Toledo invariants of Topological Quantum Field Theories
Invariants de Toledo des théories quantiques des champs
Résumé
We prove that the Fibonacci quantum representations ρ g,n : Mod g,n → PU(p, q) for (g, n) ∈ {(0, 4), (0, 5), (1, 2), (1, 3), (2, 1)} are holonomy representations of complex hyperbolic structures on some compactifications of the corresponding moduli spaces M g,n. As a corollary, the forgetful map between the corresponding compactifications of M 1,3 and M 1,2 is a surjective holomorphic map between compact complex hyperbolic orbifolds of different dimensions higher than one, giving an answer to a problem raised by Siu. The proof consists in computing their Toledo invariants: we put this computation in a broader context, replacing the Fibonacci representations with any Hermitian modular functor and extending the Toledo invariant to a full series of cohomological invariants beginning with the signature p − q. We prove that these invariants satisfy the axioms of a Cohomological Field Theory and compute the R-matrix at first order (hence the usual Toledo invariants) in the case of the SU 2 /SO 3-quantum representations at any level.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)