The property (T) conjecture for mapping class groups predicts that finite-dimensional unitary representations of mapping class groups $Mod(S_g)$ of surfaces of genus $g \geq 3$ have no infinitesimal deformations (are rigid). This rigidity question has been extensively studied for finite image representations, where it is known as the Ivanov conjecture; much less is known for infinite image representations.

 

Natural examples of infinite image unitary representations arise from unitary modular fusion categories via the Reshetikhin-Turaev construction, yielding quantum representations of mapping class groups. For closed surfaces of genus $g \geq 7$, I will discuss rigidity of quantum representations arising from SU(2) and SO(3) modular categories at prime levels.

 

The strategy is to relate infinitesimal rigidity of these representations to Ocneanu rigidity, a result asserting that modular fusion categories admit no non-trivial deformations. This reduction uses the theory of the quantum representations together with harmonic representatives in Hodge theory on twisted moduli spaces of curves--certain Kähler compact orbifolds whose fundamental groups are quotients of mapping class groups.

 

Café culturel à 13:05 par Ramanujan Santharoubane.