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Blobbed topological recursion of the quartic Kontsevich model II: Genus=0

Alexander Hock, Raimar Wulkenhaar

TL;DR

This work establishes that the genus-0 sector of the quartic Kontsevich model is entirely controlled by a global involution $ι$ acting on the spectral data, with $y(z)=-x(-z)$ and a symmetric $ω_{0,2}$ under $z\mapsto -z$. The authors derive a single involution identity that fixes $ω_{0,m+1}(I,z)$ from lower $m$ data and prove that its solution satisfies blobbed topological recursion at genus 0. The key technical advances are a careful residue calculus around ramification points, the construction of recursion kernels, and a detailed symmetry analysis that enforces pole cancellations and holomorphy at involution-fixed points. Consequently, the genus-0 correlation forms $ω_{0,n}$ coincide with the blobbed TR structure for the quartic Kontsevich model, confirming the conjectured extension of TR to this setting. The results illuminate a deep link between an involutive symmetry on the spectral curve and the universal blobbed TR framework, with potential implications for higher-genus generalizations and moduli-space geometry.

Abstract

We prove that the genus-0 sector of the quartic analogue of the Kontsevich model is completely governed by an involution identity which expresses the meromorphic differential $ω_{0,n}$ at a reflected point $ιz$ in terms of all $ω_{0,m}$ with $m\leq n$ at the original point $z$. We prove that the solution of the involution identity obeys blobbed topological recursion, which confirms a previous conjecture about the quartic Kontsevich model.

Blobbed topological recursion of the quartic Kontsevich model II: Genus=0

TL;DR

This work establishes that the genus-0 sector of the quartic Kontsevich model is entirely controlled by a global involution acting on the spectral data, with and a symmetric under . The authors derive a single involution identity that fixes from lower data and prove that its solution satisfies blobbed topological recursion at genus 0. The key technical advances are a careful residue calculus around ramification points, the construction of recursion kernels, and a detailed symmetry analysis that enforces pole cancellations and holomorphy at involution-fixed points. Consequently, the genus-0 correlation forms coincide with the blobbed TR structure for the quartic Kontsevich model, confirming the conjectured extension of TR to this setting. The results illuminate a deep link between an involutive symmetry on the spectral curve and the universal blobbed TR framework, with potential implications for higher-genus generalizations and moduli-space geometry.

Abstract

We prove that the genus-0 sector of the quartic analogue of the Kontsevich model is completely governed by an involution identity which expresses the meromorphic differential at a reflected point in terms of all with at the original point . We prove that the solution of the involution identity obeys blobbed topological recursion, which confirms a previous conjecture about the quartic Kontsevich model.

Paper Structure

This paper contains 29 sections, 34 theorems, 194 equations, 1 table.

Table of Contents

  1. Introduction and main result
  2. Overview
  3. Statement of the result
  4. Proof of Theorem \ref{['thm:flip']}
  5. Tools and conventions
  6. Poles of ω(u1...,um,z) at z=ι(uk)
  7. Symmetry of the involution identity I: q->ι uk and q->uk
  8. Linear loop equation
  9. The recursion kernel
  10. Symmetry of the involution identity II: q->β and q->ιβ
  11. Finishing the proof of Theorem \ref{['thm:flip']}
  12. The sum over all preimages
  13. A particular symmetry under the involution ι
  14. The quartic Kontsevich model
  15. Summary of previous results
  16. ...and 14 more sections

Key Result

Theorem 1.2

For $\omega_{0,|I|+1}(I,z)$ with $I=\{u_1,...,u_m\}$ of length $|I|:=m$ the following conventions are given: Then eq:flip-om is for $I=\{u_1,...,u_m\}$ with $m\geq 2$ uniquely solved by Here $\beta_1,...,\beta_r$ are the ramification points of $x$ and $\sigma_i\neq \mathrm{id}$ denotes the local Galois involution in the vicinity of $\beta_i$, i.e. $x(\sigma_i(z))=x(z)$, $\lim_{z\to \beta_i}\sigm

Theorems & Definitions (69)

  • Definition 1.1: involution identity
  • Theorem 1.2
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 59 more