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<p>Motivated by the Lawrence–Krammer–Bigelow representations of the classical braid groups, we study the homology of unordered configurations in an orientable genus-<i>g</i> 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, <i>untwisted</i> 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 <i>stably universal</i> 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.</p>

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Heisenberg homology on surface configurations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月12日

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  • Christian Blanchet,
  • Martin Palmer,
  • Awais Shaukat

摘要

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.