# Solution of all quartic matrix models

## TL;DR

This paper provides an exact solution to the quartic matrix model, extending previous results to finite and large N, and connects it to noncommutative quantum field theory without triviality issues.

## Contribution

It identifies the exact two-point function solution for the quartic matrix model using complex analysis and extends it to unbounded operators, linking to topological recursion.

## Key findings

- Exact rational function form of the two-point function for finite N
- Solution for large N involving spectral regularization of R
- Proof that the noncommutative -model is non-trivial

## Abstract

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-{N}\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitian ${N}\times{N}$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. It was previously established that the large-$N$ limit of the second moment (the planar two-point function) satisfies a non-linear integral equation. By employing tools from complex analysis, in particular the Lagrange-B\"urmann inversion formula, we identify the exact solution of this non-linear problem, both for finite $N$ and for a large-${N}$ limit to unbounded operators $E$ of spectral dimension $\leq 4$. For finite $N$, the two-point function is a rational function evaluated at the preimages of another rational function $R$ constructed from the spectrum of $E$. Subsequent work has constructed from this formula a family $\omega_{g,n}$ of meromorphic differentials which obey blobbed topological recursion. For unbounded operators $E$, the renormalised two-point function is given by an integral formula involving a regularisation of $R$. This allowed a proof, in subsequent work, that the $\lambda\Phi^4_4$-model on noncommutative Moyal space does not have a triviality problem.