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Resurgent Transseries and the Holomorphic Anomaly

Ricardo Couso-Santamaría, Jose D. Edelstein, Ricardo Schiappa, Marcel Vonk

TL;DR

This work develops a nonperturbative completion of the holomorphic anomaly equations for closed topological strings in local Calabi–Yau geometries by constructing resurgent transseries. It shows that instanton actions are holomorphic and that anti-holomorphic dependence is captured by propagators, with holomorphic ambiguities fixed at conifold points to ensure integrability. The authors derive explicit one- and multi-parameter transseries structures, including exponential factors in propagators and generating-function–encoded combinatorics, and establish large-order relations (resurgence) that connect perturbative and nonperturbative data. The framework lays the groundwork for nonperturbative topological string theory, nonperturbative topological recursion, and robust tests of large-$N$ holography, with concrete strategies for fixing ambiguities and analyzing universality near phase transitions like the conifold point.

Abstract

The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion in order to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the 't Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi-Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries, tantamount to nonperturbative topological string theory. This paper addresses this point by showing how to construct resurgent transseries solutions to the holomorphic anomaly equations. These solutions are built upon (generalized) multi-instanton sectors, where the instanton actions are holomorphic. The asymptotic expansions around the multi-instanton sectors have both holomorphic and anti-holomorphic dependence, may allow for resonance, and their structure is completely fixed by the holomorphic anomaly equations in terms of specific polynomials multiplied by exponential factors and up to the holomorphic ambiguities -- which generalizes the known perturbative structure to the full transseries. In particular, the anti-holomorphic dependence has a somewhat universal character. Furthermore, in the nonperturbative sectors, holomorphic ambiguities may be fixed at conifold points. This construction shows the nonperturbative integrability of the holomorphic anomaly equations, and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory.

Resurgent Transseries and the Holomorphic Anomaly

TL;DR

This work develops a nonperturbative completion of the holomorphic anomaly equations for closed topological strings in local Calabi–Yau geometries by constructing resurgent transseries. It shows that instanton actions are holomorphic and that anti-holomorphic dependence is captured by propagators, with holomorphic ambiguities fixed at conifold points to ensure integrability. The authors derive explicit one- and multi-parameter transseries structures, including exponential factors in propagators and generating-function–encoded combinatorics, and establish large-order relations (resurgence) that connect perturbative and nonperturbative data. The framework lays the groundwork for nonperturbative topological string theory, nonperturbative topological recursion, and robust tests of large- holography, with concrete strategies for fixing ambiguities and analyzing universality near phase transitions like the conifold point.

Abstract

The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion in order to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the 't Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi-Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries, tantamount to nonperturbative topological string theory. This paper addresses this point by showing how to construct resurgent transseries solutions to the holomorphic anomaly equations. These solutions are built upon (generalized) multi-instanton sectors, where the instanton actions are holomorphic. The asymptotic expansions around the multi-instanton sectors have both holomorphic and anti-holomorphic dependence, may allow for resonance, and their structure is completely fixed by the holomorphic anomaly equations in terms of specific polynomials multiplied by exponential factors and up to the holomorphic ambiguities -- which generalizes the known perturbative structure to the full transseries. In particular, the anti-holomorphic dependence has a somewhat universal character. Furthermore, in the nonperturbative sectors, holomorphic ambiguities may be fixed at conifold points. This construction shows the nonperturbative integrability of the holomorphic anomaly equations, and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory.

Paper Structure

This paper contains 22 sections, 7 theorems, 214 equations.

Table of Contents

  1. Introduction and Summary
  2. The Holomorphic Anomaly Equations
  3. Reviewing the Background Calabi--Yau Geometry
  4. Rewriting the Holomorphic Anomaly Equations
  5. A Word on the Transseries Framework
  6. One--Parameter Resurgent Transseries
  7. A Closer Look at the Perturbative Sector
  8. Setting the Transseries Construction
  9. Structure of the Transseries Solution
  10. Multi--Parameter Resurgent Transseries
  11. Setting the Generalized Transseries Construction
  12. Structure of the Generalized Transseries Solution
  13. Resonance and Logarithmic Sectors
  14. Genus Expansions within Transseries Solutions
  15. A Glimpse of Large--Order Analysis
  16. ...and 7 more sections

Key Result

Theorem 1

For any $\boldsymbol{n}\neq \boldsymbol{0}$ and $g\geq 0$, the structure of the nonperturbative free energies has the form where the set of numbers $\left\{ a_{\alpha\beta} \left( \boldsymbol{n}; \gamma_{\boldsymbol{n}} \right) \right\}$ and $\left\{ \lambda_b \left( \boldsymbol{n}; \gamma_{\boldsymbol{n}} \right) \right\}$ are read off from the generating functionBecause we are not focusing on t

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1
  • proof : Proof of the Theorem
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 3
  • proof
  • ...and 2 more