Cohomological field theories and generalized Seiberg-Witten equations

TL;DR

This paper develops a formalism for constructing cohomological field theories from nonlinear PDEs and applies it to generalized Seiberg-Witten equations, unifying various supersymmetric theories and suggesting a path toward manifold invariants.

Contribution

It introduces a new formalism for cohomological field theories based on PDEs and applies it to generalized Seiberg-Witten equations, aligning with physicists' proposals.

Findings

CohFT functionals match existing physicist proposals

Provides a unified perspective on supersymmetric theories

Outlines a quantization program for manifold invariants

Abstract

We introduce a formalism for constructing cohomological field theories (CohFT) out of nonlinear PDEs based on the first author's previous work (arXiv:2202.12425). We apply the formalism to the generalized Seiberg-Witten equations and show that the obtained CohFT functionals agree with the existing ones proposed by physicists. This leads to a unified perspective from which to view the full supersymmetric functionals of the Donaldson-Witten, Seiberg-Witten, and Kapustin-Witten theories and understand the relations between them. We also outline a quantization program for our framework and discuss its potential to produce manifold invariants and quantum cohomologies.