∈ a is represented by L W x .) Consider the Lie algebra cochain complex (= Chevalley-Eilenberg complex) of a (not Ca) with coefficients in W . The differential is defined as follows: All this can be defined for any a-module W . But when W is also a adifferential module (i.e. a represenatation of (Ca, d Ca )), then d CE and d (0) CE are related: Moreover, since W and U a · | are both Ca-modules, we can consider Hom(U a · | , W ) a Ca-module, as a Hom of two Ca-modules: x In other words, both (L , ι Then operation I • e ι W •α intertwines this subspace with the Chevalley-Eilenberg complex C • (g, W): Proof follows from Eq. (53). Therefore every w ∈ W defines an inhomogeneous Chevalley-Eilenberg cochain of g with coefficients in W: The map I •e ι W •α intertwines the action of Cg on ker d  ) of Proposition 1. (This action does not use ι W x , in fact there is no such thing as ι W x . Our W, unlike W , is just a g-module, not a differential g-module.)

Mapping to PDFs
Suppose that W happens to be a space of functions on some manifold M with an action of G (the Lie group corresponding to g). In this case, every w ∈ W and a point m ∈ M defines a closed PDF on G, in the following way: We will be mostly interested in the cases when this PDF descends on the G-orbit of m. For example, consider the case when W is the space of PDFs on M (the same M ) and I is the restriction of a PDF on the zero section M ⊂ ΠT M . (Remember that PDFs are functions on ΠT M . In this example, the operation I associates to every form its 0-form component.) In this case, given a PDF on M , e.g. f µ (x)dx µ , our procedure, for each x ∈ M , associates to it a PDF on G, which is just f µ (g.x)d(g.x) µ . If G acts freely, Ω w will descend to a form on the orbit of x. This is just the restriction to the orbit of the original form we started with.
As another example, consider g the Lie algebra of vector fields on some manifold N , and W the space of PDFs on N . Let M be the space of orientable p-dimensional submanifolds of N , and I the operation of integration over such a submanifold. Our construction maps closed forms on N to closed forms on M .

PDFs from representations of Dg
An analogue of Eq. (53) holds for Dg. It follows as a particular case from the results of [3]: where Φ is: Therefore, when W is a representation of Dg, we have an analogue of Eq.

BV
We will now apply the technique developed in the previous sections to the BV formalism. Let a be the Lie algebra of functions on the BV phase space with flipped statistics. Its elements are s −1 f where f is a function on the BV phase space and s the suspension: The Lie bracket is given by the odd Poisson bracket.

Half-densities as a representation of Ca
The space of half-densities on the BV phase space is a representation of Ca : We are now in the context of Section 8.1. Now g is a, W is the space of half-densities, d W Cg is −∆ can and W is Fun(LAG) -the space of functions on Lagrangian submanifolds.

Correlation functions as a Lie superalgebra cocycle
Correlation function defines a linear map: As in Section 6.3, suppose that r is an embedding of D ′ g in a. Eq. (49) becomes (cp Eqs. (29) and (30)): where i(F ) and l F were defined in Eqs. (29) and (30) Then, equivariantly closed cocycle in the Cartan model is given by: Our notations here differ from our previous papers; r i(F ) was called Φ(F ) in Section 4 of [1] and a(F ) in Section 6 of [2]. Here is the summary of notations:
Consider the deformations of the embedding r : D ′ g → a keeping r l F fixed. 3 Eq. (65) implies that a small variation δr i(F ) satisfies: Those δr which are in the image of ∆ ρ 1/2 + [r i(F ) , ] correspond to trivial deformations. This means that the cohomologies of the operator d Dg + [i(F ), ], considered in [3], in our context compute infinitesimal deformations of the equivariant half-density.
11 Integrating unintegrated vertices 11.1 Integration prescription using Cg Let us fix i(0) = 0, and consider r : Dg → a. In our discussion in Section 10.2, we assumed that deformations preserve the symmetries of ω, i.e. that f is g-invariant. In string theory, it is useful to consider more general deformations breaking g down to a smaller subalgebra g 0 . They are called "unintegrated vertex operators". As their name suggests, g-invariant deformations can be obtained by integration over the orbits of g. The procedure of integration was described in [2]. It is a particular case of Section 8.1, where M is now LAG -the space of Lagrangian submanifolds, and I the operation of integration of half-density over a Lagrangian submanifold. For this construction, we do not need the full r : Dg → a, but only its restriction on g ⊂ a: l : g → Dg For v ∈ a, consider the deformations of ω of the following form: In terms of half-densities: where v denotes (as in [2]) the BV Hamiltonian generating v.
The cone Cg acts on such deformations δρ 1/2 ; the action ofι x ,L x and d Cg is:ι Therefore the construction of Section 8.1, with M = LAG and I δω = L → L∈LAG f ρ 1/2 , gives a closed form on G for every deformation of the form Eq. (68): This is just a particular case of the general construction of Section 9.2, Eqs.
(63), (64). We restrict the general construction of Ω from all LAG to an orbit of G. In other words, we consider not all odd canonical transformations, but only a subgroup G. But now we can use G-invariance of ρ 1/2 to pull back to a fixed L: However Eq. (71) is somewhat unsatisfactory. Although it is, actually, the integrated vertex corresponding to v, this form of presenting it makes it apparently nonlocal on the string worldsheet. We would want, instead, to replace, roughly speaking, ιL with Lι: We will now explain the construction.

Deformations as a representation of Dg
In deriving Eq. (71) we have not actually used the representation of Dg, but only the representation of g ⊂ Dg; we have only used l dgg −1 and never i(dgg −1 ). Notice, however, that the whole Dg acts on deformations. (This is, ultimately, due to our requirement of ρ 1/2 being "equivariantizeable", Section 10.) Moreover, if we do not care about d Dg , then there are two ways of defining the action of just Dg. (It is easy to construct representations of free algebras.) The first way is to use the embedding Dg L → CDg. But this one does not define the action of d Dg .
There is, however, the second way, which defines the action of Dg with its differential d Dg . For δω = ι v ω or equivalently δρ 1/2 = vρ 1/2 , we define: To summarize, the space of deformations of ω can be considered as a representation of (CDg, d CDg ), or as a representation of (Dg, d Dg ). (But not of (CDg, d CDg , d Dg ) whatever that would be.) In both cases, the differential acts as ∆. That is to say, the d CDg of (CDg, d CDg ) acts as ∆, and the d Dg of (Dg, d Dg ) also acts as ∆ -see Eqs. (70) and (73), respectively.

Relation between two integration procedures
In the special case when Dg reduces to Cg (i.e. i a 1 ...an = 0 for n > 1), it was found in [2], to the second order in the expansion in powers of dgg −1 , that the two PDFs are different by an exact PDF on G. It must be true in general. We do not know the equivariant version of Eq. (74). In the pure spinor formalism, it is very likely that the form given by Eq. (74) is already base, because unintegrated vertex operator does not contain derivatives [16].

Is integration
Notice that the PDF defined in Eq. (74) does not, generally, speaking, descend to the orbit of L. In computing the average, the integration variable is g, not gL. However, the integral does not depend on the choice of L in the orbit.