{'fraction_non_alphanumeric': 0.054062516307467515, 'fraction_numerical': 0.012471951155873297, 'mean_word_length': 3.7482656095143705, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Insertion of vertex operators using BV formalism

We develop a general framework for the insertion of vertex operator on the string worldsheet, in BV formalism. Such insertions correspond to deformations of the Master Action which breaks the gauge symmetry to a subgroup, and then restoring the full gauge symmetry by integrating over a cycle in the space of Lagrangian submanifolds. We provide the general construction, global on the moduli space, which was previously conjectured in a form local on the worldsheet. We explain how the enhancement of the gauge symmetry in equivariant BV formalism can be seen as an application of the general idea of BV effective action. We derive an integral formula for the deformation of the contraction operator due to the vertex insertion.


Introduction
String amplitudes are computed in perturbation theory by integrating the CFT correlators of vertex operators over the moduli space of Riemann surfaces. A mathematical abstraction for the string measure within the BV formalism was outlined in [1] and further developed in [2], [3]. An important role is played by the equivariant BV formalism developed in [4], [5], [6], [7], [8], [9], [10], [11]. In particular, the relation to some differential graded Lie superalgebras introduced in [12] was discussed in [13].
In this paper we will continue the study of equivariant BV in the context of string measure. We will mostly concentrate on the "covariant insertion" of vertex operators on the string worldsheet [14], [15]. Our approach emphasizes the gauge symmetries of the string sigma-model, which include worldsheet diffeomorphisms. In BV formalism, gauge symmetries are exact in the following sense. If ξ is an infinitesimal gauge symmetry, then its BV Hamiltonian ξ is: where ∆ is the BV Odd Laplace operator (roughly speaking it is the odd Poisson bracket with the BV Master Action, {S BV , }) and i 1 ξ some function on the BV phase space linearly dependent on ξ as a parameter. For example, in the case of bosonic string, ξ is a vector field on the string worldsheet Σ, and i 1 is where c ⋆ is the antifield of the c-ghost. Physical states of the bosonic string correspond to the ghost number two vertex operators. For example, the tachyon vertex is: This is inserted at a point on the string worldsheet, but one should also integrate over the position of the insertion. The integration procedure involves "stripping" the c-ghosts: where the integrated vertex operator (a two-form on Σ) is defined as follows: When we insert the "unintegrated" vertex operator V , the diffeomorphism symmetry gets broken down to those maps which preserve the point of insertion. Then, when we integrate over the worldsheet metrics, we effectively integrate over the position of the insertion point, and the full diffeomorphism symmetry is restored. The relation between Eq. (5) and Eq. (2) can be generalized to other string theories where Eq. (2) is more complicated: (We use angular brackets around the argument of a function, when the function is linear.) This was previously discussed in [13], but the consideration was local on the space of metrics, and the analogue of Eq. (5) was only conjectured. Here we derive the "global" relation combining the "vertical" integration over the position of the insertion point with the integration over the moduli space of Riemann surfaces. We do not know explicit examples of string theories where i ≥2 is nonzero, but we suspect that this is what happens in the pure spinor formalism (see Section 8). One example with nonzero i 2 was recently discussed (not for a string sigma-model) in [16].

Half-densities and their deformations 2.1 Odd Laplace operator
Let M be a supermanifold with an odd symplectic structure. We call it the "BV phase space". Let a be a Lie superalgebra of functions on M , with opposite statistics, the Lie operation being the BV Poisson bracket {, }.
In Darboux coordinates: Half-density ρ 1/2 on M is a scalar function transforming under diffeomorphisms of M as a squareroot of volume element. There is a canonical second order differential operator ∆ can acting on densities [17]. A half-density ρ 1/2 is said to satisfy the Quantum Master Equation (QME) if: A small deformation of ρ 1/2 can be described by a function f on M : If ρ 1/2 satisfies QME, the deformation satisfies QME iff: The operator ∆ ρ 1/2 acts on functions. It does depend on ρ 1/2 , but its leading symbol does not depend on ρ 1/2 . In other words, if ρ 1/2 and ρ 1/2 are two half-densities, then ∆ ρ (1) is a first order differential operator. In Darboux coordinates (see Eq. (7)): In physics notations:

