标题： 精确解的 $Φ_{2}^{3}$ 有限矩阵模型 显示英文标题

标题： Exact solution of the $Φ_{2}^{3}$ finite matrix model

摘要： 我们找到了 $\Phi_{2}^{3}$ 有限矩阵模型的精确解（Grosse-Wulkenhaar 模型）。在 $\Phi_{2}^{3}$ 有限矩阵模型中，多点关联函数表示为 $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$。 记为$G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$的 $\displaystyle \sum_{i=1}^{B}N_{i}$-点函数由定义在具有 $B$-边界的黎曼曲面上的所有费曼图（带边图形）的求和给出，并且每个 $|a^{i}_{1}\cdots a^{i}_{N_{i}}|$对应于具有来自 $i$-th 边界的 $N_{i}$-外部线的费曼图。 已知任何$G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$都可以用$G_{|a^{1}|\ldots|a^{n}|}$类型的$n$点函数来表示。 因此，我们集中于严格的$G_{|a^{1}|\ldots|a^{n}|}$计算。 得到了$G_{|a^{1}|\ldots|a^{n}|}$的公式，并且通过使用由 Harish-Chandra-Itzykson-Zuber 积分计算出的配分函数$\mathcal{Z}[J]$实现了这一目标。 我们给出$G_{|a|}$, $G_{|ab|}$, $G_{|a|b|}$和$G_{|a|b|c|}$作为具体的简单例子。它们全部由使用艾里函数描述。 显示英文摘要

摘要： 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.