Topological Yang-Mills Theory with Two Fermionic Charges. A Superfield Approach on K\"ahler Manifolds

TL;DR

This paper formulates a topological Yang-Mills theory with two fermionic charges on Kähler manifolds using a superfield approach, clarifying its structure and enabling symmetry-preserving perturbation theory around instantons.

Contribution

It introduces a superspace formulation for the theory, elucidates the Faddeev-Popov sector, and constructs solutions to the cohomology problem for both fermionic charges, facilitating Donaldson's interpretation.

Findings

Superspace approach clarifies the topological structure.

Constructs solutions to cohomology problems for fermionic charges.

Enables perturbation theory around instantons while preserving symmetries.

Abstract

The four-dimensional topological Yang-Mills theory with two anticommuting charges is naturally formulated on K\"ahler manifolds. By using a superspace approach we clarify the structure of the Faddeev-Popov sector and determine the total action. This enables us to perform perturbation theory around any given instanton configuration by manifestly maintaining all the symmetries of the topological theory. The superspace formulation is very useful for recognizing a trivial observable (i.e. having vanishing correlation functions only) as the highest component of a gauge invariant superfield. As an example of non-trivial observables we construct the complete solution to the simultaneous cohomology problem of both fermionic charges. We also show how this solution has to be used in order to make Donaldson's interpretation possible.