Deformation complex
The cohomology of ∆ ρ 1/2 corresponds to deformations of ρ 1/2 preserving the QME, modulo trivial deformations. (Trivial deformations are those which can be undone with infinitesimal canonical transformations.) Eq. (10) associates to every half-density ρ 1/2 a nilpotent second order differential operator with the leading symbol ∆ (0) , which is determined by the odd Poisson structure. (The leading symbol is the odd Poisson structure.) In fact, this correspondence is one-to-one up to multiplicative constant.
For each nilpotent second order differential operator with leading symbol ∆ (0) exists a half-density ρ 1/2 such that this operator is ∆ ρ 1/2 . Therefore there is one-to-one correspondence between deformation complexes and half-densities. In some situations, we can "forget" about ∆ (0) . Then, the differential of the deformation complex is d = {S BV , }, and the half-density is e S BV . If a first order differential operator X preserves the odd Poisson bracket, then we can find the corresponding BV Hamiltonian X such that X = {X, }. Then the correspondence is: The correspondence between deformation complex and half-density is "exponentiation of the BV Hamiltonian of differential". This is only true roughly speaking, because actually X is a second order differential operator (not a vector field) and ρ is a half-density (not a function). However, locally, in Darboux coordinates, we can indeed say: We will use this "physics" notations throughout this paper, remembering that this e S BV is a halfdensity.
Half-densities (just as functions) can be pulled-back by diffeomorphisms. In case of a function f and a diffeomorphism g, the pull-back is a composition, therefore it is natural to denote it f • g. We will use the same notation for the pull-back of a half-density: If e S BV satisfies QME, and g is a canonical transformation deformable to the identity transformation, then: 3 DGLA Dg (now Dh) and equivariant BV

Cone and Weil algebra
Let h be a Lie superalgebra, Ch be the cone of h, and W h be the Weil algebra of h. (See e.g. [18] for the definitions and general .) The cone Ch is generated by L x and ι x where x ∈ h. We use angular brackets . . . instead of the usual (. . .) to emphasize linear dependence of a function on its argument. The generators of W h will be denoted θ a and t a , and its differential will be denoted d W : The action of Ch on W h is such that ι x acts as x a ∂ ∂θ a .

Definition of Dh
We use the DGLA Dh as it was defined in [12]. It is generated by l a and i a 1 ...an where n ∈ {1, 2, . . .}.
The l a generate h. The i a 1 ...an are symmetric in a 1 , . . . , a n ; they generate a free Lie superalgebra. We denote: The differential acts as follows: The characteristic property of the construction h → Dh is: • For any linear space V , to define an action of Ch on W h ⊗ V is the same as to define the action of Dh on V .
More precisely, we must assume that ι x acts on W h ⊗V as x a ∂ ∂θ a and L x acts as f a bc (17):

Action of Dh on BV phase space
Let us take V = a, where a is the Lie superalgebra of functions on BV phase space M , and require that: • l a and i a 1 ...an act by infinitesimal canonical transformations, generated by some BV Hamiltonians l a and i a 1 ...an • The differential d Dh acts as a second order differential operator, with the leading symbol defined by the odd Poisson bivector (the leading symbol of the odd Laplace operator) Then we use the "exponentiation method", Eq. (12), to obtain the equivariant half-density, Eq. (21). Eq. (17) becomes: where The action of ι x for x ∈ h is: We must stress that the action of Ch on a is not defined. We only have a representation of Dh (and the differential d Dh ) on a, and this already allows us to define (following [12]) the action of Ch on W h ⊗ a. In a special case when i a 1 ...an vanishes for n > 1, we do have the action of Ch, with its differential, on a. In this case Eqs (18) and (19) may be called "Cartan model of equivariant cohomology of a".

Action of Ch on half-densities
Although we might not have the action of Ch on a, we do have the action of Ch on half-densities Dens 1/2 (M ). Indeed, L x acts as the Lie derivative along l a x a , and ι x acts as the multiplication by l a x a . The differential of Ch acts as ∆ can . Therefore we can consider Weil (or Cartan) model of Dens 1/2 (M ).

Equivariant half-density
It turns out that the exponentiation method (see Eq. (12)) brings r W h ⊗a d Ch to a cocycle of the Weil model W h ⊗ Dens 1/2 (M ). As in Eq. (12), the equivariant Master Density ρ 1/2 (A, F ) corresponding to r W h ⊗a d Ch is defined as the unique half-density ρ 1/2 (θ, t) such that: In Darboux coordinates, the correspondence d ↔ ρ 1/2 is, literally, exponentiation: It satisfies:

Cartan model and Weil model
Notice that ρ 1/2 (θ, t) is basic in the Weil model: (remember that Ch acts on half-densities, ι x acts as multiplication by x, in this case multiplication by l a ). This is counter-intuitive, as r W h ⊗a d Ch (defined in Eq. (18)) is in the Cartan model. Indeed, an infinitesimal deformation of Therefore ι W h ⊗a l a acts on deformations as in the Cartan model: The equivariant half-density in the Cartan model is: It satisfies:

Deformation complex
Suppose that ρ 1/2 (θ, t) satisfies equivariant Master Equation, and is Ch-invariant. Let f (θ, t) be a function on M depending on θ and t. A deformation of ρ 1/2 (θ, t) of the form: preserves the equivariant Master Equation iff: where ∆ = ∆ ρ 1/2 (0,0) . Moreover, the deformation is trivial if f is in the image of d W +∆+θ a {l a , }− {i(t), }. As we also impose the Ch-invariance on ρ 1/2 , we must require: In this case we write f (t) instead of f (θ, t).

