Heisenberg homology on surface configurations

Motivated by the Lawrence–Krammer–Bigelow representations of the classical braid groups, we study the homology of unordered configurations in an orientable genus-g surface with one boundary component, over non-commutative local systems defined from representations of the discrete Heisenberg group. Mapping classes act on the local systems and for a general representation of the Heisenberg group we obtain a representation of the mapping class group that is twisted by this action. For the linearisation of the affine translation action of the Heisenberg group we obtain a genuine, untwisted representation of the mapping class group. In the case of the generic Schrödinger representation, by composing with a Stone-von Neumann isomorphism we obtain a projective representation by bounded operators on a Hilbert space, which lifts to a representation of the stably universal central extension of the mapping class group. We also discuss the finite dimensional Schrödinger representations, especially in the even case. Based on a natural intersection pairing, we show that our representations preserve a sesquilinear form.