Morse Theory in Field Theory

TL;DR

This paper explores the application of Morse theory to topological quantum mechanics, linking correlators to intersection numbers and proposing a new conjecture involving Massey products.

Contribution

It introduces a novel geometric interpretation of TQM correlators using Morse theory and conjectures a new connection with Massey products in cohomology.

Findings

Correlators expressed as intersection numbers of submanifolds.

Established a link between steepest descent paths and Morse critical points.

Proposed a conjecture relating quantum correlators to Massey products.

Abstract

We describe correlations functions of topological quantum mechanics (TQM) in terms of Morse theory. We review the basics of topological field theories and discuss geometric and algebraic interpretations of TQM. We prove that correlators in TQM can be expressed via intersection numbers of certain submanifolds of the target space with paths of steepest descent between critical points of a Morse function. In the end we conjecture another correspondence between quantum mechanics correlators and integrals of Massey products of certain cohomology classes.