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

Gaëtan Borot, Bertrand Eynard, Alexander Weisse

• We study in detail the large N expansion of SU(N) and SO(N)/Sp(2N) Chern–Simons partition function ZN (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 PSL2(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.

  Whenπ1(M)is finite—i.e., for manifolds M that are quotients of S3 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 ZN (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.