MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

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Funded by The Danish National Research Foundation

Geometric Mathematical Physics Seminar

Abstract
The work presented in this talk is joint with Toshie Takata. I will first present compact expressions for a certain family of representations $R_{r}^{frg}$ indexed by a complex finite dimensional simple Lie algebra $frg$ and a positive integer $r$ bigger than or equal to the dual coxeter number of $frg$ (the so-called shifted quantum level). These expressions are obtained using a certain (multidimensional) reciprocity formula together with certain symmetries. The representations $R_{r}^{frg}$ are known from the study of theta functions and modular forms in connection with the study of affine Lie algebras. They also play a fundamental role in conformal field theory and thereby in the Chern--Simons path integral TQFT's of Witten. I will explain how these representations enter surgery fomulas for the rigorously defined quantum invariants of $3$--manifolds, the so-called Reshetikhin--Turaev invariants. In the final part of the talk I will pay special attention to the Seifert manifolds. I will present compact expressions for the quantum invariants of these spaces using our expressions for the representations $R_{r}^{frg}$. In particular, we obtain hereby a formula for the large $r$ asymptotic expansion of the quantum invariants of lens spaces. This result is in accordance with the asymptotic expansion conjecture of J.E.Andersen. If time permits I will explain some of the problems encountered when trying to calculate these asymptotics for more general Seifert manifolds.

Contact person:Jørgen Ellegaard Andersen.