Equivariant Topological Sigma Models

TL;DR

This paper introduces equivariant topological sigma models for Kahler manifolds with real structures, extending topological quantum field theory, and explores their implications for open string theory and intersection theory.

Contribution

It generalizes topological sigma models to include real structures and develops an equivariant framework compatible with open strings and real involutions.

Findings

Explicit solution of an equivariant CP^1×CP^1 model

Correlation functions relate to intersection theory on instanton moduli spaces

Identifies a Z2 anomaly in topological open string theory

Abstract

We identify and examine a generalization of topological sigma models suitable for coupling to topological open strings. The targets are Kahler manifolds with a real structure, i.e. with an involution acting as a complex conjugation, compatible with the Kahler metric. These models satisfy axioms of what might be called ``equivariant topological quantum field theory,'' generalizing the axioms of topological field theory as given by Atiyah. Observables of the equivariant topological sigma models correspond to cohomological classes in an equivariant cohomology theory of the targets. Their correlation functions can be computed, leading to intersection theory on instanton moduli spaces with a natural real structure. An equivariant $CP^1\times CP^1$ model is discussed in detail, and solved explicitly. Finally, we discuss the equivariant formulation of topological gravity on surfaces of unoriented open and closed string theory, and find a $Z_2$ anomaly explaining some problems with the formulation of topological open string theory.