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

SIGMA 20 (2024), 073, 34 pages      arXiv:2311.17638      https://doi.org/10.3842/SIGMA.2024.073

Resurgence of Refined Topological Strings and Dual Partition Functions

Sergey Alexandrov a, Marcos Mariño b and Boris Pioline c
a) Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, F-34095, Montpellier, France
b) Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland
c) Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589, CNRS-Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris, France

Received December 13, 2023, in final form August 02, 2024; Published online August 06, 2024

Abstract
We study the resurgent structure of the refined topological string partition function on a non-compact Calabi-Yau threefold, at large orders in the string coupling constant $g_s$ and fixed refinement parameter $\mathsf{b}$. For $\mathsf{b}\neq 1$, the Borel transform admits two families of simple poles, corresponding to integral periods rescaled by $\mathsf{b}$ and $1/\mathsf{b}$. We show that the corresponding Stokes automorphism is expressed in terms of a generalization of the non-compact quantum dilogarithm, and we conjecture that the Stokes constants are determined by the refined Donaldson-Thomas invariants counting spin-$j$ BPS states. This jump in the refined topological string partition function is a special case (unit five-brane charge) of a more general transformation property of wave functions on quantum twisted tori introduced in earlier work by two of the authors. We show that this property follows from the transformation of a suitable refined dual partition function across BPS rays, defined by extending the Moyal star product to the realm of contact geometry.

Key words: resurgence; topological string theory; Borel resummation; Stokes automorphism.

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