De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology

TL;DR

This paper explores the geometric relationship between the superstring BRST operator and de Rham cohomology on supermoduli spaces, providing a formalism for computing superstring amplitudes through supermanifold integration.

Contribution

It establishes a connection between BRST cohomology in superstring theory and de Rham cohomology on supermanifolds, introducing a formalism for amplitude calculations independent of section choice.

Findings

BRST operator related to de Rham differential on supermoduli space

Superstring amplitudes computed via differential form integration

Section independence when states are BRST physical

Abstract

We show that the BRST operator of Neveu-Schwarz-Ramond superstring is closely related to de Rham differential on the moduli space of decorated super-Riemann surfaces P. We develop formalism where superstring amplitudes are computed via integration of some differential forms over a section of P over the super moduli space M. We show that the result of integration does not depend on the choice of section when all the states are BRST physical. Our approach is based on the geometrical theory of integration on supermanifolds of which we give a short review.