Martin David Palmer -- Preprint -- Heisenberg homology on surface configurations
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Heisenberg homology on surface configurations
with Christian Blanchet and Awais Shaukat
(v1: 2021, v3: 2022)

arXiv: abspdf
v3v2v1

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. For a general representation we obtain a twisted action of the mapping class group. In the case of the Schrödinger representation or its finite dimensional analogues, by composing with a Stone-von Neumann isomorphism we are able to untwist and obtain a representation to the projective unitary group which lifts to a unitary representation of the stably universal central extension of the mapping class group.

Here are the notes of a talk that I gave, based on the results of this paper, on 12 November 2021 at the IMAR Topology Seminar.

Here are the notes of another talk based on these results, on 10 March 2022, at the Fudan Topology Seminar, Shanghai.

Here is the video of another talk based on this paper, from 30 March 2022, at the Topology seminar, New York University, Abu Dhabi.