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A universal formula for the $x-y$ swap in topological recursion

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

TL;DR

This work establishes and proves a universal algebraic mechanism for the x–y swap in topological recursion, expressing swapped higher-point differentials $oldsymbol{ me}^{(g)}_{0,n}$ in terms of the original TR data $oldsymbol{ me}^{(g)}_{m,0}$ through universal graph-based expressions, vertex-operator contractions, and mixed correlators. It presents two equivalent formulations of the swap, shows their equivalence, and proves that the swapped correlators satisfy the same topological recursion when the original data do, thereby furnishing a complete duality framework. The authors also provide a simplified, shift-invariant form of the swap, extend the formalism to mixed $(m,n)$-points, and develop higher loop equations that underpin the duality; these results yield explicit closed formulas in important cases (e.g., $y(z)=z$) and recover notable instances such as Witten–Kontsevich. Collectively, the paper delivers a universal toolbox for translating TR outputs across x–y duality, with broad implications for matrix models, enumerative geometry, and related integrable systems. The methods combine Fock-space vertex operator techniques, graph-sum constructions, and a rigorous analysis of poles and loop equations to achieve computationally tractable, genus-aware dual formulas.

Abstract

We prove a recent conjecture of Borot et al. that a particular universal closed algebraic formula recovers the correlation differentials of topological recursion after the swap of $x$ and $y$ in the input data. We also show that this universal formula can be drastically simplified (as it was already done by Hock). As an application of this general $x-y$ swap result, we prove an explicit closed formula for the topological recursion differentials for the case of any spectral curve with unramified $y$ and arbitrary rational $x$.

A universal formula for the $x-y$ swap in topological recursion

TL;DR

This work establishes and proves a universal algebraic mechanism for the x–y swap in topological recursion, expressing swapped higher-point differentials in terms of the original TR data through universal graph-based expressions, vertex-operator contractions, and mixed correlators. It presents two equivalent formulations of the swap, shows their equivalence, and proves that the swapped correlators satisfy the same topological recursion when the original data do, thereby furnishing a complete duality framework. The authors also provide a simplified, shift-invariant form of the swap, extend the formalism to mixed -points, and develop higher loop equations that underpin the duality; these results yield explicit closed formulas in important cases (e.g., ) and recover notable instances such as Witten–Kontsevich. Collectively, the paper delivers a universal toolbox for translating TR outputs across x–y duality, with broad implications for matrix models, enumerative geometry, and related integrable systems. The methods combine Fock-space vertex operator techniques, graph-sum constructions, and a rigorous analysis of poles and loop equations to achieve computationally tractable, genus-aware dual formulas.

Abstract

We prove a recent conjecture of Borot et al. that a particular universal closed algebraic formula recovers the correlation differentials of topological recursion after the swap of and in the input data. We also show that this universal formula can be drastically simplified (as it was already done by Hock). As an application of this general swap result, we prove an explicit closed formula for the topological recursion differentials for the case of any spectral curve with unramified and arbitrary rational .
Paper Structure (43 sections, 23 theorems, 161 equations)

This paper contains 43 sections, 23 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Topological recursion
  3. The x-y swap
  4. Historical perspective
  5. Conjecture of Borot et al.
  6. Universal algebraic expressions
  7. The conjecture
  8. Explicit formula
  9. A reformulation of the conjecture
  10. Mixed correlation differentials
  11. Relation to the work of Eynard--Orantin
  12. Review of the paper
  13. Organization of the paper
  14. Acknowledgments
  15. Mixed correlation differentials
  16. ...and 28 more sections

Key Result

Theorem 1.8

Conjecture conj:Main holds. That is, assume the differentials $\omega^{(g)}_{m,0}$ satisfy the topological recursion for the input data $(S,x,y,B)$ with meromorphic $x$ and $y$, all zeros of $dy$ are simple, and $x$ is regular at the zeros of $dy$. Then the differentials $\omega^{(g)}_{0,n}$ given b

Theorems & Definitions (81)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Conjecture 1.7: BCGFLS-Free
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 71 more