论文标题
量子teichmüller空间和相常数的不可还原的自我辅助表示形式
Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants
论文作者
论文摘要
非紧密riemann表面的Teichmüller空间的量化已在1980年代出现,作为三维量子重力的方法。对于表面理想的三角剖分的任何选择,瑟斯顿的剪切坐标函数在边缘构成了teichmüller空间的坐标系,应将其替换为Hilbert空间上的合适的自动化联合接合操作员。改变三角形后,必须在交织量子坐标运算符的希尔伯特空间之间构建一个统一操作员,并满足乘积相位常数的组成身份。在Chekhov,Fock和Goncharov的著名建筑中,量子坐标运算符构成了一个可还原的代表家族,并且相常数都是微不足道的。在本文中,我们采用schrödinger表示的页岩 - 韦尔互穿的谐波分析理论,以及Faddeev-Kashaev的量子差异功能,构建了一个不可缩写坐标函数的不可修复表示的家族,并为相应的Intertwiners构建了Trianians cransiens chance of triansmatiens of triansmations of triansmations的杂物。相位常数是由符号矢量空间的拉格朗日子空间的Maslov指数以及三角形翻转的五角大楼关系明确计算和描述的。目前的工作可能会推广到群集$ \ mathscr {x} $ - 品种。
Quantization of the Teichmüller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichmüller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of the Shale-Weil intertwiners for the Schrödinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster $\mathscr{X}$-varieties.