Heisenberg homology on surface configurations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866918140035727360 |
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| author | Blanchet, Christian Palmer, Martin Shaukat, Awais |
| author_facet | Blanchet, Christian Palmer, Martin Shaukat, Awais |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_00515 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Heisenberg homology on surface configurations Blanchet, Christian Palmer, Martin Shaukat, Awais Geometric Topology Algebraic Topology 57K20, 55R80, 55N25, 20C12, 19C09 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. |
| title | Heisenberg homology on surface configurations |
| topic | Geometric Topology Algebraic Topology 57K20, 55R80, 55N25, 20C12, 19C09 |
| url | https://arxiv.org/abs/2109.00515 |