The Chern-Simons Functional Integral, Kauffman's Bracket Polynomial, and other link invariants

Abstract

We study Chern-Simons Gauge Theory in axial gauge on ${\mathbb R}^3.$ This theory has a quadratic Lagrangian and therefore expectations can be computed nonperturbatively by explicit formulas, giving an (unbounded) linear functional on a space of polynomial functions in the gauge fields, as a mathematically well-defined avatar of the formal functional integral. We use differential-geometric methods to extend the definition of this linear functional to expectations of products of Wilson loops corresponding to oriented links in ${\mathbb R}^3,$ and derive skein relations for them. In the case $G=SU(2)$ we show that these skein relations are closely related to those of the Kauffman bracket polynomial, which is closely related to the Jones polynomial. We also study the case of groups of higher rank. We note that in the absence of a cubic term in the action, there is no quantization condition on the coupling $\lambda,$ which can be any complex number. This is in line with the fact that the Jones polynomial, in contrast to the manifold invariants of Witten and Reshetikhin-Turaev, is defined for any value of the coupling. The appearance of the parameter $e^{\frac1{2\lambda}}$ in the expectations and skein relations is also natural. Likewise, the extension of the theory to noncompact groups presents no difficulties. Finally we show how computations similar to ours, but for gauge fields in two dimensions, yield the Goldman bracket.