2022/12/21 by Marco De Renzi, De Renzi, Marco, Jules Martel +1 · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2212.10940
We provide a homological model for a family of quantum representations of mapping class groups arising from non-semisimple TQFTs (Topological Quantum Field Theories). Our approach gives a new geometric point of view on these representations, and it gathers into one theory two of the most promising constructions for investigating linearity of mapping class groups. More precisely, if \varSigmag,1 is a surface of genus g with 1 boundary component, we consider a (crossed) action of its mapping class group Mod(\varSigmag,1) on the homology of its configuration space Confn(\varSigmag,1) with twisted coefficients in the Heisenberg quotient ℍg of its surface braid group π1(Confn(\varSigmag,1)). We show that this action intertwines an action of the quantum group of \mathfraksl2, that we define by purely homological means. For a finite-dimensional linear representation of ℍg (depending on a root of unity ζ), we tweak the construction to obtain a projective representation of Mod(\varSigmag,1). Finally, we identify, by an explicit isomorphism, a subrepresentation of Mod(\varSigmag,1) that is equivalent to the quantum representation arising from the non-semisimple TQFT associated with quantum \mathfraksl2 at ζ. In the process, we provide concrete bases and explicit formulas for the actions of all the standard generators of Mod(\varSigmag,1) and of quantum \mathfraksl2 on both sides of the equivalence, and answer a question by Crivelli, Felder, and Wieczerkowski. We also make sure that the restriction of these representations to the Torelli group I(\varSigmag,1) are integral, in the sense that the actions have coefficients in the ring of cyclotomic integers ℤ[ζ], when expressed in these bases.