Geometry and Physics Seminar

Seifert spaces are S^1 bundles over an orbifold surface. They are the simplest 3-manifolds beyond S^3, in the sense that the partition function of Chern-Simons theory on Seifert spaces can be evaluated in closed form for any compact Lie group G. It is a well-known result, obtained by many different approaches, that it takes the form of a matrix model. Observables in the matrix model represent Wilson lines of along fibers of the Seifert. As a consequence, their large rank expansion is governed by the topological recursion of Eynard and Orantin.

We studied the large N asymptotic expansion in these models for gauge group of the infinite series SU(N), SO(2N), SO(2N + 1), Sp(2N), for fixed value of u = N\hbar > 0. The initial data is the spectral curve of the matrix model, on which we can obtain a lot of information, and in certain cases fully compute. We learn that:
-  the coefficients of the perturbative expansion are not entire functions of e^{u} = q^{N}, and we can sometimes tell in which class of function of e^{u} they lie. This shows in a quantitative way how the Jones or colored HOMFLY fails to be defined as Laurent polynomial in q and q^{N} for knots in manifolds different than S^3.
- for Seifert manifolds of the form S^3/P with P = binary polyhedral group of type ADE, the spectral curve is a specialization of the spectral curve of a relavistic Toda chain of type ADE.

This is based on a joint work with B. Eynard, A. Weisse, and a current project with A. Brini and A. Klemm.