Asymptotic formulae for curve operators in TQFT

Abstract

The Reshetikhin–Turaev topological quantum field theories with gauge group ${SU}_2$ associate to any oriented surface $\Sigma$ a sequence of vector spaces $V_r\left(\Sigma \right)$ and to any simple closed curve $\gamma$ in $\Sigma$ a sequence of Hermitian operators $T_r^{\gamma }$ on the spaces $V_r\left(\Sigma \right)$. These operators are called curve operators and play a very important role in TQFT.

We show that the matrix elements of the operators $T_r^{\gamma }$ have an asymptotic expansion in orders of $1∕r$, and give a formula to compute the first two terms from trace functions, generalizing results of Marché and Paul for the punctured torus and the $4$–holed sphere to general surfaces.