Integrating unintegrated vertices

    Integration prescription using
    Integration using
        Deformations as a representation of
        Averaging procedure using
    Relation between two integration procedures
    Is integration form base with respect to ?

Integration prescription using

Let us fix , and consider . In our discussion in 〚Deformations〛, we assumed that deformations preserve the symmetries of , i.e. that is -invariant. In string theory, it is useful to consider more general deformations breaking down to a smaller subalgebra . They are called “unintegrated vertex operators”. As their name suggests, -invariant deformations can be obtained by integration over the orbits of . The procedure of integration was described in Mikhailov:2016rkp. It is a particular case of 〚PDFs from representations of 〛, where is now — the space of Lagrangian submanifolds, and the operation of integration of half-density over a Lagrangian submanifold.

For this construction, we do not need the full , but only its restriction on :

For , consider the deformations of of the following form:

where denotes (as in Mikhailov:2016rkp) the BV Hamiltonian generating .

The cone acts on such deformations ; the action of , and is:

Therefore the construction of 〚PDFs from representations of 〛, with and , gives a closed form on for every deformation of the form Eq. (34):

This is just a particular case of the general construction of 〚Correlation functions as a Lie superalgebra cocycle〛, Eqs. (30), (31). We restrict the general construction of from all to an orbit of . In other words, we consider not all odd canonical transformations, but only a subgroup . But now we can use -invariance of to pull back to a fixed :

However Eq. (36) is somewhat unsatisfactory. Although it is, actually, the integrated vertex corresponding to , this form of presenting it makes it apparently nonlocal on the string worldsheet. We would want, instead, to replace, roughly speaking, with :

Integration using

Deformations as a representation of

In deriving Eq. (36) we have not actually used the representation of , but only the representation of ; we have only used and never . Notice, however, that the whole acts on deformations. (This is, ultimately, due to our requirement of being “equivariantizeable”, 〚Equivariant BV formalism〛.) Moreover, if we do not care about , then there are two ways of defining the action of just . (It is easy to construct representations of free algebras.) The first way is to use the embedding . But this one does not define the action of .

There is, however, the second way, which defines the action of with its differential . For or equivalently , we define:

To summarize, the space of deformations of can be considered as a representation of , or as a representation of . (But not of whatever that would be.) In both cases, the differential acts as . That is to say, the of acts as , and the of also acts as — see Eqs. (35) and (37), respectively.

Averaging procedure using

Now we can apply the construction of 〚PDFs from representations of 〛. Eq. (28) gives:

There is no in Alekseev:2010gr, only .

Relation between two integration procedures

In the special case when reduces to (i.e. for ), it was found in Mikhailov:2016rkp, to the second order in the expansion in powers of , that the two PDFs are different by an exact PDF on . It must be true in general.

Is integration form base with respect to ?

Let be the stabilizer of . Eq. (36) is not base with respect to . But it can be made base, provided that one can extend to the solution of the equation:

We do not know the equivariant version of Eq. (38). In the pure spinor formalism, it is very likely that the form given by Eq. (38) is already base, because unintegrated vertex operator does not contain derivatives Flores:2019dwr.

Notice that the PDF defined in Eq. (38) does not, generally, speaking, descend to the orbit of . In computing the average, the integration variable is , not . However, the integral does not depend on the choice of in the orbit.