Abstract loop equations, topological recursion, and applications

Understand

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.