论文标题
来自横向通量中二聚体计数的拓扑弦振幅和Seiberg-witten的预属性
Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux
论文作者
论文摘要
对原理的重要说明``字符串理论中的分区函数是$τ$ - 可集成方程的函数''是一个事实，即（双）分区函数$ 4D $ $ \ $ \ nathcal {n} = 2 $ geuge理论求解painlevé方程。在本文中，我们展示了最近建议的对应关系概括的自洽证明的道路：拓扑字符串在本地Calabi-yau歧管上的分区函数求解$ q $ - 差异方程式的非自主动力学的``集群 - 代数'''集成系统。 我们详细说明了该提案的``解决方案''一侧。在最简单的非平凡示例中，我们显示了拓扑字符串分区函数的$ 3D $ box计数如何从二分图上的二聚体计数中出现，并具有``flux''$ q $的离散仪表字段。这是拓扑字符串/光谱理论类型对应关系的一种新形式，因为可以将二聚体的分区函数计算为线性$ q $ -Difference kasteleyn运算符的决定因素。在``熔化''$ q \ to 1 $限制中使用WKB方法，我们获得了相应$ 5D $量规理论的Seiberg-Witten Prepitient的封闭积分公式。对应关系的``方程式''方面仍然是进一步研究的有趣主题。
Important illustration to the principle ``partition functions in string theory are $τ$-functions of integrable equations'' is the fact that the (dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve $q$-difference equations of non-autonomous dynamics of the ``cluster-algebraic'' integrable systems. We explain in details the ``solutions'' side of the proposal. In the simplest non-trivial example we show how $3d$ box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of ``flux'' $q$. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear $q$-difference Kasteleyn operator. Using WKB method in the ``melting'' $q\to 1$ limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding $5d$ gauge theory. The ``equations'' side of the correspondence remains the intriguing topic for the further studies.
