Toledo invariants of Topological Quantum Field Theories

TL;DR

This paper proves that certain Fibonacci quantum representations are holonomy representations of complex hyperbolic structures on moduli spaces, extending the concept of Toledo invariants within a cohomological framework and computing related invariants.

Contribution

It introduces a broader context for Toledo invariants by replacing Fibonacci representations with Hermitian modular functors and extends their application to cohomological invariants.

Findings

Fibonacci quantum representations are holonomy representations of complex hyperbolic structures.

The paper extends Toledo invariants to a series of cohomological invariants.

Computed the R-matrix and Toledo invariants for SU(2)/SO(3) quantum representations.

Abstract

We prove that the Fibonacci quantum representations $\rho_{g,n}:\rm{Mod}_{g,n}\to \rm{PU}(p,q)$ for $(g,n)\in\{(0,4),(0,5),(1,2),(1,3),(2,1)\}$ are holonomy representations of complex hyperbolic structures on some compactifications of the corresponding moduli spaces $\mathcal{M}_{g,n}$. 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 $\rm{SU}_2/\rm{SO}_3$-quantum representations at any level.