Communications in Number Theory and Physics
Volume 9 (2015)
Abstract loop equations, topological recursion and new applications
Pages: 51 – 187
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.