Geometric quantization and asymptotics of pairings in TQFT

Detcherry, Renaud

doi:10.24033/asens.2382

Geometric quantization and asymptotics of pairings in TQFT
[Quantification géométrique et asymptotique de produits scalaires en TQTC]

Detcherry, Renaud

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1599-1630.

Résumé
Abstract

Dans ce papier, nous construisons un isomorphisme explicite entre les espaces vectoriels $V_{r} (Σ)$ des TQTC de Reshetikhin-Turaev de groupe de gauge $SU (2)$ et des espaces de sections holomorphes de fibrés en droites complexes sur une certaine variété kählerienne, suivant l'approche de la quantification géométrique. Les opérateurs courbes deviennent ainsi des opérateurs de Toeplitz de symboles principaux correspondant aux fonctions traces sur l'espace des modules. Nous en déduisons que les vecteurs propres de ces opérateurs se concentrent sur les lignes de niveaux de ces fonctions traces, et obtenons une formule asymptotique pour les produits scalaires de ces vecteurs propres. Ceci permet d'obtenir une asymptotique pour les coefficients de matrice des représentations quantiques satisfaisant une hypothèse de généricité.

This paper presents an explicit mapping between the $SU (2)$ -Reshetikhin-Turaev TQFT vector spaces $V_{r} (Σ)$ of surfaces and spaces of holomorphic sections of complex line bundles on some Kähler manifold, following the approach of geometric quantization. We explain how curve operators in TQFT correspond to Toeplitz operators with symbols some trace functions. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic estimates of pairings of such eigenvectors. This yields under some genericity assumptions an asymptotic for the matrix coefficients of quantum representations.

MR Zbl

DOI : 10.24033/asens.2382

Classification : 57M27, 57R56, 53D50.
Keywords: Quantum invariants, TQFT, geometric quantization, Witten's asymptotic expansion conjecture.
Mot clés : Invariants quantiques, théories quantiques topologiques des champs, quantification géométrique, conjecture asymptotique de Witten.

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     author = {Detcherry, Renaud},
     title = {Geometric quantization and asymptotics of pairings in {TQFT}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1599--1630},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {6},
     year = {2018},
     doi = {10.24033/asens.2382},
     mrnumber = {3940905},
     zbl = {1426.57034},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2382/}
}

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Detcherry, Renaud. Geometric quantization and asymptotics of pairings in TQFT. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1599-1630. doi : 10.24033/asens.2382. http://www.numdam.org/articles/10.24033/asens.2382/

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