论文标题
对单一矩阵模型的可整合性的新见解
New Insights into Superintegrability from Unitary Matrix Models
论文作者
论文摘要
某些特征值矩阵模型具有有趣的属性：可以明显地定义可以明确计算所有平均值的基础。例如，在高斯乡村和矩形复杂模型中，Schur函数的平均值再次通过Schur函数表达。但是，到目前为止，该特性仍然仅限于非常特殊的（例如高斯）措施。在本文中，我们将此观察结果扩展到统一的矩阵积分，在那里人们可以期望这种限制更容易取消。我们证明了情况确实如此，只有这次，Schur平均值才是Schur函数的线性组合。总和完全分解为总和仅出现在MIWA基因座上，其中至少有一个时间变量的一半是通过相同大小的矩阵表达的。对于单一积分，这是De-Wit-t'Hooft异常的体现，它阻止答案在矩阵尺寸$ n $中完全分析。一旦实现，这种理解就可以扩展到Hermitian模型，在该模型中，现象看起来非常相似：超越高斯措施可累加性需要额外的求和。
Some eigenvalue matrix models possess an interesting property: one can manifestly define the basis where all averages can be explicitly calculated. For example, in the Gaussian Hermitian and rectangular complex models, averages of the Schur functions are again expressed through the Schur functions. However, so far this property remains restricted to very particular (e.g. Gaussian) measures. In this paper, we extend this observation to unitary matrix integrals, where one could expect that this restriction is easier to lift. We demonstrate that this is indeed the case, only this time the Schur averages are linear combinations of the Schur functions. Full factorization to a single item in the sum appears only on the Miwa locus, where at least one half of the time-variables is expressed through matrices of the same size. For unitary integrals, this is a manifestation of the de Wit-t'Hooft anomaly, which prevents the answer to be fully analytic in the matrix size $N$. Once achieved, this understanding can be extended back to the Hermitian model, where the phenomenon looks very similar: beyond Gaussian measures superintegrability requires an additional summation.
