论文标题
矩阵模型的扩展和$ \ hbar $扩展KP层次结构
Genus expansion of matrix models and $\hbar$ expansion of KP hierarchy
论文作者
论文摘要
我们研究了Takasaki-Takebe Arxiv之后的KP层次结构的$ \ hbar $扩展：HEP-TH/9405096考虑了几个具有自然属扩张的矩阵模型$τ$函数的示例。在示例中，有特殊兴趣的KP方程的解决方案，例如为简单的Hurwitz数字，Hermitian Matrix模型，Kontsevich模型和Brezin-Gross-witten模型生成功能。我们表明，所有这些带有参数$ \ hbar $的模型都是$ \ hbar $ -KP层次结构的$τ$ - functions，对于$ \ hbar $ in $ \ hbar $ -kp的扩展与这些模型的扩展相吻合。此外，考虑到KP层次结构ARXIV的$ \ hbar $成型：1509.04472，ARXIV：1512.07172，我们展示了最近论文的联系。我们发现，在这种方法中，$τ$ - 功能的枚举几何含义的恢复是简单而算法的。
We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $τ$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $τ$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the $\hbar$-formulation of the KP hierarchy arXiv:1509.04472, arXiv:1512.07172 with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of $τ$-functions is straightforward and algorithmic.
