arXiv 1301.01922013-01hep-thmath.AGmath.GTmath.QAmath.SG

K-分解与三维规范理论

K-Decompositions and 3d Gauge Theories

Tudor Dimofte, Maxime Gabella, Alexander B. Goncharov

备注：: 121 pages + 2 appendices, 80 figures; Version 2: reorganized mathematical perspective, swapped Sections 3 and 4

摘要：

本文结合了数学和物理中的几种新结构。在数学上，我们研究了一类带边界的3-流形M上的框架平坦PGL（K，C）-联络。我们定义了一个空间L_K（M），它是由M的边界上延伸到M的框架平坦联络构成的。我们的目标是将L_K（M）的开部分理解为边界上框架平面联络的辛空间中的拉格朗日量，并且理解为K_2-拉格朗日量，这意味着辛形式的K_2-化身限制为零。本文利用M的理想三角剖分的超单纯K-分解的数据构造了L_K（M）的一个开部分，推广了Thurston在三维双曲几何中的胶合方程，并将它们与曲面上框架平坦PGL（K）-联络的簇坐标相结合。利用从修饰旗构形复形到Bloch复形的正则映射，我们证明了L_K（M）的任何一般分支都是K_2-迷向的，如果其边界满足某些拓扑约束（定理4.2）。在某些情况下，这意味着L_K（M）是K_2-拉格朗日量。对于一般M，我们推广了Neumann-Zagier关于PGL（2）胶合方程的辛性质的一个经典结果，将K_2-Lagrange性质化为一个组合命题.在物理上，我们利用K-分解的辛性质构造了3d N=2超共形场论T_K[M]，它对应于M上K个M ~ 5-膜的紧化.这扩展了K=2的已知构造。当K=2时，理论T_K[M]被描述为交换Chern-Simons物质理论的IR不动点。三角剖分的变化（2-3步）导致阿贝尔镜像对称，这些对称都是由N_f=1 SQED和XYZ模型之间的初等对偶产生的。在大K极限下，我们发现了T_K[M]的自由度在K中立方增长的证据。

英文摘要：

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.

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