Integration over families of Lagrangian submanifolds
Let us fix a Lagrangian submanifold L, and consider all Lagrangian submanifolds which can be obtained from L by canonical transformations. This is an "infinite-dimensional supermanifold". We will call it LAG: Here G is the group of canonical transformations of M . Consider the following pseudo-differential form (PDF) on LAG: This is a closed form. This can be seen by rewriting it equivalently as follows: The h-equivariant analogue of Ω is: The construction of Ω, as we now presented it, has the following defect. We want to define a PDF on LAG, not on G. Therefore g is only defined up to g ∼ gg 0 where g 0 L = L. Therefore the restriction of g −1 dg on L is only defined up to a constant. This can be remedied in a number of ways, as described in [2], [3]. Here we will assume that there is a ghost number symmetry, and consider only those Lagrangian submanifolds which are invariant under the ghost number symmetry. Then g −1 dg has ghost number −1, and cannot be constant. We will use g to parametrize LAG, keeping an eye on g ∼ gg 0 .

Some lagrangian submanifolds in ΠT * ΠT Y
Let us consider a PDF α on a supermanifold Y . Let d ∈ Fun(ΠT * (ΠT Y )) be the generating function of the de Rham differential d (considered as an odd vector field on ΠT Y ).
For an oriented submanifold X ⊂ Y , we can define X α as L X e d α where L X ⊂ ΠT * (ΠT Y ) is the conormal bundle to ΠT Y | X ⊂ ΠT Y . In local coordinates, L X can be described as follows. We let y ∈ Y and its antifield y ⋆ run in the conormal bundle of X ⊂ Y . At the same time, dy runs free and (dy) ⋆ = 0.
To summarize:

Some canonical transformations
Let us consider, in this context, canonical transformations. For any β ∈ Fun(ΠT Y ), we can consider β to be a function on ΠT * ΠT Y (as the pullback under the projection) and compute the Lie derivative of the half-density e d α under its Hamiltonian flow: (This flow deforms L X away from (dy) ⋆ = 0, showing that there is nothing we can get from this construction, beyond an integral of a differential form.) For a vector field v ∈ Vect(Y ), let L v be the Lie derivative (a vector field on ΠT Y ), and L v its BV Hamiltonian. The Lie derivative of the half-density along {L v , } is:

Universal canonical transformation
Let us consider the extended BV phase space: where G is the group of canonical transformations of M (or perhaps a subgroup of it). We define the universal canonical transformation Γ as follows: where α is such that: (Remember that ξ denotes the BV Hamiltonian of a vector field ξ. The left-invariant form g −1 dg is a map ΠT G → Vect(M ), therefore g −1 dg is a function on ΠT G × M .) The Ω of Eq. (30) can be interpreted as a half denisty on M 6 Restriction of gauge symmetry to a subgroup 6

.1 General story
Consider a manifold X with an action of the group H, and a subgroup H 0 ⊂ H. The Weil algebra of h 0 is the factoralgebra of the Weil algebra of h by a differential ideal: Let Ω h (θ, t) be an h-equivariantly closed form on X, and Ω h 0 (θ 0 , t 0 ) its restriction on h 0 ⊂ h: In terms of connections and base forms, suppose that: To see this, let us introduce a parameter τ and the following family of h 0 -connections on X × R τ and their curvatures:â We observe thatâ andf satisfy the Weil algebra and we can substitute them for θ and t, obtaining a closed form: In particular: This implies Eq. (42).
We will now develop this idea in the BV context.

