Cohomological Yang-Mills Theories on Kahler 3-Folds

TL;DR

This paper develops a topological gauge theory on Kahler 3-folds, extending holomorphic bundles with a 3-form, leading to complex 3D invariants and connections to string theory compactifications.

Contribution

It introduces a novel cohomological Yang-Mills theory on Kahler 3-folds incorporating a holomorphic 3-form, generalizing Donaldson-Witten invariants and linking to Seiberg-Witten monopoles.

Findings

Path integral sums over stable bundles and monopoles.

Reduces to Vafa-Witten theory on Kahler 2-folds.

Enhanced supersymmetry on Calabi-Yau 3-folds.

Abstract

We study topological gauge theories with N=(2,0) supersymmetry based on stable bundles on general Kahler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson-Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg-Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. Afterdimensional reduction to a Kahler 2-fold, the theory reduces to Vafa-Wittentheory. On a Calabi-Yau 3-fold, the supersymmetry is enhanced to N=(2,2). This model may be used to describe classical limits of certain compactifications of(matrix) string theory.