Exact solution of the $\Phi_{2}^{3}$ finite matrix model

TL;DR

This paper derives exact solutions for the $ ext{Φ}_2^3$ finite matrix model, expressing multipoint correlation functions via Feynman diagrams and Airy functions, advancing precise analytical understanding of this quantum field theory.

Contribution

It provides the first rigorous formulas for multipoint correlation functions in the $ ext{Φ}_2^3$ finite matrix model using Harish-Chandra-Itzykson-Zuber integrals and Airy functions.

Findings

Explicit formulas for $G_{|a|}$, $G_{|ab|}$, $G_{|a|b|}$, and $G_{|a|b|c|}$.

Correlation functions expressed with Airy functions.

Exact solutions for multipoint functions in the finite matrix model.

Abstract

We find the exact solutions of the $\Phi_{2}^{3}$ finite matrix model (Grosse-Wulkenhaar model). In the $\Phi_{2}^{3}$ finite matrix model, multipoint correlation functions are expressed as $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$. The $\displaystyle \sum_{i=1}^{B}N_{i}$-point function denoted by $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$ is given by the sum over all Feynman diagrams (ribbon graphs) on Riemann surfaces with $B$-boundaries, and each $|a^{i}_{1}\cdots a^{i}_{N_{i}}|$ corresponds to the Feynman diagrams having $N_{i}$-external lines from the $i$-th boundary. It is known that any $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$ can be expressed using $G_{|a^{1}|\ldots|a^{n}|}$ type $n$-point functions. Thus we focus on rigorous calculations of $G_{|a^{1}|\ldots|a^{n}|}$. The formula for $G_{|a^{1}|\ldots|a^{n}|}$ is obtained, and it is achieved by using the partition function $\mathcal{Z}[J]$ calculated by the Harish-Chandra-Itzykson-Zuber integral. We give $G_{|a|}$, $G_{|ab|}$, $G_{|a|b|}$, and $G_{|a|b|c|}$ as the specific simple examples. All of them are described by using the Airy functions.