Homological Construction of Quantum Representations of Mapping Class Groups

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 $\varSigma_{g,1}$ is a surface of genus $g$ with $1$ boundary component, we consider a (crossed) action of its mapping class group $\mathrm{Mod}(\varSigma_{g,1})$ on the homology of its configuration space $\mathrm{Conf}_n(\varSigma_{g,1})$ with twisted coefficients in the Heisenberg quotient $\mathbb{H}_g$ of its surface braid group $\pi_1(\mathrm{Conf}_n(\varSigma_{g,1}))$. We show that this action intertwines an action of the quantum group of $\mathfrak{sl}_2$, that we define by purely homological means. For a finite-dimensional linear representation of $\mathbb{H}_g$ (depending on a root of unity $\zeta$), we tweak the construction to obtain a projective representation of $\mathrm{Mod}(\varSigma_{g,1})$. Finally, we identify, by an explicit isomorphism, a subrepresentation of $\mathrm{Mod}(\varSigma_{g,1})$ that is equivalent to the quantum representation arising from the non-semisimple TQFT associated with quantum $\mathfrak{sl}_2$ at $\zeta$. In the process, we provide concrete bases and explicit formulas for the actions of all the standard generators of $\mathrm{Mod}(\varSigma_{g,1})$ and of quantum $\mathfrak{sl}_2$ 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 $\mathcal{I}(\varSigma_{g,1})$ are integral, in the sense that the actions have coefficients in the ring of cyclotomic integers $\mathbb{Z}[\zeta]$, when expressed in these bases.