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

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].