Deformation quantization as the origin of D-brane non-Abelian degrees of freedom

TL;DR

This paper links deformation quantization to the non-Abelian degrees of freedom in D-branes, providing a geometric interpretation of gauge theories via K-homology and deformation quantization.

Contribution

It introduces a map from the Grothendieck group of coherent sheaves to K-homology, connecting D-brane configurations with deformation quantization of cycles.

Findings

Explicit realization of K-homology cycles for D-branes.

Non-Abelian degrees of freedom from deformation quantization.

Large N limit interpreted as deformation quantization of formal completions.

Abstract

I construct a map from the Grothendieck group of coherent sheaves to $K$-homology. This results in explicit realizations of $K$-homology cycles associated with D-brane configurations. Non-Abelian degrees of freedom arise in this framework from the deformation quantization of $N$-tuple cycles. The large $N$ limit of the gauge theory on D-branes wrapped on a subvariety $V$ of some variety $X$ is geometrically interpreted as the deformation quantization of the formal completion of $X$ along $V.$