Introduction

BV formalism is a generalization of the BRST formalism, based on the mathematical theory of odd symplectic supermanifolds. In this formalism the path integral is interpreted as an integral of a density of weight over a Lagrangian submanifold. It turns out that this “standard” formulation is not sufficient to describe string worldsheet theory. One has to also consider integration over families of Lagrangian submanifolds. Indeed, the idea of Schwarz:2000ct,Mikhailov:2016myt,Mikhailov:2016rkp was to interpret integration over the worldsheet metrics as a particular case of integration over the space of gauge fixing conditions. Varying the worldsheet metric is a particular case of varying the Lagrangian submanifold. Taking into account the worldsheet diffeomorphism invariance requires an equivariant version of this integration procedure. (In a sense, worldsheet metric is not necessarily a preferred, or “special”, object. Varying the worldsheet metric is just one way to build an integration cycle, there are others. The worldsheet diffeomorphisms, however, are special.)

The construction of equivariant form involves a map of some differential graded Lie algebra (DGLA) into the algebra of functions on the BV phase space of the string sigma-model. To the best of our knowledge, was first introduced, or at least clearly presented, in Alekseev:2010gr. Here we will rederive some constructions of Mikhailov:2016myt,Mikhailov:2016rkp using an algebraic language which emphasizes the DGLA structure, and apply some results of Alekseev:2010gr to the study of worldsheet vertex operators. In a sense, is a “universal structure” in equivariant BV formalism, i.e. the “worst-case scenario” in terms of complexity. The construction of is a generalization of the construction of the “cone” superalgebra (which is called “supersymmetrized Lie superalgebra” in Cordes:1994fc).

We will now briefly outline these constructions, and the results of the present paper.

The cone of Lie superalgebra

For every Lie superalgebra

, we can define a DGLA

(the “cone” of

) as follows. We consider vector superspace

as a graded vector space, such that the grade of all elements is zero. Then, we denote

the vector space

with flipped statistics at degree

In our conventions, “grade” corresponds to the “ghost number”; statistics is not grade mod 2.

Consider a graded vector space:

where

is at grade zero, and

at grade

. (The letter

means “suspension”, the standard terminology in linear algebra.)

The commutator is defined as follows. The commutator of two elements of is the commutator of , the commutator of two elements of is zero, is an ideal, the action of on corresponds to the adjoint representation of . The differential is zero on and maps elements of to the elements of , i.e.: .

This construction has an important application in differential geometry. If

acts on a manifold

, then

acts on differential forms on

. The same applies to supermanifolds and pseudo-differential forms (PDFs) on

. The elements of

act as Lie derivatives. For each

we denote

the corresponding Lie derivative. The elements of

act as “contractions”. For

, the contraction will be denoted

. (We use angular brackets

when

is a linear function, to highlight linear dependence on

.) We have:

(1)

The definition of

is similar to the definition of

. Essentially, we replace the commutative ideal

with a free Lie superalgebra of the linear space

where

is the space of symmetric tensors of

. Instead of defining the commutators to be zero, we only require that some linear combinations of commutators are

-exact. Eq. (1) is replaced with:

(2)

In particular, if

is a linear function of

, then

becomes

. In this case,

becomes

and

becomes

. In general,

is a nonlinear function of

(but

remains linear). In 〚

〛 we explain the details of the construction, and why it is natural. We slightly generalize it, by allowing

to be a Lie superalgebra (while in Alekseev:2010gr it was a Lie algebra).

String measure

In BV formalism, to every half-density

satisfying the Quantum Master Equation corresponds a closed PDF on the space of Lagrangian submanifolds, which we denote

Mikhailov:2016myt,Mikhailov:2016rkp. Besides being closed, it satisfies the following very special property:

(3)

where

is the algebra of infinitesimal odd canonical transformations, and

is some differential on

, which is associated to the half-density

. This form

is inhomogeneous, i.e. does not have a definite rank. It is, generally speaking, a pseudo-differential form (PDF). Otherwise, Eq. (3) would not make sense. We rederive

and explain its meaning as a Lie superalgebra cocycle in 〚Correlation functions as a Lie superalgebra cocycle〛.

Equivariant string measure

Let be the algebra of vector fields on the worldsheet. In the BV approach to string worldsheet theory, worldsheet diffeomorphisms are symmetries of , and therefore is -invariant. We are interested in constructing the -equivariant version of . Generally speaking, there is no good algorithm for constructing an equivariant PDF out of an invariant PDF. But in our case, since satisfies Eq. (3), we can reduce the construction of equvariant form to the construction of an embedding — see 〚Ansatz for equivariant form〛.

Vertex operators

Consider deformations of . In string theory context they are called “vertex operators”. It is useful to consider deformations which break some of the symmetries. Typically, we insert some operators at some points on the wordsheet, breaking the diffeomorphisms down to the subgroup preserving that set of points. This is the “unintegrated vertex operator”. Then, there exists an averaging procedure which restores the symmetry group back to all diffeomorphisms. The result of this averaging is effectively an insertion of “integrated vertex operator” which preserves all the diffeomorphisms. This relation between unintegrated and integrated vertex operators is important in string theory.

As we show in 〚Integrating unintegrated vertices〛, this averaging procedure requires an action of

. Just to define the action of symmetries, we only need

. But the averaging procedure, which is needed to compute string amplitudes, does involve

. In previously studied cases, such as bosonic or NSR string,

reduces to

, and

is a very simple expression. It is basically the contraction of

with the ghost antifield,

using the notations of Mikhailov:2016rkp (see also 〚Ghost number〛). The averaging procedure consists in this case of simply removing the ghost fields from the vertex, and then integrating over the insertion point. In 〚Integrating unintegrated vertices〛 we derive the general formula, which is rather nontrivial and uses some intertwining operator constructed in Alekseev:2010gr.

At this time, we do not have concrete examples of string worldsheet theories where would not enter only through the projection to . It is likely that pure spinor superstring in AdS background is an example, but we only have a partial construction Mikhailov:2017mdo.

Previous work

Previous work on equivariant BV formalism includes Nersessian:1993me, Nersessian:1993eq, Nersessian:1995yt, Getzler:2015jrr, Getzler:2016fek, Cattaneo:2016zrn, Getzler:2018sbh, Bonechi:2019dqk. Similar algebraic structures appeared recently (in a different context?) in Bonezzi:2019bek and references therein.

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