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Symmetry of meromorphic differentials produced by involution identity, and relation to integer partitions

Alexander Hock, Sergey Shadrin, Raimar Wulkenhaar

TL;DR

This work proves that the meromorphic differentials $ω^{(0)}_n(z_1,...,z_n)$, defined recursively through an involution identity on multiple copies of $\mathbb{P}^1$, are symmetric in all their arguments. By analyzing poles at $z_1=\iota z_2$ and at ramification points $β_j$, the authors reduce the symmetry question to a pure combinatorial statement about integer partitions, Theorem conj main, which they prove using a detailed partition-based counting argument. The central result links analytic residue calculus and loop equations with a combinatorial identity involving multinomial coefficients of partitions, thereby underpinning symmetry and supporting blobbed topological recursion in this setting. The work also notes connections to quartic Kontsevich-type models and outlines avenues for extending the symmetry to genus-one objects $ω^{(1)}_n$ in future research.

Abstract

We prove that meromorphic differentials $ω^{(0)}_n(z_1,...,z_n)$ which are recursively generated by an involution identity are symmetric in all their arguments $z_1,...,z_n$. The proof involves an intriguing combinatorial identity between integer partitions into given number of parts.

Symmetry of meromorphic differentials produced by involution identity, and relation to integer partitions

TL;DR

This work proves that the meromorphic differentials , defined recursively through an involution identity on multiple copies of , are symmetric in all their arguments. By analyzing poles at and at ramification points , the authors reduce the symmetry question to a pure combinatorial statement about integer partitions, Theorem conj main, which they prove using a detailed partition-based counting argument. The central result links analytic residue calculus and loop equations with a combinatorial identity involving multinomial coefficients of partitions, thereby underpinning symmetry and supporting blobbed topological recursion in this setting. The work also notes connections to quartic Kontsevich-type models and outlines avenues for extending the symmetry to genus-one objects in future research.

Abstract

We prove that meromorphic differentials which are recursively generated by an involution identity are symmetric in all their arguments . The proof involves an intriguing combinatorial identity between integer partitions into given number of parts.
Paper Structure (7 sections, 9 theorems, 93 equations)

This paper contains 7 sections, 9 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The pole at $z_1= \iota z_2$
  4. The pole at $z_1=\beta_j$
  5. Translation into a combinatorial problem
  6. Combinatorial argument
  7. Outlook

Key Result

Theorem 1

Let $\mathsf{P}_k(n)$ be the set of partitions of an integer $n\geq 1$ into $1\leq k\leq n$ parts. For any given integers $(s,k,l)$ with $k,l\geq 0$ and $s\geq \max(k+l,1)$ and any given partition $\nu \in \mathsf{P}_s(2s-k-l)$ one has By $\binom{\nu}{\mu_1,...,\mu_l}$ we denote the multinomial coefficient of partitions, where $\binom{h_k}{g_{k,1},...,g_{k,l}}$ are the usual multinomial coeffici

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Hock:2021tbl
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['thm:symmetry-differentials']}
  • Example 7
  • Example 8
  • ...and 9 more