论文标题
m2-branes和$ \ m athfrak {q} $ - painlevé方程
M2-branes and $\mathfrak{q}$-Painlevé equations
论文作者
论文摘要
在本文中，我们研究了$（\ Mathbb {C}^2/\ Mathbb {Z} _2 \ Times \ Mathbb {C}^2/\ Mathbb {Z} _2 _2 _2）/\ Mathbb {z} _k $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath $ nath of $（\ mathbb {c}^2/\ mathbb {c}^2/\ mathbb {z}^2/\ mathbb {z} _2 _2 _2）/\ Mathbb {z} _k $}通过提出相应的四节点的大规范分区函数圆形Quiver $ \ Mathcal {n} = 4 $ Chern-Simons Matter理论解决了$ \ Mathfrak {q} $ - PANCHELEVUR-painlevéVI方程。我们分析了如何描述本地$ \ text {dp} _5 $上拓扑字符串的模量空间，并且通过几何工程，五维$ n_f = 4 $ $ $ \ text {su}（su}（2）$ $ $ $ \ nathcal {n} = 1 $ gauge理论。我们发现的结果扩展了ABJM理论，$ \ Mathfrak {q} $ - painlevé$ \ text {iii} _3 $与本地$ {\ mathbb p}^1 \ times {\ mathbb p}^1 $的拓扑之间的关系。从数学角度来看，Quiver Chern-Simons理论提供了$ \ Mathfrak {Q} $ - PACHLEVéVI$τ$ - 功能的猜想的弗雷姆决定源。我们通过分析和数值检查为该建议提供了证据，并详细讨论了连续的解耦限制至$ n_f = 0 $，对应于$ \ m athfrak {q} $ - pachlelevé$ \，\，\，$ iii $ {} _ 3 $。
In this paper we investigate a novel connection between the effective theory of M2-branes on $(\mathbb{C}^2/\mathbb{Z}_2\times \mathbb{C}^2/\mathbb{Z}_2)/\mathbb{Z}_k$ and the $\mathfrak{q}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver $\mathcal{N}=4$ Chern-Simons matter theory solves the $\mathfrak{q}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local $\text{dP}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text{SU}(2)$ $\mathcal{N}=1$ gauge theory on a circle. The results we find extend the known relation between ABJM theory, $\mathfrak{q}$-Painlevé $\text{III}_3$, and topological strings on local ${\mathbb P}^1\times{\mathbb P}^1$. From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the $\mathfrak{q}$-Painlevé VI $τ$-function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to $N_f=0$, corresponding to $\mathfrak{q}$-Painlevé$\,\,$III${}_3$.
