Abstract : The Fibonacci TQFT gives interesting representations of mapping class groups into pseudo-unitary groups. In some exceptional cases, they correspond to holonomy representation of complex hyperbolic structures on some compactifications of the moduli spaces of curves. The proof uses a computation of the Toledo invariant which fits in the framework of Cohomological Field theories. This is joint work with Bertrand Deroin.

Keywords : mapping class groups; quantum representations; moduli spaces of curves; complex hyperbolic geometry