High order perturbation theory for difference equations and Borel summability of quantum mirror curves

2017/09/04 by Jie Gu, Tin Sulejmanpasic
Computer Science · Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Fano plane #Mathematical functions and polynomials #Order (exchange) #Perturbation (astronomy) #Perturbation theory (quantum mechanics) #Polynomial and algebraic computation #Quantum #Quantum Computing Algorithms and Architecture #hep-th #quant-ph

paper · pdf · doi:10.1007/jhep12(2017)014

39 pages, 4 figures, and 4 tables. Bundled with the source files of this document are the Mathematica notebooks for the package BenderWu, including the new function BWDifference

arxiv created 2017/09/04 · openalex created_date 2017/09/15 · openalex publication_date 2017/12/01 · arxiv updated 2018/01/17 · openalex updated_date 2026/08/06

Abstract

We adapt the Bender-Wu algorithm to solve perturbatively but very efficiently the eigenvalue problem of "relativistic" quantum mechanical problems whose Hamiltonians are difference operators of the exponential-polynomial type. We implement the algorithm in the function BWDifference in the updated Mathematica package BenderWu. With the help of BWDifference, we survey quantum mirror curves of toric fano Calabi-Yau threefolds, and find strong evidence that not only are the perturbative eigenenergies of the associated 1d quantum mechanical problems Borel summable, but also that the Borel sums are exact.

Citations

• Resurgence and trans-series in Quantum Field Theory: the ℂℙ^N-1 model
• Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local ℂℙ² C P 2
• Lectures on mirror symmetry
• Mirror Symmetry and Algebraic Geometry
• Multi-instantons and multicuts
• Quantum geometry of refined topological strings
• Direct integration and non-perturbative effects in matrix models