Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 064, 44 pages      arXiv:2201.11594      https://doi.org/10.3842/SIGMA.2023.064

Exponential Networks, WKB and Topological String

Alba Grassi ab, Qianyu Hao c and Andrew Neitzke d
a) Section de Mathématiques, Université de Genève, 1211 Genève 4, Switzerland
b) Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
c) Department of Physics, University of Texas at Austin, 2515 Speedway, C1600, Austin, TX 78712-1992, USA
d) Department of Mathematics, Yale University, PO Box 208283, New Haven, CT 06520-8283, USA

Received March 07, 2023, in final form August 23, 2023; Published online September 13, 2023

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. 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 and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau $X$ to be either $\mathbb{C}^3$ 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, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane.

Key words: difference equation; Stokes phenomenon; BPS states; topological string; exponential network.

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