Introduction

The -ghost of the pure spinor formalism in a general curved background is only holomorphic up to a -exact expression Berkovits:2010zz. The construction of the string measure for such theories was suggested in Mikhailov:2016myt,Mikhailov:2016rkp. It requires the knowledge of the action of the group of worldsheet diffeomorphisms on the BV phase space. For a vector field on the worldsheet (= infinitesimal diffeomorphism) let be the BV Hamiltonian generating the action of on the BV phase space. Then, the string measure is, schematically:

It is not completely trivial to construct for the pure spinor superstring in AdS. One of the complications is the somewhat unusual form of the pure spinor part of the action. Schematically:

where is a linear combination of Ramond-Ramond field strengths. Notice that the conjugate momenta and only enter through their and component, respectively. We can try to integrate out , ending up with a “standard” kinetic term for ghosts:

Notice that landed in the denominator. It would seem that the theory depends quite irregularly on the Ramond-Ramond field, but this is not true. All physics sits at , and the term is in some sense subleading.

In this paper we will show, closely following Berkovits:2008ga,Tonin:2013uec, that the pure spinor terms (2) can actually be removed by reduction to a smaller BV phase space, keeping intact all the symmetries of . The resulting action is degenerate, and therefore can not be immediately used for quantization. On the other hand, it is simpler than the original action. In particular, the action of worldsheet diffeomorphisms in this reduced BV phase space is rather transparent, although the explicit expression Eq. (46) is somewhat involved. We then explain how to lift this action to an action on some quantizable theory which is basically the same as the original pure spinor sigma-model of Berkovits:2000fe.

Formal application of BV formalism

Here, as in Mikhailov:2016rkp, we formally apply the formalism of odd symplectic manifolds in the infinite-dimensional case (the field space of two-dimensional sigma-models). This should be proven in perturbation theory, but in this paper we restrict ourselves with purely formal manipulations. We believe that supersymmetry will play crucial role in controlling quantum anomalies; therefore it is important that our constructions preserve supersymmetries.

Plan of the paper

We begin in Section Master Actions quadratic-linear in antifields with the general discussion of the reduction procedure when a BV Master Action is a quadratic-linear functional of antifields. In Section Pure spinor superstring in we apply this to the case of pure spinor superstring in . In Sections Action of diffeomorphisms we construct the action of diffeomorphisms in the minimalistic sigma-model. Then in Section Regularization we construct the action of diffeomorphisms on the BV phase space of the non-degenerate theory, which is essentially equivalent (quasiisomorphic) to the original sigma-model. Sections Taking apart the AdS sigma model and Generalization contain summary and generalizations, and Section Open problems open problems.