On the construction of integrated vertex in the pure spinor formalism in curved background

TL;DR

This paper extends the construction of integrated vertex operators in the pure spinor formalism to arbitrary curved backgrounds with nondegenerate RR bispinors, using Lie algebroid structures despite the lack of integrability.

Contribution

It generalizes previous methods from AdS5xS5 to arbitrary curved backgrounds, introducing a Lie algebroid framework for vertex operator construction.

Findings

The construction applies to non-integrable backgrounds.

Lie algebroid provides a consistent framework.

Clarifies previous descriptions of vertex operators.

Abstract

We have previously described a way of describing the relation between unintegrated and integrated vertex operators in AdS5xS5 which uses the interpretation of the BRST cohomology as a Lie algebra cohomology and integrability properties of the AdS background. Here we clarify some details of that description, and develop a similar approach for an arbitrary curved background with nondegenerate RR bispinor. For an arbitrary curved background, the sigma-model is not integrable. However, we argue that a similar construction still works using an infinite-dimensional Lie algebroid.