Exact solution of the $\Phi_{2}^{3}$ 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.