January 15, 2021
TITLE: Geometric description of topological string partition functions from quantum curves and integrability
I will give a progress update on work relating topological string partition functions Ztop for a class of supersymmetric gauge theories to quantum Seiberg-Witten curves through integrability. In particular, I will discuss a geometric characterisation of the Ztop functions in terms of a line bundle over the moduli space of quantum curves, providing evidence for this picture through examples. Part of this discussion will review earlier results which show how the Ztop functions enter certain series expansions of isomonodromic tau functions associated to quantised SW curves. New insight then concerns the existence of certain preferred coordinates on the moduli space of quantum curves, which are defined from the curves via exact WKB analysis and which enter theta-series expansions of appropriately normalised tau functions, in a way that allows to extract the functions Ztop. Understanding these coordinates, how they are related on different patches as a consequence of Stokes phenomena, leads to the proposed geometric characterisation of the tau functions and Ztop.