BRST Cohomology Ring in ${\hat c_M}<1$ NSR String Theory

TL;DR

This paper analyzes the structure of the cohomology ring in ${ ilde c_M}<1$ NSR string theory, identifying generators and their sectors, and provides explicit constructions and examples within supergravity models.

Contribution

It demonstrates that the cohomology ring is generated by three elements with specific sector properties, extending known results from bosonic to supersymmetric string theories.

Findings

Cohomology ring generated by three elements x, y, w

Ground ring generators x, y are non-invertible and in the Ramond sector

Higher ghost number operators generated by an invertible element w

Abstract

The full cohomology ring of the Lian-Zuckerman type operators (states) in ${\hat c_M}<1$ Neveu-Schwarz-Ramond (NSR) string theory is argued to be generated by three elements $x$, $y$ and $w$ in analogy with the corresponding results in the bosonic case. The ground ring generators $x$ and $y$ are non-invertible and belong to the Ramond sector whereas the higher ghost number operators are generated by an invertible element $w$ with ghost number one less than that of the ground ring generators and belongs to either Neveu-Schwarz (NS) or Ramond (R) sector depending on whether we consider (even, even) or (odd, odd) series coupled to $2d$ supergravity. We explicitly construct these operators (states) and illustrate our result with an example of pure Liouville supergravity.