Hilbert space of curved \beta\gamma systems on quadric cones

TL;DR

This paper investigates the structure of the Hilbert space for curved eta systems constrained by a quadratic surface, using intrinsic and BRST methods, revealing key cohomological properties and mappings.

Contribution

It provides a detailed analysis of the Hilbert space structure and BRST cohomology for curved eta systems on quadric cones, with implications for pure spinor superstring theory.

Findings

BRST cohomology is non-empty only at ghost numbers 0 and 1

Partition functions from intrinsic and BRST methods agree

One-to-one mapping between ghost number 0 and 1 sectors

Abstract

We clarify the structure of the Hilbert space of curved \beta\gamma systems defined by a quadratic constraint. The constraint is studied using intrinsic and BRST methods, and their partition functions are shown to agree. The quantum BRST cohomology is non-empty only at ghost numbers 0 and 1, and there is a one-to-one mapping between these two sectors. In the intrinsic description, the ghost number 1 operators correspond to the ones that are not globally defined on the constrained surface. Extension of the results to the pure spinor superstring is discussed in a separate work.