Asymptotic formulas for curve operators in TQFT

Abstract

Topological quantum field theories with gauge group $\textrm{SU}_2$ associate to each surface with marked points $Σ$ and each integer $r>0$ a vector space $V_r (Σ)$ and to each simple closed curve $γ$ in $Σ$ an Hermitian operator $T_r^γ$ acting on that space. We show that the matrix elements of the operators $T_r^γ$ 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 Marché and Paul.