Exponential Networks, WKB and the Topological String

Abstract

We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds X: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes, and the central charges of 5d BPS KK-modes are related to the singularities in the Borel planes of the closed topological string free energy on X. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify this picture in detail in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau X to be either C 3 or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on X with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on X and the corresponding non-perturbative effects.

On the resurgent structure of quantum periods

“…We refer to the companion paper [29] for a lightning review of alien calculus and some references on the subject. We also mention that the resurgent structure of the NS free energy of the resolved conifold has been recently studied in [56][57][58].…”

Quantum Curves, Resurgence and Exact WKB

“…Remark 5.1. We remark that a similar study of the Borel summability of S(ϵ, x, t) is performed in the recent work [52,Section 4]. An analogous discussion of the Borel transform of F NS is written down in [52, Section 4.5.1].…”

“…presented her ideas on the relations of the Borel analysis of [63] to that of [9] at this colloquium in November 2021, triggering our joint work. We would like to thank Alba Grassi, Qianyu Hao and Andy Neitzke for discussions and for informing us about their related work [52]. We would like to furthermore thank Sergei Alexandrov, Tom Bridgeland, Boris Pioline, Arpan Saha and Jörg Teschner for discussions and correspondence.…”

“…The resulting 5d BPS spectrum should however be independent of this choice. A brief discussion of the closed spectrum, in our choice of framing, as well as an comprehensive analysis of the open spectrum of 3d BPS states has furthermore appeared recently in [52]. We will therefore keep our discussion concise and merely emphasize key aspects, as well as introduce two distinguished networks that play an important role later.…”

A Panorama Of Physical Mathematics c. 2022

“…Curiously, similar ideas have recently been applied in the context of nonperturbative definitions of JT gravity [381]. Some important progress in the theory of quantum curves and the relation to topological string theory was made in [382][383][384][385]. The relation between the quantization of a classical system based on a spectral curve, and the topological string, a B model on a local Calabi-Yau which is a complexification of a handlebody whose boundary is that curve, or an A model on a local Calabi-Yau mirror, involves a version of a blow-up formula and is discussed in [18].…”