#### Volume 20, issue 6 (2016)

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Asymptotic formulae for curve operators in TQFT

### Renaud Detcherry

Geometry & Topology 20 (2016) 3057–3096
##### 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.

##### Keywords
TQFT, skein calculus, moduli spaces
Primary: 57R56
##### Publication
Revised: 11 September 2015
Accepted: 25 March 2016
Published: 21 December 2016
Proposed: Vaughan Jones
Seconded: Robion Kirby, Leonid Polterovich
##### Authors
 Renaud Detcherry Institut de Mathématiques de Jussieu Université Paris 6 4 place Jussieu 75005 Paris France