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Strong-weak symmetry and quantum modularity of resurgent topological strings on local $\mathbb{P}^2$

Veronica Fantini, Claudia Rella

TL;DR

This work develops a unified resurgence framework for resurgent topological strings on local $P^2$ via the TS/ST correspondence, revealing a strong-weak symmetry that exchanges perturbative and non-perturbative contributions between holomorphic and anti-holomorphic blocks. It constructs dual $L$-functions and $q$-series from the Stokes constants, showing these generating functions are holomorphic quantum modular forms for Gamma1(3) and can be reconstructed by median resummation. The analysis provides exact results for the first fermionic spectral trace, links resurgent data across weak and strong coupling, and exposes arithmetic twists connecting Stokes data across regimes. The results point to a modular resurgence paradigm linking spectral theory, topological strings, and number theory, with potential extensions to a broader class of toric Calabi–Yau geometries and higher fermionic traces, and implications for BPS state counting and mirror-symmetric geometry.

Abstract

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent formal power series in the Planck constant and its inverse. These are conjecturally captured by the Nekrasov-Shatashvili and standard topological string free energies, respectively, via the TS/ST correspondence. The resurgent structures of the first fermionic spectral trace of local $\mathbb{P}^2$ in both weak and strong coupling limits were solved exactly by the second author in [1]. Here, we argue that a full-fledged strong-weak resurgent symmetry is at play, exchanging the perturbative/non-perturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a global net of relations connecting the perturbative series and the discontinuities in the dual regimes, which is built upon the analytic properties of the $L$-functions with coefficients given by the Stokes constants and the $q$-series acting as their generating functions. Then, we show that the latter are holomorphic quantum modular forms for $Γ_1(3)$ and are reconstructed by the median resummation of their asymptotic expansions.

Strong-weak symmetry and quantum modularity of resurgent topological strings on local $\mathbb{P}^2$

TL;DR

This work develops a unified resurgence framework for resurgent topological strings on local via the TS/ST correspondence, revealing a strong-weak symmetry that exchanges perturbative and non-perturbative contributions between holomorphic and anti-holomorphic blocks. It constructs dual -functions and -series from the Stokes constants, showing these generating functions are holomorphic quantum modular forms for Gamma1(3) and can be reconstructed by median resummation. The analysis provides exact results for the first fermionic spectral trace, links resurgent data across weak and strong coupling, and exposes arithmetic twists connecting Stokes data across regimes. The results point to a modular resurgence paradigm linking spectral theory, topological strings, and number theory, with potential extensions to a broader class of toric Calabi–Yau geometries and higher fermionic traces, and implications for BPS state counting and mirror-symmetric geometry.

Abstract

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent formal power series in the Planck constant and its inverse. These are conjecturally captured by the Nekrasov-Shatashvili and standard topological string free energies, respectively, via the TS/ST correspondence. The resurgent structures of the first fermionic spectral trace of local in both weak and strong coupling limits were solved exactly by the second author in [1]. Here, we argue that a full-fledged strong-weak resurgent symmetry is at play, exchanging the perturbative/non-perturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a global net of relations connecting the perturbative series and the discontinuities in the dual regimes, which is built upon the analytic properties of the -functions with coefficients given by the Stokes constants and the -series acting as their generating functions. Then, we show that the latter are holomorphic quantum modular forms for and are reconstructed by the median resummation of their asymptotic expansions.
Paper Structure (17 sections, 14 theorems, 210 equations, 2 figures)

This paper contains 17 sections, 14 theorems, 210 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Resurgence of the spectral trace of local P2
  3. Geometric setup and the fermionic spectral traces
  4. Resurgent structures in the weak and strong coupling regimes
  5. Exact solutions for the spectral trace
  6. Resurgent L-functions
  7. Generating functions of the Stokes constants
  8. Strong-weak resurgent symmetry
  9. Functional equation from a unified perspective
  10. Arithmetic twist of the Stokes constants
  11. Summability and quantum modularity
  12. Borel--Laplace sums
  13. Median resummation
  14. Quantum modularity of the generating functions
  15. Conclusions
  16. ...and 2 more sections

Key Result

Lemma 3.1

The weak and strong coupling $L$-functions $L_0(s)$, $L_\infty(s)$, $s \in {\mathbb C}$, in Eqs. eq: convolution2-0 and eq: convolution2-infty are given by the Mellin transform of the generating functions $f_0(y)$, $f_\infty(y)$, $y \in {\mathbb H}$, of the corresponding Stokes constants, that is,

Figures (2)

  • Figure 1: Toric diagram of the local ${\mathbb P}^2$ geometry. We show the vectors in Eq. \ref{['eq: vecP2']} and the polyhedron $\Delta_{{\mathbb P}^2}$. The complex modulus $\kappa$ corresponds to the internal vertex of the diagram.
  • Figure 2: On the left, the first few singularities of the Borel transform of the asymptotic series $\phi(\hbar)$, defined in Eq. \ref{['eq: phiP2']}, and the associated integer constants $\alpha_n$, $n \in {\mathbb Z}_{\ne 0}$, defined in Eq. \ref{['eq: intStokesP2zero']}. On the right, the first few singularities of the Borel transform of the asymptotic series $\psi(\hbar)$, defined in Eq. \ref{['eq: phiP2infty']}, and the associated integer constants $\beta_n$, $n \in {\mathbb Z}_{\ne 0}$, defined in Eq. \ref{['eq: intStokesP2infty']}.

Theorems & Definitions (45)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • proof
  • ...and 35 more