On the resurgent structure of quantum periods

Abstract

Quantum periods appear in many contexts, from quantum mechanics to local mirror symmetry. They can be described in terms of topological string free energies and Wilson loops, in the so-called Nekrasov-Shatashvili limit. We consider the trans-series extension of the holomorphic anomaly equations satisfied by these quantities, and we obtain exact multi-instanton solutions for these trans-series. Building on this result, we propose a unified perspective on the resurgent structure of quantum periods. We show for example that the Delabaere-Pham formula, which was originally obtained in quantum mechanical examples, is a generic feature of quantum periods. We illustrate our general results with explicit calculations for the double-well in quantum mechanics, and for the quantum mirror curve of local ${\mathbb{P}}^2$.

arXiv comment40 pages, 8 figures; typos corrected, refs added

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• Exact multi-instantons in topological string theory
• Exact instanton transseries for quantum mechanics
• Resurgence of Refined Topological Strings and Dual Partition Functions
• Resurgent Structure of the Topological String and the First Painlevé Equation
• Resurgence, Stokes constants, and arithmetic functions in topological string theory
• Lattice random walks and quantum A-period conjecture