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 SU(N ) and SO(N )/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 PSL 2 (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 π 1 (M) is finite-i.e., for manifolds M that are quotients of 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 SU/SO/Sp perturbative invariants of knots along fibers in M.Mathematics Subject Classification 14Hxx · 45Exx · 51Pxx · 57M27 · 60B20 · 81T45

Blobbed topological recursion: properties and applications

“…We prove this theorem in by constructing BTR C : ϕ Þ Ñ ω by necessary conditions, and checking in § 2.6 it indeed defines a solution of abstract loop equations. To describe this inverse map, we assume that ω is a solution of loop equations, and we decompose it: (5) ω g,n pz 1 , . .…”

“…The solutions in M 0 have the property that the holomorphic part of ω g,n is normalized uniformly for all n and g. The topological recursion can be studied per se and enjoys beautiful properties: Seiberg-Witten like formulas for infinitesimal variations of the initial data [20], symplectic invariance [19], representation in terms of intersection numbers on M g,n [15,14], etc. It has received many applications in algebraic geometry [18,26,13,9,28,21], in relation with integrable systems [3,25], in knot theory [12,11,4,5], and we also refer to the examples given in [20,6] for applications to matrix models and statistical physics.…”

E8 spectral curves

“…) on the double symmetric product (minus the diagonal) of the smooth completion Γ LMO τ of the algebraic † plane curve y = W LMO 0,1 (x): the LMO spectral curve. A strategy to determine the family of Riemann surfaces Γ LMO τ as the base parameter τ is varied was put forward in the extensive analysis of Chern-Simons-type matrix models of [17], and is summarised in the next section.…”

On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds

“…That a dual curve counting theory exists is encouraged by the successful test of this proposal for the case of L(p, 1) lens spaces in [5,53]. The case of more general 3-manifolds was considered in [15,16,25,27], and we review it below.…”

“…(32) and (33) of S 3 /Γ and fibre knots therein around the exceptional fibres fall squarely in this category [12,14,71]. When v = 0 is the trivial flat connection, the resulting matrix model turns out to be a trigonometric deformation of the gauged gaussian matrix model of Example 2.1: it is a canonical ensemble with gaussian 1-body interaction and a sum of q-deformed Vandermonde 2-body interactions, with coefficients determined by the orders of the exceptional fibres of the Seifert fibration [11,16,24,71]: this restriction gives rise to the so-called LMO invariant. This presentation is amenable to a large N analysis via loop equations, akin to that of Point iii) in Example 2.1, leading for all Γ to a singular integral equation to be solved by the input datum of the recursionthe planar disk function W 0,1 .…”