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Complete solution of the LSZ Model via Topological Recursion

Johannes Branahl, Alexander Hock

TL;DR

The paper analyzes the Langmann–Szabo–Zarembo model with quartic interaction on noncommutative space and proves it is exactly solvable via topological recursion. Using Ward identities, the authors derive closed Dyson–Schwinger equations and define generalized boundary correlators that feed into a genus-zero spectral curve with explicit $x(z)$ and $y(z)$, from which the meromorphic forms $ω_{g,n}$ are generated by the Chekhov–Eynard–Orantin recursion; the correlators $Ω_n^{(g)}$ are tied to derivatives of $\log \mathcal{Z}$. They compare the LSZ framework to the Hermitian Grosse–Wulkenhaar/QKM model, showing that blobbed TR naturally arises in that setting due to holomorphic data while LSZ remains within standard TR. The authors validate their construction through cross-checks, perturbative expansions, and combinatorial limits, and discuss broader implications for integrability, noncommutative QFT, and connections to Hurwitz numbers. The work thus bridges noncommutative QFT toy models with enumerative geometry by embedding the LSZ model into the TR formalism and providing a complete recursive description of all correlation functions.

Abstract

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials $ω_{g,n}$ of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fieldss) and their consequences for the resulting particular type of topological recursion that governs the models.

Complete solution of the LSZ Model via Topological Recursion

TL;DR

The paper analyzes the Langmann–Szabo–Zarembo model with quartic interaction on noncommutative space and proves it is exactly solvable via topological recursion. Using Ward identities, the authors derive closed Dyson–Schwinger equations and define generalized boundary correlators that feed into a genus-zero spectral curve with explicit and , from which the meromorphic forms are generated by the Chekhov–Eynard–Orantin recursion; the correlators are tied to derivatives of . They compare the LSZ framework to the Hermitian Grosse–Wulkenhaar/QKM model, showing that blobbed TR naturally arises in that setting due to holomorphic data while LSZ remains within standard TR. The authors validate their construction through cross-checks, perturbative expansions, and combinatorial limits, and discuss broader implications for integrability, noncommutative QFT, and connections to Hurwitz numbers. The work thus bridges noncommutative QFT toy models with enumerative geometry by embedding the LSZ model into the TR formalism and providing a complete recursive description of all correlation functions.

Abstract

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fieldss) and their consequences for the resulting particular type of topological recursion that governs the models.
Paper Structure (22 sections, 27 theorems, 145 equations, 6 figures, 1 table)

This paper contains 22 sections, 27 theorems, 145 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction and Main Result
  2. The Setup
  3. Topological Recursion
  4. Correlation Functions
  5. Ward Identities
  6. Dyson-Schwinger Equations
  7. The 2-Point Function
  8. Generalised Correlation Functions
  9. Topological Recursion of the LSZ Model
  10. Analytic Continuation and Initial Data
  11. Complexified Dyson-Schwinger Equations
  12. Proof of Topological Recursion
  13. Complete Solution of the LSZ Model
  14. Cross Checks
  15. Recovering the Results of Langmann-Szabo-Zarembo Langmann:2003if
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(\overline{\mathbb{C}},x,y,B= \frac{dz_1\, dz_2}{(z_1-z_2)^2})$ be a spectral curve with where $x(\varepsilon_k)=E_k$ and $y(\tilde{\varepsilon}_k)=\tilde{E}_k$. Then, topological recursion generates meromorphic forms $\omega_{g,n}(z_1,...,z_n)$ according to eq. (eq:TR-intro), that solve the Langmann-Szabo-Zarembo model for quartic field interaction. In particular, $\omega_{g,n}$ relate From

Figures (6)

  • Figure 1: A guide through the main steps in this article towards the complete solution of the LSZ model. In the sense of matrix models, we arrive there in the fourth step by reconstructing the partition function with the free energies of topological recursion. TR will be however only an intermediate step to formulate the complete solution in the quantum field theoretical sense: Exact solutions as rational functions of all arbitrarily complicated correlation functions.
  • Figure 2: This is a graphical description of the topological recursion formula eq. (\ref{['eq:TR-intro']}). There are two different ways to obtain the left hand side $\omega_{g,n}$ by gluing the recursion kernel $K$ (a pair of pants, built by the initial data) with something of lower topology: Either one glues one object with one genus less and one boundary more $(g-1,n+2)$ along two boundaries of the kernel creating the missing genus, or one glues two objects with the kernel along its boundaries (causing no genus change): Then one has to account any $(g_1,n_1),(g_2,n_2)$ conform with the left hand side -- the sum over all possible partitions in the master formula eq. (\ref{['eq:TR-intro']}) arises.
  • Figure 3: Graphical interpretation of blobbed topological recursion: The usual recursion formula is enriched by a holomorphic (at ramification points) term $\mathcal{H}_z\omega_{g,n+1}$ (coloured) that appears as a surplus structure in the solution of the loop equations. It has to be seen as additional data that has to be taken into account at each further recursion step.
  • Figure 4: Ribbon graphs contributing to the planar 2-point function of the quartic LSZ model up to order $\mathcal{O}(\lambda^2)$. At order $\lambda^k$, a total amount of $\frac{2 \cdot 3^k(2k)!}{k!(k+2)!}$ ribbon graphs contributes.
  • Figure 5: Ribbon graphs contributing to the planar four-point function of the quartic LSZ model up to order $\mathcal{O}(\lambda^2)$, where the first two graphs of order $\lambda^2$ contribute with four different labellings and the last graph with two different labellings (after probably inverting the orientation).
  • ...and 1 more figures

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Abstract loop equations
  • Theorem 2.2: Borot:2013lpa
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • ...and 52 more