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Blobbed topological recursion from extended loop equations

Alexander Hock, Raimar Wulkenhaar

TL;DR

The paper proves that the quartic Kontsevich matrix model, under the external covariance $\langle M_{kl}M_{mn}\rangle=\frac{\delta_{kn}\delta_{lm}}{N(E_k+E_l)}$ and potential $\frac{\lambda N}{4}\mathrm{tr}(M^4)$, yields a blobbed topological recursion structure. It builds a six-function framework from Dyson–Schwinger equations, derives global linear and quadratic loop equations valid beyond ramification neighborhoods, and, for genus up to $1$, constructs a recursion kernel that expresses $\omega^{(g)}_{n}$ in terms of lower-genus data with residues at ramification points, opposite diagonals, and $z=0$. The work connects to the classic Kontsevich setting via the entangled spectral curve $x,y$ with $y(z)=-x(-z)$ and extends topological recursion to include blob-like corrections arising from poles away from ramification points. The results lay groundwork for a full blobbed TR proof at all genera and suggest links to $x$-$y$ duality and generalized TR in higher-genus spectral geometries, with potential implications for enumerative geometry and integrable hierarchies.

Abstract

We consider the $N\times N$ Hermitian matrix model with measure $dμ_{E,λ}(M)=\frac{1}{Z} \exp(-\frac{λN}{4} \mathrm{tr}(M^4)) dμ_{E,0}(M)$, where $dμ_{E,0}$ is the Gaussian measure with covariance $\langle M_{kl}M_{mn}\rangle=\frac{δ_{kn}δ_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. It was previously understood that this setting gives rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by $y(z)=-x(-z)$ and a family $ω^{(g)}_{n}$ of meromorphic differentials conjectured to obey blobbed topological recursion due to Borot and Shadrin. We develop a new approach to this problem via a system of six meromorphic functions which satisfy extended loop equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. An expansion at $\infty$ gives global linear and quadratic loop equations for the $ω^{(g)}_{n}$. These global equations provide the $ω^{(g)}_{n}$ not only in the vicinity of the ramification points of $x$ but also in the vicinity of all other poles located at opposite diagonals $z_i+z_j=0$ and at $z_i=0$. We deduce a recursion kernel representation valid at least for $g\leq 1$.

Blobbed topological recursion from extended loop equations

TL;DR

The paper proves that the quartic Kontsevich matrix model, under the external covariance and potential , yields a blobbed topological recursion structure. It builds a six-function framework from Dyson–Schwinger equations, derives global linear and quadratic loop equations valid beyond ramification neighborhoods, and, for genus up to , constructs a recursion kernel that expresses in terms of lower-genus data with residues at ramification points, opposite diagonals, and . The work connects to the classic Kontsevich setting via the entangled spectral curve with and extends topological recursion to include blob-like corrections arising from poles away from ramification points. The results lay groundwork for a full blobbed TR proof at all genera and suggest links to - duality and generalized TR in higher-genus spectral geometries, with potential implications for enumerative geometry and integrable hierarchies.

Abstract

We consider the Hermitian matrix model with measure , where is the Gaussian measure with covariance for given . It was previously understood that this setting gives rise to two ramified coverings of the Riemann sphere strongly tied by and a family of meromorphic differentials conjectured to obey blobbed topological recursion due to Borot and Shadrin. We develop a new approach to this problem via a system of six meromorphic functions which satisfy extended loop equations. Two of these functions are symmetric in the preimages of and can be determined from their consistency relations. An expansion at gives global linear and quadratic loop equations for the . These global equations provide the not only in the vicinity of the ramification points of but also in the vicinity of all other poles located at opposite diagonals and at . We deduce a recursion kernel representation valid at least for .
Paper Structure (18 sections, 20 theorems, 103 equations)

This paper contains 18 sections, 20 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Historical comments
  3. Main result
  4. Setup
  5. Summary of previous results
  6. Auxiliary functions
  7. Motivation of a loop insertion operator
  8. Definition of a loop insertion operator on meromorphic functions
  9. Solution for genus $g=0$
  10. Auxiliary functions for genus $g = 0$
  11. Linear and quadratic loop equations for genus $g=0$
  12. Computation of $\hat{Q}^{(0)}$
  13. Solution for $g=1$
  14. The case $I=\emptyset$
  15. The case $I\neq \emptyset$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $I=\{u_1,...,u_n\}$ and $x$ be a generic ramified cover of $\hat{\mathbb{C}}$. Let $x,y$ be related via glInvolution and $\omega^{(0)}_2(z,w)=\frac{dz dw}{(z-w)^2}+\frac{dz dw}{(z+w)^2}$. If the seven functions $W^{(g)}_{n+1}$, $(P^{(g)},H^{(g)},U^{(g)})$ and $(Q^{(g)},M^{(g)},V^{(g)})$ satisfy where $\beta_i$ are the ramification points of $x$, $\omega^{(0)}_{2}(q,q)$ should be replaced by $

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 32 more