The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its algebro--geometric interpretation has been unveiled in recent works of Norbury. We prove that a suitably generalized Brezin-Gross-Witten tau function is the isomonodromic tau function of a $2\times 2$ isomonodromic system and consequently present a study of this tau function purely by means of this isomonodromic interpretation. Within this approach we derive effective formul\ae\ for the generating functions of the correlators in terms of simple generating series, the Virasoro constraints, and discuss the relation with the Painlev\'{e} XXXIV hierarchy.

中文翻译：

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Brezin-Gross-Witten tau 函数和等单向变形

Brezin-Gross-Witten tau 函数是在 Brezin-Gross-Witten 模型的弱耦合阶段出现的 KdV 层次结构的 tau 函数。它属于广义 Kontsevich 矩阵积分家族，其代数 - 几何解释已在 Norbury 最近的作品中公开。我们证明了适当广义的 Brezin-Gross-Witten tau 函数是 $2\times 2$ 等单系统的等单 tau 函数，因此纯粹通过这种等单向解释来研究该 tau 函数。在这种方法中，我们根据简单的生成序列、Virasoro 约束推导出相关器的生成函数的有效公式，并讨论与 Painlev\'{e} XXXIV 层次结构的关系。