A Noncommutative Deformation of Topological Field Theory

TL;DR

This paper formulates a noncommutative version of topological field theory using a $ heta$-deformation, enabling the definition of deformed Donaldson invariants and exploring their mathematical properties.

Contribution

It introduces a noncommutative deformation of cohomological Yang-Mills theory and interprets the resulting invariants via Chevalley-Eilenberg homology and cohomology.

Findings

Noncommutative Donaldson invariants are defined via $ heta$-deformation.

Quantum theory localizes at noncommutative instanton moduli space in weak coupling.

The framework links noncommutative geometry with topological invariants.

Abstract

Cohomological Yang-Mills theory is formulated on a noncommutative differentiable four manifold through the $\theta$-deformation of its corresponding BRST algebra. The resulting noncommutative field theory is a natural setting to define the $\theta$-deformation of Donaldson invariants and they are interpreted as a mapping between the Chevalley-Eilenberg homology of noncommutative spacetime and the Chevalley-Eilenberg cohomology of noncommutative moduli of instantons. In the process we find that in the weak coupling limit the quantum theory is localized at the moduli space of noncommutative instantons.