Algebraic Geometry Seminar

The property (T) conjecture for mapping class groups predicts that finite dimensional unitary local systems on moduli stacks of curves $\mathcal{M}_{g,n}$ for $g\geq 3$ are rigid (in the sense that they admit no infinitesimal deformations). While extensively studied for local systems with finite monodromy, a special case known as the Ivanov conjecture, much less is known when the monodromy is infinite.

We establish rigidity of local systems of conformal blocks arising from SU(2) and SO(3) modular categories, over $\mathcal{M}_g$ for $g\geq 7$ and at conformal levels $\ell$ such that $\ell+2$ is prime and at least $5$. These are natural infinite monodromy examples arising in quantum topology via the Witten-Reshetikhin-Turaev construction or alternatively in algebraic geometry via non-abelian theta functions.

The core of our argument is a proof that any infinitesimal deformation of a conformal block local system, within the space of all flat unitary local systems, necessarily remains a conformal block local system. This then implies triviality, since conformal block local systems admit no such internal deformations by a result known as Ocneanu rigidity. The proof combines the factorization property of conformal blocks with elementary Hodge theory on certain root stacks over $\overline{\mathcal{M}}_{g,n}$, over which conformal block local systems extend.