Varying the gauge group in BV formalism
Let a 0 be a connection in the H 0 -principal bundle LAG −→ H 0 \LAG: We will start by using the trick of Section 5 to replace Ω with the half-density on (ΠT * ΠT G) × M : This defines ρ as a half-density on (ΠT * ΠT LAG) × M . But LAG does not have closed cycles. We use a closed cycle in H 0 \LAG, not in LAG. Therefore, we need a half-density on (ΠT * ΠT (H 0 \LAG))×M . Indeed, ρ can be considered as a half-density on (ΠT * ΠT (H 0 \LAG))×M , by the BV Hamiltonian reduction. We first put to zero the BV Hamiltonians generating the action of ΠT H 0 (a constraint on g ⋆ and [dg] ⋆ ), and then factor out by this action. This is possible because ρ is H 0 -base as a PDF on LAG. Let us introduce an extra parameter τ ∈ R and denote: where the differential d in da ′ includes dτ ∂ ∂τ . We will now use the following general construction. Consider an odd symplectic manifold X and add a pair of Darboux conjugate coordinates τ and τ ⋆ . We get the new odd symplectic manifold: For any function F ∈ Fun(X × R τ ), there is the canonical transformation: where γ τ is a family of canonical transformations of M , parameterized by τ , defined from the equations: More explicitly: This construction is a special case of Section 5.3, namely the restriction from G to a curve {γ τ |τ ∈ R} ⊂ G.
Indeed, a ′ (τ ) is an h 0 -connection for all values of τ , and therefore F • g is h 0 -base as a PDF on G.
Therefore it descends to a PDF on H 0 \G. Moreover, it can be considered a PDF on H 0 \LAG, since a ′ and f ′ are defined as forms on LAG (see Eq. (45)). We observe that: Therefore exp S BV − l a ′ + i(f ′ ) + d G×Rτ • Φ −1 • Γ contains dτ and τ ⋆ only through the multiplicative factor e dτ τ * . Then QME implies: Restricting to dτ = 0 and taking τ = 0 then τ = 1, this implies: On the other hand: where ∆ can is the canonical operator on ΠT * ΠT (H 0 \G) × M . Therefore γ −1 1 can be replaced with id, in the following sense:

Unintegrated and integrated vertices
Suppose that V ∈ Fun(M ) satisfies the following conditions: Then, the product: satisfies the h 0 -equivariant Master Equation. This may be understood as the first derivative of the measure with the S BV deformed to S BV + ǫV w.r.to ǫ at ǫ = 0. Eq. (56) implies: Similarly to Eq. (58): Given an integration cycle C ⊂ H 0 \LAG, where L C ⊂ ΠT * ΠT (H 0 \LAG) is the Lagrangian submanifold corresponding to C by the construction of Section 5. Going back to the language of differential forms: The Each g ∈ H 0 \LAG can be written as a product: With this local trivialization, the h-connection a can be written as follows:

Unperturbed measure in terms of local trivialization
Eq. (32) becomes: Gauge invariance is manifested by the h-independence of this measure.

Local integrated vertex
Locally on LAG we can choose connection so that A = 0. Then: This is a PDF in H 0 \H, and has to be integrated over a closed cycle in H 0 \H. The result is the integrated vertex operator corresponding to V : It satisfies the Master equation: We will now give a direct proof of this. The restriction of Ω h 0 −base V to H 0 \HL ⊂ H 0 \LAG is: We know by construction this is a closed form, i.e. annihilated by d H 0 \H . Let us consider the halfdensity on (ΠT * ΠT (H 0 \H)) × M corresponding to Ω h 0 −base V | H 0 \HL by the construction of Section 5. It is equal to: This satisfies the Master Equation by construction. As γ 1 is a canonical transformation, and Γ −1 • γ 1 • Γ is a canonical transformation, the following half-density on (ΠT * ΠT (H 0 \H)) × M also satisfies the Master Equation: Then pick a Lagrangian submanifold L c ∈ ΠT * ΠT (H 0 \H) corresponding to a closed cycle c ⊂ H 0 \H and integrate over it, as a half-density on ΠT * ΠT (H 0 \H). This is essentially a BV push-forward:

Computation from the defining equation
We have to find δ V i(t) such that: Here we will restrict ourselves with the case when dim(H 0 \H) = 1. The integration is onedimensional, let u denote the coordinate on H 0 \H. Then the integrated vertex is: Given a t ∈ h such that t = 0 mod h 0 , let us defineÃ u so that: We can expand i(t) in powers of t: We claim that the variation of i from the insertion of U equals to: where • means symmetrized tensor product: This agrees with Eq. (67).

Some observations
The case dim(H 0 \H) = 1 has the following property. If i(t) is a polynomial of degree n, it will remain a polynomial of degree n after the deformation. Moreover, the highest order term i n t ⊗n does not receive correction. Suppose that i n = 0 for n > 1, i.e. i(t) is linear in t. Then δ V i(t) = 0, even when dim(H 0 \H) = 1. This can be seen from Eq. (67). The curvatures would all come from the F ′ in Eq. (67), but they only enter through i n≥2 . At the same time, in this case {U, i t } = 0, and therefore no need to correct i(t). This observation might have an implication for the pure spinor superstring. Although the BV formalism for the pure spinor superstring has not been fully developed (only the i 1 has been constructed, in [19], for the AdS background), we do have some hints. The i 1 must be related to OPE with the b-ghost. The pure spinor b-ghost (unlike the bosonic string b-ghost) does receive corrections when we deform the background [20], [21]. This seems to indicate that higher i n are nonzero in the pure spinor formalism.