Heisenberg homology on surface configurations

TL;DR

This paper explores homology of surface configurations with non-commutative local systems derived from the Heisenberg group, leading to novel representations of the mapping class group with connections to quantum mechanics.

Contribution

It introduces new homological representations of surface mapping class groups using Heisenberg group actions, including projective and untwisted representations linked to quantum physics.

Findings

Constructed twisted and untwisted representations of the mapping class group.

Connected the Schr"odinger representation to projective operators on Hilbert space.

Showed these representations preserve a sesquilinear form.

Abstract

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 Schr\"odinger 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 Schr\"odinger representations, especially in the even case. Based on a natural intersection pairing, we show that our representations preserve a sesquilinear form.