We prove that level $5$ Witten–Reshetikhin–Turaev $\mathrm{SO}(3)$ quantum representations, also known as the Fibonacci representations, of mapping class groups are locally rigid. More generally, for any prime level $\ell $, we prove that the level $\ell $ $\mathrm{SO}(3)$ quantum representations are locally rigid on all surfaces of genus $g\ge 3$ if and only if they are locally rigid on surfaces of genus $3$ with at most $3$ boundary components. This reduces local rigidity in prime level $\ell $ to a finite number of cases.
Nous démontrons la rigidité locale des représentations quantiques $\mathrm{SO}(3)$ de Witten–Reshetikhin–Turaev de niveau $5$ des groupes modulaires de surfaces, aussi connues sous le nom de représentations Fibonacci. Plus généralement, pour tout niveau premier $\ell $, nous démontrons que les représentations quantiques $\mathrm{SO}(3)$ de Witten–Reshetikhin–Turaev de niveau $\ell $ sont localement rigides pour toutes les surfaces de genre $g\ge 3$ si et seulement si elles sont localement rigides pour les surfaces de genre $3$ avec au plus $3$ composantes de bord. Cela réduit la rigidité en niveau premier $\ell $ à un nombre fini de cas.
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Keywords: Quantum Representations, Mapping Class Groups, Rigidity, TQFT
Mots-clés : Représentations quantiques, groupes modulaires de surfaces, rigidité, TQCT
Godfard, Pierre 1
CC-BY-ND 4.0
@article{AIF_2025__75_6_2529_0,
author = {Godfard, Pierre},
title = {Rigidity of {Fibonacci} representations of mapping class groups},
journal = {Annales de l'Institut Fourier},
pages = {2529--2563},
year = {2025},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {75},
number = {6},
doi = {10.5802/aif.3676},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3676/}
}
TY - JOUR AU - Godfard, Pierre TI - Rigidity of Fibonacci representations of mapping class groups JO - Annales de l'Institut Fourier PY - 2025 SP - 2529 EP - 2563 VL - 75 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3676/ DO - 10.5802/aif.3676 LA - en ID - AIF_2025__75_6_2529_0 ER -
%0 Journal Article %A Godfard, Pierre %T Rigidity of Fibonacci representations of mapping class groups %J Annales de l'Institut Fourier %D 2025 %P 2529-2563 %V 75 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3676/ %R 10.5802/aif.3676 %G en %F AIF_2025__75_6_2529_0
Godfard, Pierre. Rigidity of Fibonacci representations of mapping class groups. Annales de l'Institut Fourier, Volume 75 (2025) no. 6, pp. 2529-2563. doi: 10.5802/aif.3676
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