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Topological Strings and Quantum Spectral Problems

Min-xin Huang, Xian-fu Wang

TL;DR

This work analyzes quantum spectral problems from local Calabi–Yau geometries using Bohr–Sommerfeld quantization with deformed periods in the Nekrasov–Shatashvili limit to probe non-perturbative physics. It confirms that the Hatsuda–Kallen–Marino singularity-cancellation mechanism largely constrains the non-perturbative sector, while also uncovering higher-order non-singular corrections through high-precision numerics and precision spectroscopy, enabling exact determination of early non-perturbative coefficients for models such as local \\mathbb{P}^2, \\mathbb{P}^1\\times\\mathbb{P}^1, and \\mathbb{F}_1. The authors derive explicit forms for leading non-perturbative terms, connect them to refined GV invariants, and validate the results against extensive numerical spectra, thereby strengthening the non-perturbative framework of refined topological strings and their quantum spectral problems. The findings illuminate the structure of phase-space volumes, deformed periods, and the interplay between perturbative and non-perturbative sectors, with potential links to ABJM-like matrix models and broader topological-string formulations.

Abstract

We consider certain quantum spectral problems appearing in the study of local Calabi-Yau geometries. The quantum spectrum can be computed by the Bohr-Sommerfeld quantization condition for a period integral. For the case of small Planck constant, the periods are computed perturbatively by deformation of the Omega background parameters in the Nekrasov-Shatashvili limit. We compare the calculations with the results from the standard perturbation theory for the quantum Hamiltonian. There have been proposals in the literature for the non-perturbative contributions based on singularity cancellation with the perturbative contributions. We compute the quantum spectrum numerically with some high precisions for many cases of Planck constant. We find that there are also some higher order non-singular non-perturbative contributions, which are not captured by the singularity cancellation mechanism. We fix the first few orders formulas of such corrections for some well known local Calabi-Yau models.

Topological Strings and Quantum Spectral Problems

TL;DR

This work analyzes quantum spectral problems from local Calabi–Yau geometries using Bohr–Sommerfeld quantization with deformed periods in the Nekrasov–Shatashvili limit to probe non-perturbative physics. It confirms that the Hatsuda–Kallen–Marino singularity-cancellation mechanism largely constrains the non-perturbative sector, while also uncovering higher-order non-singular corrections through high-precision numerics and precision spectroscopy, enabling exact determination of early non-perturbative coefficients for models such as local \\mathbb{P}^2, \\mathbb{P}^1\\times\\mathbb{P}^1, and \\mathbb{F}_1. The authors derive explicit forms for leading non-perturbative terms, connect them to refined GV invariants, and validate the results against extensive numerical spectra, thereby strengthening the non-perturbative framework of refined topological strings and their quantum spectral problems. The findings illuminate the structure of phase-space volumes, deformed periods, and the interplay between perturbative and non-perturbative sectors, with potential links to ABJM-like matrix models and broader topological-string formulations.

Abstract

We consider certain quantum spectral problems appearing in the study of local Calabi-Yau geometries. The quantum spectrum can be computed by the Bohr-Sommerfeld quantization condition for a period integral. For the case of small Planck constant, the periods are computed perturbatively by deformation of the Omega background parameters in the Nekrasov-Shatashvili limit. We compare the calculations with the results from the standard perturbation theory for the quantum Hamiltonian. There have been proposals in the literature for the non-perturbative contributions based on singularity cancellation with the perturbative contributions. We compute the quantum spectrum numerically with some high precisions for many cases of Planck constant. We find that there are also some higher order non-singular non-perturbative contributions, which are not captured by the singularity cancellation mechanism. We fix the first few orders formulas of such corrections for some well known local Calabi-Yau models.

Paper Structure

This paper contains 15 sections, 147 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The local $\mathbb{P}^2$ model
  3. Classical ground state energy
  4. Quantum perturbative contributions
  5. Quantum non-perturbative contributions
  6. Numerical calculations of the spectrum
  7. Higher order non-perturbative contributions from precision spectroscopy
  8. The local $\mathbb{P}^1\times \mathbb{P}^1$ model
  9. Classical and perturbative contributions
  10. Non-perturbative contributions
  11. The local $\mathbb{F}_1$ model
  12. Classical and perturbative contributions
  13. Non-perturbative contributions
  14. Conclusion
  15. The refined Gopakumar-Vafa invariants

Figures (3)

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  • Figure 3: