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

Abstract

We study in detail the large N expansion of {mathrm {SU}}(N) and {mathrm {SO}}(N)/{mathrm {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. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a certain scalar, linear, non-local Riemann-Hilbert problem (RHP). We develop tools necessary to address a class of such RHPs involving finite subgroups of mathrm{PSL}_{2}({mathbb {C}}). We associate with such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. These techniques are applied to the RHP relevant for Chern–Simons theory on Seifert spaces. 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—we find that the Weyl group associated with 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 {mathrm {SU}}/{mathrm {SO}}/{mathrm {Sp}} perturbative invariants of knots along fibers in M.