Heisenberg homology on surface configurations
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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 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. |
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. Here is the video of yet another talk based on this paper, from 11 October 2022, at the Workshop on Cobordisms, Strings, and Thom Spectra, Oaxaca/online. |