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Resurgence in the Universal Structures in B-model Topological String Theory

Yong Li, David Sauzin, Shanzhong Sun

TL;DR

The paper develops a rigorous resurgence-analytic framework for perturbative all-genus free energies in the B-model of topological string theory under two regimes: AyZ’s double scaling limit and Couso-Santamaría’s large-radius limit. It demonstrates that the perturbative expansions are simple $2 ext{Z}$-resurgent and constructs two-parameter transseries solving the holomorphic anomaly equations via alien calculus and the Bridge Equation, with Borel-Laplace summation yielding real-analytic solutions and explicit Stokes phenomena. A key achievement is the precise linking of the double-scaling transseries to the large-radius transseries through explicit changes of variables, uncovering a common resurgent structure that governs both limits. The results provide a rigorous mathematical backbone for non-perturbative completions in topological string theory and open avenues for interpreting the resulting transseries coefficients and connection formulas in geometric/enumerative terms.

Abstract

We propose a systematic analysis of Alim-Yau-Zhou's double scaling limit and Couso-Santamaría's large radius limit for the perturbative free energies in B-model topological string theory based on Écalle's Resurgence Theory. Taking advantage of the known resurgent properties of the formal solutions to the Airy equation and of the stability of resurgent series under exponential/logarithm and nonlinear changes of variable, we show how to rigorously derive the non-perturbative information from the perturbative one by means of alien calculus in this context, spelling out the notions of formal integral and Bridge Equation, typical of the resurgent approach to ordinary differential equations. We also discuss the Borel-Laplace summation of the obtained resurgent transseries, including a study of real analyticity based on the connection formulas stemming from the resummation of the Bridge Equation.

Resurgence in the Universal Structures in B-model Topological String Theory

TL;DR

The paper develops a rigorous resurgence-analytic framework for perturbative all-genus free energies in the B-model of topological string theory under two regimes: AyZ’s double scaling limit and Couso-Santamaría’s large-radius limit. It demonstrates that the perturbative expansions are simple -resurgent and constructs two-parameter transseries solving the holomorphic anomaly equations via alien calculus and the Bridge Equation, with Borel-Laplace summation yielding real-analytic solutions and explicit Stokes phenomena. A key achievement is the precise linking of the double-scaling transseries to the large-radius transseries through explicit changes of variables, uncovering a common resurgent structure that governs both limits. The results provide a rigorous mathematical backbone for non-perturbative completions in topological string theory and open avenues for interpreting the resulting transseries coefficients and connection formulas in geometric/enumerative terms.

Abstract

We propose a systematic analysis of Alim-Yau-Zhou's double scaling limit and Couso-Santamaría's large radius limit for the perturbative free energies in B-model topological string theory based on Écalle's Resurgence Theory. Taking advantage of the known resurgent properties of the formal solutions to the Airy equation and of the stability of resurgent series under exponential/logarithm and nonlinear changes of variable, we show how to rigorously derive the non-perturbative information from the perturbative one by means of alien calculus in this context, spelling out the notions of formal integral and Bridge Equation, typical of the resurgent approach to ordinary differential equations. We also discuss the Borel-Laplace summation of the obtained resurgent transseries, including a study of real analyticity based on the connection formulas stemming from the resummation of the Bridge Equation.

Paper Structure

This paper contains 25 sections, 20 theorems, 321 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A brief compendium of Resurgence theory
  3. Borel transform, convolution and Laplace transform
  4. Borel-Laplace summation of formal power series
  5. Extension to non-integer powers
  6. Asymptotic expansion property, compatibility with composition
  7. Example: Asymptotics of the solutions to the Airy equation
  8. Alien calculus for simple Ω-resurgent series
  9. Extension to more general singularities
  10. Resurgence in the Airy equation
  11. Simple Ω-resurgent transseries
  12. Resurgent analysis of the free energy in the double scaling limit
  13. Link with the series $\widetilde{\psi}(z)$ of § \ref{['exa1']} and first summability result
  14. Resurgent structure, formal integral and Bridge Equation
  15. Another view on the formal integral
  16. ...and 10 more sections

Key Result

Theorem 2.6

Suppose $H(t) = \sum\limits_{n=0}^\infty H_n t^n \in \mathbb{C}\{t\}$, $\widetilde{\varphi}_*\in\mathscr{N}(I)$, and $\widetilde{\varphi},\widetilde{\psi} \in \mathbb{C}\oplus\mathscr{N}(I)$. Then the formal series are 1-summable in the directions of I, with

Figures (9)

  • Figure 1: Left: Directions for Laplace integration with $\theta \in J=[\theta_1,\theta_2]$. Right: The union of half-planes $\mathscr{D}_J$.
  • Figure 2: Left: The arcs of directions $I^+$ and $I^-$ in the Borel plane. Right: The domains $\mathscr{D}_{I^+}$ and $\mathscr{D}_{I^-}$ in the plane of the variable $z=\frac{1}{3}\lambda_s^{-2}$.
  • Figure 3: The domains $\mathscr{D}_{I^+}\cap\mathscr{D}_{I^-}$ and $\mathscr{D}_{I^-}\cap(e^{-2\pi i}\mathscr{D}_{I^+})$.
  • Figure 4: Analytic continuation along $\gamma$, for $\zeta$ near $\omega$.
  • Figure 5: An example of a path $\gamma(\varepsilon)$, with $r=4$ and $\varepsilon=(+,+,-).$
  • ...and 4 more figures

Theorems & Definitions (60)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: [M.S.]
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 50 more