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Physics and Geometry of Knots-Quivers Correspondence
Communications in Mathematical Physics ( IF 2.2 ) Pub Date : 2020-09-18 , DOI: 10.1007/s00220-020-03840-y
Tobias Ekholm , Piotr Kucharski , Pietro Longhi

The recently conjectured knots-quivers correspondence relates gauge theoretic invariants of a knot $K$ in the 3-sphere to representation theory of a quiver $Q_{K}$ associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of the large $N$ duality of Ooguri and Vafa, that relates knot invariants to counts of holomorphic curves with boundary on $L_{K}$, the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d ${\mathcal N}=2$ theory $T[Q_{K}]$ whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of $Q_{K}$. From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on $L_{K}$ is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver $Q_{K}$ and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to $Q_{K}$ and (doubly) refined version of LMOV invariants.

中文翻译:

结-箭袋对应的物理学和几何学

最近推测的结-箭袋对应关系将 3 球体中结 $K$ 的规范理论不变量与与结相关的箭袋 $Q_{K}$ 的表示理论联系起来。在本文中,我们在 Ooguri 和 Vafa 的大 $N$ 对偶性的框架内为这个猜想提供了几何和物理背景,它将结不变量与边界在 $L_{K}$ 上的全纯曲线的计数相关联,这是已解析的圆锥形中的结,以及相应的 M 理论考虑。从物理方面,我们证明了 quiver 编码了一个 3d ${\mathcal N}=2$ 理论 $T[Q_{K}]$ 其低能量动力学出现在一个 M5 膜的 worldvolume 上将该理论的(K 理论)涡旋分配函数与 $Q_{K}$ 的动机生成序列相匹配。从几何方面,我们认为 $L_{K}$ 上的(广义)全纯曲线谱是由一组有限的基本磁盘生成的。这些圆盘对应箭袋 $Q_{K}$ 的节点及其边界与箭袋箭头的链接。我们进一步扩展了这个基本字典,并提出了颤抖数据与基本磁盘的拓扑和几何属性之间的详细映射,这再次导致匹配的分区函数。我们还研究了与 $Q_{K}$ 和 LMOV 不变量的(双重)精炼版本相关的 A 多项式的推广。我们进一步扩展了这个基本字典,并提出了颤抖数据与基本磁盘的拓扑和几何属性之间的详细映射,这再次导致匹配的分区函数。我们还研究了与 $Q_{K}$ 和 LMOV 不变量的(双重)精炼版本相关的 A 多项式的推广。我们进一步扩展了这个基本字典,并提出了颤抖数据与基本磁盘的拓扑和几何属性之间的详细映射,这再次导致匹配的分区函数。我们还研究了与 $Q_{K}$ 和 LMOV 不变量的(双重)精炼版本相关的 A 多项式的推广。
更新日期:2020-09-18
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