Abstract loop equations, topological recursion, and 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 O(n) 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 SU(N) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

89 pages, 8 figures

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Cited by in corpus (13)

• The Tensor Track, III
• Large-N correlation functions in $\mathcal{N}=2$ superconformal QCD
• Tensor models from the viewpoint of matrix models: the case of loop models on random surfaces
• Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
• Weighted Hurwitz numbers and topological recursion: an overview
• Higher Airy structures and topological recursion for singular spectral curves
• Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces
• Integrable differential systems of topological type and reconstruction by the topological recursion
• Torus knot polynomials and susy Wilson loops
• Notes on Mayer Expansions and Matrix Models
• Global Asymptotics for the Christoffel-Darboux Kernel of Random Matrix Theory
• Loop equations from differential systems
• $\mathcal{N}=1$ Super Topological Recursion