Holomorphic Anomaly and Quantum Mechanics

Abstract

We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory.

30 pages, 5 figures; comments and references added

References in corpus (8)

• Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
• Holomorphic anomaly and matrix models
• Extended Holomorphic Anomaly in Gauge Theory
• Holomorphic Anomaly in Gauge Theories and Matrix Models
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• RG-Whitham dynamics and complex Hamiltonian systems
• Direct Integration and Non-Perturbative Effects in Matrix Models
• Non-Perturbative Quantum Geometry II

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• On the resurgent structure of quantum periods
• TBA Equations and Quantization Conditions
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• WKB Method and Quantum Periods beyond Genus One
• WKB periods for higher order ODE and TBA equations
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• More on Homological Supersymmetric Quantum Mechanics
• Resurgence, Painleve Equations and Conformal Blocks
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• Partition functions of non-Lagrangian theories from the holomorphic anomaly
• Exact instanton transseries for quantum mechanics
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB Analysis
• Recursive Generation of The Semi-Classical Expansion in Arbitrary Dimension
• Exact-WKB Analysis of Two-level Floquet Systems
• Exact WKB in all sectors II: Potentials with non-degenerate saddles