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

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 A () of (smooth) connections on the trivialized principal G-bundle on a compact, oriented two manifold. 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 A (). This relation echoes the relation between the manifold invariants of Witten 20 and Reshetikhin-Turaev 16 and 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.