Toledo invariants of Topological Quantum Field Theories

2022/07/20 by Deroin, Bertrand, Marché, Julien · 4 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2207.09952

Abstract

We prove that the Fibonacci quantum representations ρ_g,n:\rmMod_g,n→ \rmPU(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 \mathcal M_1,3 and \mathcal 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 \rmSU₂/\rmSO₃-quantum representations at any level.

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