Abstract loop equations, topological recursion, and applications

TL;DR

This paper introduces a universal framework of abstract loop equations solved by topological recursion, connecting various models and applications in mathematical physics and geometry.

Contribution

It formulates abstract loop equations and demonstrates their solutions via topological recursion, unifying multiple models and applications.

Findings

Solution of abstract loop equations via topological recursion

Connection of loop equations to Virasoro constraints

Applications to matrix models, knot invariants, and Liouville theory

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.