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Holomorphic Anomaly and Quantum Mechanics

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arxiv 1612.07687 v3 pith:A32PJD75 submitted 2016-12-22 hep-th math-phmath.MP

classification hep-thmath-phmath.MP
keywords quantumanomalydoubleequationsexpansionholomorphicoscillatorsperiods
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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.

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Cited by 3 Pith papers

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  1. Analytic approaches to perturbations of strongly coupled Yang-Mills plasma

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    Seiberg-Witten instanton expansions combined with exact WKB period integrals allow analytic computation and continuation of quasinormal modes from large q to q=0.

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    For generic one-dimensional potentials, the exact spectrum decomposes into as many trans-series sectors as there are distinct local-minimum energy levels, with continuous transitions across barrier tops and discontinu...

  3. Many-faced Painlev\'e I: irregular conformal blocks, topological recursion, and holomorphic anomaly approaches

    math-ph 2025-05 conditional novelty 7.0 of 10

    The paper proves the conifold gap property for Weierstrass elliptic topological recursion and gives an algebraic construction of the rank 5/2 Virasoro Whittaker state, linking the Painlevé I tau function to CFT and ho...

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