Asymptotic formulas for curve operators in TQFT

Topological quantum field theories with gauge group $\textrm{SU}_2$ associate to each surface with marked points $\Sigma$ and each integer $r>0$ a vector space $V_r (\Sigma)$ and to each simple closed curve $\gamma$ in $\Sigma$ an Hermitian operator $T_r^{\gamma}$ acting on that space. We show that the matrix elements of the operators $T_r^{\gamma}$ have an asymptotic expansion in orders of $\frac{1}{r}$, and give a formula to compute the first two terms in terms of trace functions, generalizing results of March\'e and Paul.