Issues in Topological Gauge Theory

TL;DR

This paper explores topological gauge theories derived from twisted $ ext{N}=2$ theories, analyzing their observables, dualities, and correlation functions within a Gromov-Witten framework, and examining their relations to sigma models.

Contribution

It introduces a formalism for studying topological gauge theories, fixes contact terms uniquely, and derives correlation functions for various manifolds, extending understanding of these theories.

Findings

Derived a formula for correlation functions on manifolds of generalized simple type.

Presented a twisted superfield formalism that clarifies duality transformations.

Explored properties of universal instantons and relations to sigma models.

Abstract

We discuss topological theories, arising from the general $\mathcal{N}=2$ twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various $p$-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of $p$-observables on the manifolds of generalized simple type for $0 \leq p \leq 4$ and on some manifolds with $b_{2}^{+} =1$. We study the theories with matter and explore the properties of universal instanton. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type $A$ and $B$ are pointed out and exploited.