Abstract
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 Σ g,1 is a surface of genus g with 1 boundary component, we consider a (crossed) action of its mapping class group Mod(Σ g,1 ) on the homology of its configuration space Confn(Σ g,1 ) with twisted coefficients in the Heisenberg quotient Hg of its surface braid group π 1 (Confn(Σ g,1 )). We show that this action intertwines an action of the quantum group of sl 2 , 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(Σ g,1 ). Finally, we identify, by an explicit isomorphism, a subrepresentation of Mod(Σ g,1 ) that is equivalent to the quantum representation arising from the non-semisimple TQFT associated with quantum sl 2 at ζ. In the process, we provide concrete bases and explicit formulas for the actions of all the standard generators of Mod(Σ g,1 ) and of quantum sl 2 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(Σ g,1 ) are integral, in the sense that the actions have coefficients in the ring of cyclotomic integers Z[ζ], when expressed in these bases.