Cohomological Yang-Mills theories on Kähler 3-folds

Abstract

We study topological gauge theories with Nc = (2; 0) supersymmetry based on stable bundles on general Kähler 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. After dimensional reduction to a Kähler 2-fold, the theory reduces to Vafa-Witten theory. On a Calabi-Yau 3-fold, the supersymmetry is enhanced to Nc = (2; 2). This model may be used to describe classical limits of certain compactifications of (matrix) string theory.