Abstract
Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\).
Highlights
This is an unplanned sequel to [1,2]
We will show that the Chern-Simons path integral can be written as a linear functional defined on C(B1× × B2×, C) and this linear functional is similar to the
We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter κ
Summary
This is an unplanned sequel to [1,2]. Let M be a 3-manifold and G be a compact connected semisimple Lie group. The main purpose of this article is to define a Chern-Simons path integral in S 2 × S 1 using torus gauge fixing and non-abelian gauge group. After ‘scaling’ the truncated link and embed it inside C3 , the Wilson Loop observable will be defined on this quantum space. We will explicitly work out this integral for the truncated link embedded inside the quantum space. For gauge group SO(N ), the Wilson Loop observable will satisfy a Conway-type skein relation, with z = 2i sin(πq 2 /2) For both cases, q 2 is quantized to take only a discrete number of values. We will define the Wilson Loop observable given in Equation (2) and compute it. We always use h·, ·i to denote an inner product
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