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In the present paper we extend the "torus gauge fixing" approach by Blau and Thompson, which was developed in [10] for the study of Chern-Simons models with base manifolds M of the form M = Σ × S 1 , in a suitable way. We arrive at a heuristic path integral formula for the Wilson loop observables associated to general links in M . We then show that the right-hand side of this formula can be evaluated explicitly in a non-perturbative way and that this evaluation naturally leads to the face models in terms of which Turaev's shadow invariant is defined. 2 we remark that while the final perturbation series appearing in approach (A1) is rigorous (cf.[5]) the path integral expressions that are used for its derivation are not 1 2 AN ANALYTIC APPROACH TO TURAEV'S SHADOW INVARIANTThe aim of the present paper is to give a partial solution of problems (P1) and (P2)'. In order to do so we will concentrate on the special situation where the base manifold M of the Chern-Simons model is of the form M = Σ×S 1 and then apply the so-called "torus gauge fixing" procedure which was successfully used in [10] for the computation of the partition function of Chern-Simons models on such manifolds (cf. eq. (7.1) in [10]) and for the computation of the Wilson loop observables of a special type of links in M , namely links L that consist of "vertical" loops (cf. eq. (7.24) in [10], see also our Subsec. 6.2). The first question which we study in the present paper is the question whether is is possible to generalize the formulae (7.1) and (7.24) in [10] to general links L in M . The answer to this questions turns out to be "yes", cf. Eq. (3.31) below.Next we study the question whether it is possible to give a rigorous meaning to the heuristic path integral expressions on the right-hand side of Eq. (3.31). Fortunately, it is very likely that also this question has a positive answer (cf. [25] and point (4) in Subsec. 7.2). In fact, due to the remarkable property of Eq. (3.31) that all the heuristic measures that appear there are of "Gaussian type" we can apply similar techniques as in the axial gauge approach to Chern-Simons models on R 3 developed in [19,4,21,22]. In particular, we can make use of white noise analysis and of the two regularization techniques "loop smearing" and "framing".Finally, we study the question if and how the right-hand side of Eq. (3.31) can be evaluated explicitly and if, by doing this, one arrives at the same algebraic expressions for the corresponding quantum invariants as in the shadow version of approach (A2). It turns out that also this question has a positive answer, at least in all the special cases that we will study in detail.The present paper is organized as follows. In Sec. 2 we recall and extend the relevant definitions and results from [10,12,23,24] on Blau and Thompson's torus gauge fixing procedure. In Secs 3.1-3.3, we then apply the torus gauge fixing procedure to Chern-Simons models with compact base manifolds of the form M = Σ × S 1 . After introducing a suitable decomposition A ⊥ =Â ⊥ ⊕ A ⊥ c in Su...

This is the second of a series of papers in which we introduce and study a rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral for manifolds M of the form M = Sigma x S1 and arbitrary simply-connected compact structure groups G. More precisely, we introduce, for general links L in M, a rigorous simplicial version WLO_{rig}(L) of the corresponding Wilson loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson (Nucl. Phys. B408(2):345-390, 1993). For a simple class of links L we then evaluate WLO_{rig}(L) explicitly in a non-perturbative way, finding agreement with Turaev's shadow invariant |L|.Comment: 53 pages, 1 figure. Some minor changes and corrections have been mad

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This is the first of a series of papers in which we introduce and study a rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral for manifolds M of the form M = Σ × S 1 and arbitrary simply-connected compact structure groups G. More precisely, we will introduce, for general links L in M , a rigorous simplicial version WLO rig (L) of the corresponding Wilson loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson (Nucl. Phys. B408(2): 1993). For a simple class of links L we then evaluate WLO rig (L) explicitly in a non-perturbative way, finding agreement with Turaev's shadow invariant |L|. AMS subject classifications: 57M27, 81T08, 81T452 Chern-Simons theory on M = Σ × S 1 in the torus gauge Chern-Simons theoryLet us fix a simply-connected compact Lie group 2 G with Lie algebra g.For every smooth manifold M , every real vector space V and every n ∈ N 0 we will denote by Ω n (M, V ) the space of V -valued n-forms on M and we setBy G M we will denote the "gauge group" C ∞ (M, G). We will usually write A instead of A M = Ω 1 (M, g) and G instead of G M .In the following we will restrict ourselves to the special case where M is an oriented closed 3-manifold. Moreover, we will consider only the special case where G is simple (cf. Remark 2.2 1 or, rather, Abelian BF3-models, cf. the beginning of Sec. 5 below 2 cf. part A of the Appendix for concrete formulas in the special case G = SU (2) 3 More precisely, the normalization is chosen such that α,α = 2 for every short real corootα w.r.t. any fixed Cartan subalgebra of g. Observe that after making the identification t ∼ = t * which is induced by ·, · we have α, α = 2 for every long root α. Thus the normalization here coincides with the one in [68]. This normalization guarantees that the exponential exp(iSCS) is "gauge invariant", i.e. invariant under the standard G-operation on A 4 i.e. a smooth embedding S 1 → M 5 observe that if G is simple then every Ad-invariant scalar product on g is proportional to the Killing form ad(B(σ))) = Ad(exp(B(σ))) and that k is an Ad |T -invariant subspace of g 13 the notation [Σ, G/T ] is motivated by the fact that C ∞ (Σ, G/T )/GΣ coincides with the set of homotopy classes of maps Σ → G/T , cf. Proposition 3.2 in [40] 14 observe that for b ∈ t the valueḡbḡ −1 will not depend on the special choice of g