Exact solution of matricial $\Phi^3_2$ quantum field theory

TL;DR

This paper presents an exact solution to a specific matrix model related to 2D quantum field theory, revealing detailed correlation functions and their analytic properties in the coupling constant.

Contribution

It provides the first exact solution of the $ ext{Phi}^3_2$ matrix model using integral equations derived from Ward-Takahashi identities and Schwinger-Dyson equations.

Findings

Correlation functions are exactly solvable in the large-N limit.

Correlation functions are analytic in the coupling constant.

The model arises from noncommutative field theory with specific properties.

Abstract

We apply a recently developed method to exactly solve the $\Phi^3$ matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward-Takahashi identities and Schwinger-Dyson equations lead in a special large-$\mathcal{N}$ limit to integral equations that we solve exactly for all correlation functions. Remarkably, these functions are analytic in the $\Phi^3$ coupling constant, although bounds on individual graphs justify only Borel summability. The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.