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Topological recursion, symplectic duality, and generalized fully simple maps

Alexander Alexandrov, Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin

TL;DR

The paper develops a broad framework connecting topological recursion on different spectral curves via a $\psi$-dependent symplectic duality, providing an explicit universal formula that expresses dual $n$-point functions in terms of the original TR data and vice versa. By constructing a large class of dual spectral curves and proving that the resulting $0,n$-point functions satisfy the loop equations and projection property on the dual curve, the authors show that generalized fully simple maps also satisfy topological recursion. The results generalize prior duality formulas and establish TR for generalized fully simple maps as a corollary, reinforcing a deep link between dual spectral-data and enumerative geometries encoded by Fock-space correlators. This duality offers a powerful tool for identifying TR structures in new combinatorial and geometric settings and for verifying TR when an explicit duality relation is known. The work thus broadens the scope of TR applicability and provides concrete, checkable formulas for translating TR data across dual spectral curves with potential applications in knot invariants and related combinatorial theories.

Abstract

For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the $n$-point functions produced by the topological recursion on these curves via the $n$-point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions.

Topological recursion, symplectic duality, and generalized fully simple maps

TL;DR

The paper develops a broad framework connecting topological recursion on different spectral curves via a -dependent symplectic duality, providing an explicit universal formula that expresses dual -point functions in terms of the original TR data and vice versa. By constructing a large class of dual spectral curves and proving that the resulting -point functions satisfy the loop equations and projection property on the dual curve, the authors show that generalized fully simple maps also satisfy topological recursion. The results generalize prior duality formulas and establish TR for generalized fully simple maps as a corollary, reinforcing a deep link between dual spectral-data and enumerative geometries encoded by Fock-space correlators. This duality offers a powerful tool for identifying TR structures in new combinatorial and geometric settings and for verifying TR when an explicit duality relation is known. The work thus broadens the scope of TR applicability and provides concrete, checkable formulas for translating TR data across dual spectral curves with potential applications in knot invariants and related combinatorial theories.

Abstract

For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the -point functions produced by the topological recursion on these curves via the -point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions.
Paper Structure (15 sections, 9 theorems, 42 equations)

This paper contains 15 sections, 9 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Two formulations of topological recursion
  3. Original definition
  4. Loop equation definition
  5. Spectral curve symplectic duality
  6. Generating functions of generalized fully simple maps
  7. Organization of the paper
  8. Acknowledgments
  9. Expansion and the (m,n)-point functions
  10. Expansion of the m-point functions
  11. (m,n)-point functions
  12. Topological recursion
  13. Loop equations
  14. Projection property
  15. Proof of Theorem \ref{['thm:mainthm']}

Key Result

Theorem 1.6

Let where $R$ is an arbitrary rational function and $P$ is an arbitrary polynomial, such that $\hat{\psi}(0,0)=0$, and let Let $(\Sigma,X,y,B)$ be a spectral curve with the meromorphy condition, that is, we assume that $d\log X$ and $y$ are meromorphic, and let $(\Sigma,w,y,B)$ be its $\psi$-symplectic dual curve according to Definition def:psidual. We also further assume the generality conditio

Theorems & Definitions (32)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 22 more