Chern–Simons theory on R3 in axial gauge: a rigorous approach

We study pure Chern–Simons models on M=R3 using a functional integral quantization approach which is based on axial gauge fixing. It is well-known (see, e.g., Comm. Math. Phys. 126 (1989) 167; Comm. Math. Phys. 186 (1997) 563) that in axial gauge the Chern–Simons action function is quadratic and that the Faddeev-Popov determinant of this gauge fixing procedure is a constant function. This means that the Wilson loop observables (WLOs) of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of “Gaussian type”. We demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from White noise analysis with certain regularization techniques like “loop smearing” and “framing”. For the special case G=U(1), G being the group of the model, we carry out the details of this approach, which is also applicable to the case of non-Abelian G, and find well-known linking number expressions for the WLOs.