On deformation quantization of the space of connections on a two manifold and Chern Simons Gauge Theory

TL;DR

This paper provides explicit formulas for deformation quantization of the space of connections on a 2-manifold, linking knot invariants from Chern-Simons theory to geometric quantization.

Contribution

It introduces explicit star product formulas on the space of connections, connecting knot invariants with deformation quantization in a rigorous mathematical framework.

Findings

Explicit star product formulas for connection space

Connection between knot invariants and deformation quantization

Relation to geometric quantization and manifold invariants

Abstract

We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space ${\mathcal A}(\Sigma)$ of (smooth) connections on the trivialized principal $G$-bundle on a compact, oriented two manifold $\Sigma.$ These formulas give a close relation between knot invariants, such as the Kauffman bracket polynomial, and the Jones and HOMFLY polynomials, arising in Chern Simons gauge theory, and deformation quantization of ${\mathcal A}(\Sigma).$ This relation echoes the relation between the manifold invariants of Witten [20] and Reshetikhin-Turaev [16] and {\em geometric} quantization of this space (or its symplectic quotient by the action of the gauge group). In our case this relation arises from explicit algebraic formulas arising from the (mathematically well-defined) functional integrals of [18].