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Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

Kohei Iwaki, Omar Kidwai

Abstract

We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices $Ω$ in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's $τ$-function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function $Z_{\rm TR}$, up to a simple factor.

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

Abstract

We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's -function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function , up to a simple factor.

Paper Structure

This paper contains 47 sections, 24 theorems, 159 equations, 3 tables.

Table of Contents

  1. Introduction
  2. BPS structures
  3. Topological recursion and spectral curves of hypergeometric type
  4. Brief review of results of Part I
  5. Results of Part II
  6. Voros solution to the (almost-doubled) BPS Riemann-Hilbert problem
  7. Relation between the Voros solution and other solutions
  8. Voros potential, BPS $\tau$-function and TR partition function function
  9. Remarks
  10. Organization
  11. Acknowledgements
  12. Review of BPS structures and the BPS Riemann-Hilbert problem
  13. Definition of BPS structure
  14. Twisted torus and BPS automorphism
  15. BPS Riemann-Hilbert problem
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $(\Gamma, Z, \Omega)$ denote a BPS structure obtained from any spectral curve of hypergeometric type. The Borel sum of cycle and path Voros symbols of the corresponding quantum curve ${\bm E}({\bm \nu})$ provide a meromorphic solution to the BPS Riemann-Hilbert problem associated to the almost-d

Theorems & Definitions (64)

  • Theorem 1.1: Theorem \ref{['thm:voros-soln']}
  • Theorem 1.2: Theorem \ref{['thm:taupotential']}
  • Corollary 1.3
  • Theorem 1.4: Theorem \ref{['thm:tauhol']}
  • Definition 2.1: Bri19
  • Remark 2.3
  • Theorem 2.4: BarBri19
  • Definition 2.5: Bri19
  • Definition 2.6
  • Definition 2.7: Bri19
  • ...and 54 more