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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2022

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.
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Dates et versions

hal-03852379 , version 1 (15-11-2022)

Identifiants

  • HAL Id : hal-03852379 , version 1

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Bertrand Deroin, Julien Marché. Toledo invariants of Topological Quantum Field Theories. 2022. ⟨hal-03852379⟩
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