Exact solution of the Φ23 finite matrix model

We find the exact solutions of the Φ23 finite matrix model (Grosse-Wulkenhaar model). In the Φ23 finite matrix model, multipoint correlation functions are expressed as G|a11…aN11|…|a1B…aNBB|. The ∑i=1BNi-point function denoted by G|a11…aN11|…|a1B…aNBB| is given by the sum over all Feynman diagrams (ribbon graphs) on Riemann surfaces with B-boundaries, and each |a1i⋯aNii| corresponds to the Feynman diagrams having Ni-external lines from the i-th boundary. It is known that any G|a11…aN11|…|a1B…aNBB| can be expressed using G|a1|…|an| type n-point functions. Thus we focus on rigorous calculations of G|a1|…|an|. The formula for G|a1|…|an| is obtained, and it is achieved by using the partition function 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.