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Toeplitz operators in TQFT via skein theory


Authors: Julien Marché and Thierry Paul
Journal: Trans. Amer. Math. Soc. 367 (2015), 3669-3704
MSC (2010): Primary 47B35, 57M27
DOI: https://doi.org/10.1090/S0002-9947-2014-06322-8
Published electronically: December 4, 2014
MathSciNet review: 3314820
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Abstract: Topological quantum field theory associates to a punctured surface $ \Sigma $, a level $ r$ and colors $ c$ in $ \{1,\ldots ,r-1\}$ at the marked points a finite dimensional Hermitian space $ V_r(\Sigma ,c)$. Curves $ \gamma $ on $ \Sigma $ act as Hermitian operator $ T_r^\gamma $ on these spaces. In the case of the punctured torus and the 4-times punctured sphere, we prove that the matrix elements of $ T_r^\gamma $ have an asymptotic expansion in powers of $ \frac {1}{r}$ and we identify the two first terms using trace functions on representation spaces of the surface in $ \mathrm {SU}_2$. We conjecture a formula for the general case. Then we show that the curve operators are Toeplitz operators on the sphere in the sense that $ T_r^{\gamma }=\Pi _r f^\gamma _r\Pi _r$ where $ \Pi _r$ is the Toeplitz projector and $ f^\gamma _r$ is an explicit function on the sphere which is smooth away from the poles. Using this formula, we show that under some assumptions on the colors associated to the marked points, the sequence $ T^\gamma _r$ is a Toeplitz operator in the usual sense with principal symbol equal to the trace function and with subleading term explicitly computed. We use this result and semi-classical analysis in order to compute the asymptotics of matrix elements of the representation of the mapping class group of $ \Sigma $ on $ V_r(\Sigma ,c)$. We recover in this way the result of Taylor and Woodward on the asymptotics of the quantum 6j-symbols and treat the case of the punctured S-matrix.


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Additional Information

Julien Marché
Affiliation: Centre de mathématiques Laurent Schwartz (UMR 7640), Ecole Polytechnique, 91128 Palaiseau, France

Thierry Paul
Affiliation: Centre de mathématiques Laurent Schwartz (UMR 7640), Ecole Polytechnique, 91128 Palaiseau, France

DOI: https://doi.org/10.1090/S0002-9947-2014-06322-8
Received by editor(s): November 2, 2012
Received by editor(s) in revised form: September 17, 2013
Published electronically: December 4, 2014
Additional Notes: The first author was supported by the Agence Nationale de la Recherche ANR-08-JCJC-0114-01 and the second is a member of the CNRS
Article copyright: © Copyright 2014 American Mathematical Society