BRS Cohomology in Topological String Theory and Integrable Systems

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.