Chern-Simons models on S2×S1, torus gauge fixing, and link invariants I

We study Abelian and non-Abelian Chern-Simons models on manifolds M of the form M=Σ×S1, where Σ is a compact oriented surface. By applying the “torus gauge fixing” procedure of Blau and Thompson we derive a heuristic integral formula for the corresponding Wilson loop observables (WLOs) which has some features that make it a promising starting point for the search of a rigorous path integral representation for the WLOs. For the special case Σ=S2 and G=U(1), G being the structure group of the model, we indeed obtain a rigorous version of the right-hand side of the aforementioned heuristic formula and thus a rigorous path integral representation of the WLOs in terms of infinite-dimensional oscillatory integrals. This is achieved by combining certain regularization procedures like “loop smearing” and “framing” with methods from white noise analysis. We expect that similar considerations will eventually lead to a rigorous path integral representation of the WLOs also for non-Abelian Chern-Simons models on M=S2×S1 and to a new and purely geometric derivation of the R-matrices of Jones and Turaev.