Mathematical structures of cohomological field theories

TL;DR

This paper develops a new mathematical framework for cohomological field theories using bigraded manifolds, unifying various topological models and generalizing existing formalisms like Mathai-Quillen.

Contribution

It introduces a novel formalism for CohFTs, including gauge theories, and provides a unified approach to constructing examples such as topological quantum mechanics and Yang-Mills.

Findings

Algebraic properties of operators in CohFTs are characterized.

A generalized Mathai-Quillen formalism is established.

Multiple topological models are derived uniformly within the new framework.

Abstract

A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism.