Higher Nahm transform in non commutative geometry

TL;DR

This paper introduces a noncommutative geometric extension of the Nahm transform, linking ASD connections on four-manifolds with a generalized Connes-Yang-Mills functional using Dixmier trace.

Contribution

It develops a novel noncommutative geometric framework for the Nahm transform, extending classical ASD connection correspondence to noncommutative spaces.

Findings

Formulation of a noncommutative Nahm transform

Generalization of the Connes-Yang-Mills action functional

Potential applications in noncommutative gauge theory

Abstract

Anti-self-dual (ASD) connections for a compact smooth four manifold arise as critical values for the Yang-Mills action functional. Nahm transform is a nice correspondence between a vector bundle with ASD connections and a vector bundle with ASD connections over Picard torus associated to X. In this talk we propose a noncommutative geometric version of the Nahm transform that generalises the Connes-Yang-Mills action functional formulated using Dixmier trace.