Asymptotic formulas for curve operators in TQFT

TL;DR

This paper derives asymptotic formulas for curve operators in SU(2) topological quantum field theories, providing explicit calculations of their matrix elements' leading terms in the large r limit.

Contribution

It generalizes previous results by providing a detailed asymptotic expansion for curve operators in TQFT, connecting matrix elements to trace functions.

Findings

Matrix elements have an asymptotic expansion in 1/r.

Explicit formulas for the first two terms of the expansion.

Generalization of Marché and Paul's results.

Abstract

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.