论文标题
$φ_{2}^{3} $有限矩阵模型的精确解决方案
Exact solution of the $Φ_{2}^{3}$ finite matrix model
论文作者
论文摘要
我们找到$φ_{2}^{3} $有限矩阵模型（Grosse-Wulkenhaar模型）的确切解决方案。在$φ_{2}^{3} $有限矩阵模型中，多点相关函数表示为$ g_ {| a_ {1}^{1}^{1} \ ldots a_ {n_ {n_ {1}}}}}}^{1}^{1} {1} a_ {n_ {b}}^{b} |} $。 $ \ displayStyle \ sum_ {i = 1}^{b} n_ {i} $ - 点函数由$ g_ {| a_ {| a_ {1}^{1}^{1} \ ldots a_ { a_ {n_ {b}}^{b} |} $由总和在riemann表面上的所有feynman图（色带图）上给出，带有$ b $ bub-boundaries，每个$ | a^{i} {i} _ {1} _ {1} _ {1} \ cdots a^i} $ n_ {i} $ - $ i $ th边界的外部线。众所周知，任何$ g_ {| a_ {1}^{1} \ ldots A_ { $ g_ {| a^{1} | \ ldots | a^{n} |} $ type $ n $ - 点函数。因此，我们专注于$ g_ {| a^{1} | \ ldots | a^{n} |} $的严格计算。获得$ g_ {| a^{1} | \ ldots | a^{n} |} $的公式，并通过使用分区函数$ \ MATHCAL {Z} [J] $来实现。我们给出$ g_ {| a |} $，$ g_ {| ab |} $，$ g_ {| a | b |} $，$ g_ {| a | a | b | c |} $作为特定的简单示例。所有这些都是通过使用通风函数来描述的。
We find the exact solutions of the $Φ_{2}^{3}$ finite matrix model (Grosse-Wulkenhaar model). In the $Φ_{2}^{3}$ 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.
