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Complex Chern-Simons Theory: Knot Invariants and Mapping Class Group Representations

by Simone Marzioni
PhD Dissertations | October 2016

In this thesis we investigate some properties of Complex Chern-Simons theory. Contrary to the compact situation which was the object of a lot of focus for more than 20 years, not much was known until few years ago, regarding rigorous computations of invariants via Chern-Simons theory with complex gauge group. In the recent years, in parallel to an increased interest from physics, the works of Andersen and Kashaev, and of Andersen and Gammelgaard opened the way to a rigorous mathematical investigation of such theories. Andersen and Kashaev provided the theory to compute invariants of knot complements (and actually a more general class of cusped 3 manifolds) starting from the quantization of the Teichmüller space. The work of Andersen and Gammelgaard provides a general differential geometric setting for the ideas of Witten [Wit91], regarding techniques to quantize the 2 dimensional part of Chern-Simons theory with gauge group \(\mathrm{SL}(n;\mathbb{C})\). In general we are still missing \(2+1\) functorial interpretation, like the Witten-Reshetikhin-Turaev TQFT for the compact theory. In this thesis we try to have a closer look to some of the most elementary aspects of these constructions. We focused particularly in computing and studying explicit expressions for the simplest examples of knot invariants and mapping class group representations. We first construct invariants of hyperbolic knots, showing their relation with some new representations of Quantum Teichmüller Theory. Then we focus in a couple of examples. The study of the asymptotic behavior of such knot invariants requires a generalization of the theory of Andersen and Kashaev to a non obviously unitary one, the existence of which was claimed again by Witten in [Wit91]. In this setting some parallel with other previously known invariants is discussed. Afterwards we follow the approach of Andersen and Gammelgaard in the example of a genus 1 surface, and study the mapping class group representations that this quantum theory defines. We give explicit formulas for the representations and show how the representations from Chern-Simons theory with gauge group \(\mathrm{SU}(2)\) appear in these.

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Thesis advisor: Jørgen Ellegaard Andersen