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 ${\mathbb{P}}^2$.

40 pages, 8 figures; typos corrected, refs added

References in corpus (14)

• Extended Holomorphic Anomaly in Gauge Theory
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• Lagrange Inversion
• Topological Strings and Quantum Spectral Problems
• Peacock patterns and new integer invariants in topological string theory
• Exact WKB methods in $SU(2)$ $N_f=1$
• Mathematical structures of non-perturbative topological string theory: from GW to DT invariants
• Quantum Curves, Resurgence and Exact WKB
• Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
• On the BPS spectrum of 5d SU(2) super-Yang-Mills
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• New Instantons for Matrix Models
• Topological strings and Wilson loops
• The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

Cited by in corpus (6)

• 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