Holomorphic anomaly and non-perturbative quantum mechanics

by Dr Santiago Codesido Sanchez (UNIGE)

Tuesday 13 Mar 2018, 14:00 → 16:00 Europe/Zurich

It has been recently observed that the all-orders WKB periods of certain quantum mechanical problems are governed by the (refined) holomorphic anomaly equations of topological string theory. The relation between topological strings and the spectrum of the operators obtained by quantizing their mirror curves is certainly known. Remarkably, the holomorphic anomaly applies also to problems without a stringy counterpart. This allows the efficient computation of all-orders WKB, but also its non-perturbative corrections. This is achieved by upgrading the holomorphic anomaly to a transseries framework. It reproduces known exact quantization conditions and the large order behaviour of the perturbative WKB series. Such tools are also relevant for quantum mirror curves in so far as the corresponding topological string theory does not have a direct non-perturbative definition. The talk will discuss a few examples of this framework, taken from both quantum mechanics and quantum mirror curves.