$\operatorname{SL}(2,\mathbb{C})$ Chern-Simons theory, flat connections, and four-dimensional quantum geometry
$\operatorname{SL}(2,\mathbb{C})$ Chern-Simons theory, flat connections, and four-dimensional quantum geometry
复制标题
DOI：
10.4310/atmp.2019.v23.n4.a3
复制
发表时间：
2015-12
影响因子：
1.5
通讯作者：
Hal M. Haggard;Muxin Han;W. Kamiński;A. Riello
中科院分区：
文献类型：
--
作者：
Hal M. Haggard;Muxin Han;W. Kamiński;A. Riello
本文分析了一类图补3-流形上的SL（2，C）Chern-Simons理论及其与四维流形上经典几何和量子几何的关系。在经典理论中，我们解释了三维流形上一类SL（2，C）平坦联络与四维Lorentz单纯几何之间的对应关系。图补3-流形上的平坦连通类由一定的边界条件确定。相应的单形四维几何由常曲率四维单形构成。在Chern-Simons理论中，四维单纯几何的量子化可以通过三维流形上平坦联络的量子化来实现。在量子SL（2，C）Chern-Simons理论中，物理波函数的基是（全纯）3d块类，由Lefschetz套管上解析连续的Chern-Simons路径积分定义。在这里，我们建议（全纯）3D块与适当的边界条件施加给单纯的4维几何的量化。有趣的是，在（全纯）3d块的半经典渐近展开中，主要贡献给出了四维单纯Einstein-Hilbert引力的经典作用量，即具有宇宙常数的常曲率4-单纯体上的Lorentzian 4d Regge作用量。这一结果表明了一类三维流形上的SL（2，C）Chern-Simons理论与四维流形上的单纯量子引力之间的关系。本文详细介绍了（1）中报道的结果。
The present paper analyze SL(2,C) Chern-Simons theory on a class of graph complement 3- manifolds, and its relation with classical and quantum geometries on 4-dimensional manifolds. In classical theory, we explain the correspondence between a class of SL(2,C) flat connections on 3-manifold and the Lorentzian simplicial geometries in 4 dimensions. The class of flat connections on the graph complement 3-manifold is specified by a certain boundary condition. The corresponding simplicial 4-dimensional geome- tries are made by constant curvature 4-simplices. The quantization of 4d simplicial geometry can be carried out via the quantization of flat connection on 3-manifold in Chern-Simons theory. In quantum SL(2,C) Chern-Simons theory, a basis of physical wave functions is the class of (holomorphic) 3d block, defined by analytically continued Chern-Simons path integral over Lefschetz thimbles. Here we propose that the (holomorphic) 3d block with the proper boundary condition imposed gives the quantization of simplicial 4- dimensional geometry. Interestingly in the semiclassical asymptotic expansion of (holomorphic) 3d block, the leading contribution gives the classical action of simplicial Einstein-Hilbert gravity in 4 dimensions, i.e. Lorentzian 4d Regge action on constant curvature 4-simplices with a cosmological constant. Such a result suggests a relation between SL(2,C) Chern-Simons theory on a class of 3-manifolds and simplicial quantum gravity on 4-dimensional manifolds. This paper presents the details for the results reported in (1).