Exact Solutions v.s. Perturbative Calculations of Finite $\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model

There is a matrix model corresponding to a scalar field theory called Grosse-Wulkenhaar model, which is renormalizable by adding a harmonic oscillator potential to scalar $\Phi^{4}$ theory on Moyal spaces. There are more unknowns in $\Phi^{4}$ matrix model than in $\Phi^{3}$ matrix model, for example, in terms of integrability. We then construct a one-matrix model ($\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model) with multiple potentials, which is a combination of a $3$-point interaction and a $4$-point interaction, where the $3$-point interaction of $\Phi^{3}$ is multiplied by some positive definite diagonal matrix $M$. This model is solvable due to the effect of this $M$. In particular, the connected $\displaystyle\sum_{i=1}^{B}N_{i}$-point function $G_{|a_{N_{1}}^{1}\cdots a_{N_{1}}^{1}|\cdots|a_{1}^{B}\cdots a_{N_{B}}^{B}|}$ of $\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model is studied in detail. This $\displaystyle\sum_{i=1}^{B}N_{i}$-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with $B$ boundaries (punctures). Each $|a_{1}^{i}\cdots a_{N_{i}}^{i}|$ represents $N_{i}$ external lines coming from the $i$-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for $\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function $\mathcal{Z}[J]$ and use the result to compute exact solutions for $1$-point function $G_{|a|}$ with $1$-boundary, $2$-point function $G_{|ab|}$ with $1$-boundary, $2$-point function $G_{|a|b|}$ with $2$-boundaries, and $n$-point function $G_{|a^{1}|a^{2}|\cdots|a^{n}|}$ with $n$-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.