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.

Quantum geometry of resurgent perturbative/nonperturbative relations

“…In the gauge theory formalism, this "quantum Matone relation" generalizes the classical Matone relation [87] with the inclusion of gravitational couplings [88][89][90][91][92], and it also has a natural interpretation in all-orders WKB [27,60,63]. At the classical level, → 0, the relation (1.5) is a simple consequence of the associated classical Picard-Fuchs equation, which characterizes the energy dependence of the classical action variables (see section 3.1.1 below).…”

“…We stress that this manifestation of resurgence is completely constructive: given a certain number of terms of the expansion of u pert ( , N ), the expression (1.4) generates a similar number of terms in the fluctuations about the one-instanton sector, P inst ( , N ). Furthermore, these relations propagate throughout the entire trans-series, so that perturbation theory encodes the fluctuations about each nonperturbative sector [17][18][19][20][21][22][23][24][25]27].…”

“…In so doing, it proves useful to adopt some of the language and ideas from supersymmetric gauge theories, integrability, conformal field theory and wall-crossing [37][38][39][40][41][42][43][44][45][46][47][48], given the close connection of such theories with exact WKB [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. This is similar to the philosophy of [27], where the refined holomorphic anomaly equation of topological string theory is shown to describe these P/NP relations for some 1d quantum oscillator systems. Further motivation along these lines comes from the ODE/Integrable Model correspondence [66][67][68][69][70], which provides explicit mappings between monodromy operators in certain Schrödinger systems and Yang-Baxter operators in integrable models.…”

“…There is a systematic procedure in classical geometry for reduction to standard forms, determining the geometric invariants from the elliptic curve [115,116], summarized in appendix A. The examples in [16][17][18][19][20][21][22][23][24][25]27] are all genus 1 (recall that the Hydrogenic Stark problem separates in parabolic coordinates to anharmonic oscillator problems [117]). The general proof of the perturbative/non-perturbative relation for genus 1 systems proceeds in three simple steps:…”

“…Later studies found further examples of such P/NP relations in the periodic cosine (Mathieu), supersymmetric double-well, radial anharmonic oscillator, and supersymmetric Mathieu potentials [21][22][23][24][25], and more recently in quasi-exactly soluble models [26]. In a recent paper, Codesido and Mariño have demonstrated the precise connection of these P/NP relations for some 1d quantum oscillator systems with the refined holomorphic anomaly equation of topological string theory [27]. Connections between similar quantum mechanical systems and topological invariants of local Calabi-Yau spaces are also explored in [28,29].…”