Topological recursion, BPS structures, and quantum curves


BPS structures were introduced by Bridgeland to describe certain outputs of both Donaldson-Thomas theory and four-dimensional N=2 supersymmetric quantum field theory. To solve a totally different problem, the so-called loop equations in the theory of matrix models, Chekhov and Eynard-Orantin introduced the topological recursion, which takes initial data called a spectral curve and recursively produces an infinite tower of geometric invariants, often with an enumerative interpretation. $\newline$

We will describe recent work and ambitions connecting these two different theories. In the simplest cases, we describe an explicit formula for the “free energies” of the topological recursion in terms of a corresponding BPS structure constructed from the same initial data. We will sketch the relation to the WKB analysis of quantum curves, and gesture towards a wide range of generalizations.