Action of subgroups of the mapping class group on Heisenberg homologies

Abstract

In previous work we constructed twisted representations of mapping class groups of surfaces, depending on a choice of representation $V$ of the Heisenberg group $\mathcal{H}$. For certain $V$ we were able to untwist these mapping class group representations. Here, we study the restrictions of our twisted representations to different subgroups of the mapping class group. In particular, we prove that these representations may be untwisted on the Torelli group for any given representation $V$ of $\mathcal{H}$. When $V$ is the Schrödinger representation, we also construct untwisted representations of subgroups defined as kernels of crossed homomorphisms studied by Earle and Morita.

21 pages, 5 figures. Final version, to appear in Contemporary Mathematics, volume based on the conference "Quantum symmetries: Tensor categories, Topological quantum field theories, Vertex algebras". arXiv admin note: text overlap with arXiv:2109.00515

References in corpus (1)

• A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \mathcal{M}_{g,*}

Cited by in corpus (1)

• Heisenberg homology on surface configurations