Abstract loop equations, topological recursion and new applications

Abstract

We formulate a notion of "abstract loop equations", and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpN q Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

Formal multidimensional integrals, stuffed maps, and topological recursion

“…T h k " 0 whenever pk, hq ‰ p1, 0q, p2, 0q) in a recent work with Eynard and Orantin [BEO13]. As we explain in Section 2, the dependence in N of the measure (1.2) is the natural choice in order to have an expansion of topological nature.…”

“…A computation done in the proof of Proposition 3.8 in [BEO13] shows that ω 0 2 pz 1 , z 2 q has its only singularities at z 1 " z 2 , and it is a double pole with leading coefficient 1 and no residues. Let us define the local Cauchy kernel : is such that ϕpz 0 q´r ϕpz 0 q is holomorphic for z 0 P Ω 1 .…”

“…In the terminology of [BEO13], Theorem 4.2 means that ω For usual maps (or for the usual 1-hermitian matrix model), the relation between W g n px 1 , . .…”

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

“…(1.1) captures the contribution of the trivial flat connection in perturbative Chern-Simons theory, or equivalently, the evaluation of the LMO invariants of M in the weight system defined by g [2,3,3,4,39]. Moreover, the correlation functions in General arguments [15] show that the large N asymptotic expansion of models of the type (1.1)-once it is proven to exist using the tools of [16]-can be computed by the topological recursion of [23]. It then remains to compute the initial data (ω 0,1 , ω 0,2 ) of the recursion : ω 0,1 is related to the equilibrium measure of (2.1), also called spectral curve, and ω 0,2 is related to the large N limit of the 2-point correlation function.…”

“…When Q is positive definite and is fixed, (2.18) characterizes W among the holomorphic functions in C\ satisfying (2.15) and whose discontinuity on defines an integrable measure (see, for example, the argument of [15,Lemma 3.10]). When Q is not positive definite, we do not know how to address the question of uniqueness.…”