On the Mathai-Quillen Formalism of Topological Sigma Models

TL;DR

This paper interprets topological sigma models using the Mathai-Quillen formalism, connecting geometric structures with quantum field theory, and extends the approach to gauged models, providing a new perspective on their mathematical foundation.

Contribution

It introduces a Mathai-Quillen framework for topological sigma models and their gauged versions, linking infinite-dimensional geometry with physical models.

Findings

Identifies a natural connection in the space of maps from Riemann surfaces to almost complex manifolds.

Shows the covariant derivative of the pseudo-holomorphic curve equation's section equals its linearization.

Extends the formalism to gauged topological sigma models.

Abstract

We present a Mathai-Quillen interpretation of topological sigma models. The key to the construction is a natural connection in a suitable infinite dimensional vector bundle over the space of maps from a Riemann surface (the world sheet) to an almost complex manifold (the target). We show that the covariant derivative of the section defined by the differential operator that appears in the equation for pseudo-holomorphic curves is precisely the linearization of the operator itself. We also discuss the Mathai-Quillen formalism of gauged topological sigma models.