Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

Abstract

We study a class of scalar, linear, non-local Riemann-Hilbert problems (RHP) involving finite subgroups of PSL(2,C). We associate to such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. As an application, we study in detail the large N expansion of SU(N) or SO(N) or Sp(2N) Chern-Simons partition function Z_N(M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. It has a matrix model-like representation, whose spectral curve can be characterized in terms of a RHP as above. When pi_1(M) is finite (i.e. for manifolds M that are quotients of \mathbb{S}_{3} by a finite isometry group of type ADE), the Weyl group associated to the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large $N$ expansion of Z_N(M) is computed by the topological recursion. This has consequences for the analyticity properties of SU/SO/Sp perturbative invariants of knots along fibers in $M$.