BRS Cohomology in Topological String Theory and Integrable Systems

TL;DR

This paper explores the role of equivariant cohomology in topological string theory, linking it to integrable systems and gravitational descendants within a purely algebraic framework.

Contribution

It demonstrates how $U(1)$ equivariant cohomology in topological string theory produces gravitational descendants and connects cohomological field theories to integrable systems.

Findings

Equivariant cohomology explains gravitational descendants in topological strings.

Deformation family of cohomological theories is governed by integrable systems.

Landau-Ginzburg approach reveals contact term interactions as integrable system features.

Abstract

In cohomological field theory we can obtain topological invariants as correlation functions of BRS cohomology classes. A proper understanding of BRS cohomology which gives non-trivial results requires the equivariant cohomology theory. Both topological Yang-Mills theory and topological string theory are typical examples of this fact. After reviewing the role of the equivariant cohomology in topological Yang-Mills theory, we show in purely algebraic framework how the $U(1)$ equivariant cohomology in topological string theory gives the gravitational descendants. The free energy gives a generating function of topological correlation functions and leads us to consider a deformation family of cohomological field theories. In topological strings such a family is controlled by the theory of integrable system. This is most easily seen in the Landau-Ginzburg approach by looking at the contact term interactions between topological observables.