Toeplitz operators in TQFT via skein theory

Abstract

Topological quantum field theory associates to a punctured surface $\Sigma$, a level $r$ and colors $c$ in $\{1,...,r-1\}$ at the marked points a finite dimensional hermitian space $V_r(\Sigma,c)$. Curves $\gamma$ on $\Sigma$ act as Hermitian operator $T_r^\gamma$ on these spaces. In the case of the punctured torus and the 4 times punctured sphere, we prove that the matrix elements of $T_r^\gamma$ 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 $T_r^{\gamma}=\Pi_r f^\gamma_r\Pi_r$ where $\Pi_r$ is the Toeplitz projector and $f^\gamma_r$ 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 $T^\gamma_r$ 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 $\Sigma$ on $V_r(\Sigma,c)$. 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 $\Sigma$ is a genus 2 surface without marked points.