Asymptotic formulae for curve operators in TQFT

Detcherry, Renaud

doi:10.2140/gt.2016.20.3057

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Asymptotic formulae for curve operators in TQFT

Renaud Detcherry

Geometry & Topology 20 (2016) 3057–3096

DOI: 10.2140/gt.2016.20.3057

Abstract

The Reshetikhin–Turaev topological quantum field theories with gauge group ${SU}_{2}$ associate to any oriented surface $Σ$ a sequence of vector spaces $V_{r} (Σ)$ and to any simple closed curve $γ$ in $Σ$ a sequence of Hermitian operators $T_{r}^{γ}$ on the spaces $V_{r} (Σ)$ . These operators are called curve operators and play a very important role in TQFT.

We show that the matrix elements of the operators $T_{r}^{γ}$ 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

Mathematical Subject Classification 2010

Primary: 57R56

References

Publication

Received: 3 September 2012

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

Volume 20, issue 6 (2016)