多矩阵模型与由对称群和Brauer代数得到的非交换Frobenius代数
Multi-matrix models and Noncommutative Frobenius algebras obtained from symmetric groups and Brauer algebras
摘要：
已经理解，${\cal N}=4$ SYM中的多迹算子的关联函数可以使用对称群的群代数或有壁Brauer代数来巧妙地计算。另一方面，已知这样的代数构造2D拓扑场论（TFT）。在回顾了基于对称群的二维TFT的构造之后，我们基于有壁Brauer代数构造了二维TFT。在构造中，对偶基础的引入体现了两种理论之间的相似性。接下来，我们构造了一类二维场论，其物理算子具有与由某些矩阵构造的多迹算子相同的对称性。这样的场论对应于非交换Frobenius代数。矩阵结构的出现是由于非交换性。高斯复多矩阵模型的相关函数可以转化为二维场论的关联函数。
英文摘要：
It has been understood that correlation functions of multi-trace operators in ${\cal N}=4$ SYM can be neatly computed using the group algebra of symmetric groups or walled Brauer algebras. On the other hand such algebras have been known to construct 2D topological field theories (TFTs). After reviewing the construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based on walled Brauer algebras. In the construction, the introduction of a dual basis manifests a similarity between the two theories. We next construct a class of 2D field theories whose physical operators have the same symmetry as multi-trace operators constructed from some matrices. Such field theories correspond to non-commutative Frobenius algebras. A matrix structure arises as a consequence of the noncommutativity. Correlation functions of the Gaussian complex multi-matrix models can be translated into correlation functions of the two-dimensional field theories.
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