Brezin-Gross-Witten矩阵模型的枚举几何和组合学意义

Topological string theory was introduced by Witten in the 1980’s as a simplified model of string theory. It captures topological information of the target space. Ever since then, it has been intensively studied by mathematicians and physicists. One of the most important lessons we have learned about topological strings is that, in some cases, the space-time description of open topological strings using string field theory reduces to very simple gauge theories. In fact, a particular class of topological string theories reduces to a matrix model in the gauge theory description. This is why matrix models attracts mathematicians and physicists...Matrix models are the simplest examples of quantum gauge theories, that is, they are quantum gauge theories in zero dimensions. They are usually matrix integrals with he basic field being a Hermitian matrix. The matrix integrals can have different integration domains and measures. And what we see the most are the integrals over ensembles of unitary, Hermitian, complex and normal matrices. So many matrix integrals have arisen in the study of topological strings. Among those, Kontsevich’s matrix integral became an inevitable part of mathematical physics. It was used for the description of the intersection numbers on the moduli spaces of smooth complete complex curves with marked points. These intersections are the two-dimensional topological gravity investigated by Witten. In 1991, Witten introduced a conjecture about intersection numbers of stable classes on the moduli space of curves, known as the Witten conjecture. And later, it was proven by Kontsevich using the matrix model. ..The tau-function derived from Kontsevich’s matrix model is called the Kontsevich-Witten tau-function. It is usually considered as an elementary building block for a great number of more complicated partition functions. As a consequence of its special role, the Kontsevich-Witten tau-function is also very well studied. Besides its matrix integral representation, this tau-function has many important properties, such as Virasoro constraints, integrable system, topological recursion relation, moment variables description, random partitions representation, spectral curve description, and so on. ..The Brezin-Gross-Witten model was first introduced in the lattice gauge theory in the 1980’s. This model has many similar properties as the well-known Kontsevich-Witten tau function. For example, it is a tau-function of the KdV hierarchy and can be described by the generalized Kontsevich model. Furthermore, in the weak coupling phase, this model satisfies the Virasoro constraints. However, unlike the Kontsevich-Witten tau function, the geometrical interpretation of the Brezin-Gross-Witten model, including the enumerative invariants and combinatorics, is still unknown. This project, is to study the geometrical meaning of the Brezin-Gross-Witten model, including its one parametric deformation, namely the generalized Brezin-Gross-Witten model.

拓扑弦理论是一种简化了的弦理论模型，由Witten在80年代的时候提出。其中有一项重要结果是，对开拓扑弦的时空描述可以简化为单规范场理论。事实上一类特殊的拓扑弦可以简化为规范场论描述下的矩阵模型。这是矩阵模型吸引学者们去研究的重要原因。矩阵模型是量子规范场论里的基础研究对象。Kontsevich矩阵模型在数学物理中起着不可忽视的作用。此模型被Kontsevich用于解决著名的Witten猜想，在几何意义上它描述了曲线模空间上的相交数，而这些相交数对应的就是Witten研究的二维拓扑重力。此矩阵模型对应的Kontsevich-Witten tau函数具有相当多的性质，如Virasoro约束，拓扑递推关系，可积系统等。而Brezin-Gross-Witten模型也具有类似的诸多性质，但它的几何意义却还未得知。此项目将借重点研究Brezin-Gross-Witten 模型的枚举几何与组合学意义。

矩阵模型是量子规范场论里的基础研究对象。Brezin-Gross-Witten模型是矩阵模型中很重要很基础的一个模型。在Kontsevich 阶段下，它是KdV上的tau函数，而且满足Virasoro约束形式。一般化Brezin-Gross-Witten模型tau_N是在原版模型下进行单变量参数化的一个变化形式。对任意复数N，其同样满足KdV可积系统。本项目重点研究的是一般化Brezin-Gross-Witten 模型的枚举几何意义。 我们知道Kontsevich矩阵模型在数学物理中起着不可忽视的作用。此模型被Kontsevich用于解决著名的Witten猜想，在几何意义上它描述了曲线模空间上的相交数，而这些相交数对应的就是Witten研究的二维拓扑重力。此矩阵模型对应的Kontsevich-Witten tau函数具有相当多的性质，如Virasoro约束，拓扑递推关系，可积系统等。一般化Brezin-Gross-Witten模型具有类似的诸多性质，但它的几何意义却还未得知。在本项目的研究中，我们已经得到了一个用无穷维李代数算子连接这两个tau函数的公式及其单独的表达式。这个公式的证明方法可以普及到类似所有的KP可积梯队上的tau函数中。并且可以利用一般化Brezin-Gross-Witten模型的表达式证明其亏格零的自由能表达式。这对今后一般化Brezin-Gross-Witten模型的几何意义研究有着重要的作用。

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