基于扭群理论的TQFT中的Toeplitz算子
Toeplitz operators in TQFT via skein theory
摘要：
拓扑量子场论将穿孔表面$Σ$、能级$r$和在标记点处的$\{1,...,r-1\}$中的颜色$c$关联到有限维厄米空间$V_r(Σ,c)$。$Σ$上的曲线$γ$充当这些空间上的埃尔米特算子$T_r^γ$。在穿孔环面和4次穿孔球面的情况下，我们证明了$T_r^γ$的矩阵元具有$\frac{1}{r}$幂的渐近展开式，并利用$\su$中曲面表示空间上的迹函数确定了前两项.我们为一般情况推测了一个公式。然后证明了曲线算子是球面上的Toeplitz算子，即$T_r^γ=Π_r f^γ_rΠ_r$，其中$Π_r$是Toeplitz投影算子，$f^γ_r$是球面上的显式函数，它在远离极点处是光滑的.利用这个公式，我们表明，在一些假设的颜色相关联的标记点，序列$T^γ_r$是一个Toeplitz运营商在通常意义下的主要符号等于跟踪功能和subleading项明确计算。我们使用这个结果和半经典的分析，以计算矩阵元素的渐近表示的映射类群的$Σ$ $V_r(Σ,c)$。用这种方法我们恢复了Taylor和Woodward关于量子6 j-符号的渐近性的结果，并处理了穿孔S-矩阵的情况。当$Σ$是没有标记点的亏格2曲面时，我们得出了一些部分结果。
英文摘要：
Topological quantum field theory associates to a punctured surface $Σ$, a level $r$ and colors $c$ in $\{1,...,r-1\}$ at the marked points a finite dimensional hermitian space $V_r(Σ,c)$. Curves $γ$ on $Σ$ act as Hermitian operator $T_r^γ$ on these spaces. In the case of the punctured torus and the 4 times punctured sphere, we prove that the matrix elements of $T_r^γ$ have an asymptotic expansion in powers of $\frac{1}{r}$ and we identify the two first terms using trace functions on representation spaces of the surface in $\su$. We conjecture a formula for the general case. Then we show that the curve operators are Toeplitz operators on the sphere in the sense that $T_r^γ=Π_r f^γ_rΠ_r$ where $Π_r$ is the Toeplitz projector and $f^γ_r$ is an explicit function on the sphere which is smooth away from the poles. Using this formula, we show that under some assumptions on the colors associated to the marked points, the sequence $T^γ_r$ is a Toeplitz operator in the usual sense with principal symbol equal to the trace function and with subleading term explicitly computed. We use this result and semi-classical analysis in order to compute the asymptotics of matrix elements of the representation of the mapping class group of $Σ$ on $V_r(Σ,c)$. We recover in this way the result of Taylor and Woodward on the asymptotics of the quantum 6j-symbols and treat the case of the punctured S-matrix. We conclude with some partial results when $Σ$ is a genus 2 surface without marked points.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
