Classification of Consistent Systems of Handlebody Group Representations

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

  • Lukas Müller
  • Lukas Woike
The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory
(we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in
⁠. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.
TidsskriftInternational Mathematics Research Notices
Udgave nummer6
Sider (fra-til)4767-4803
StatusUdgivet - 2024

Bibliografisk note

Funding Information:
L.M. gratefully acknowledges support by the Max Planck Institute for Mathematics in Bonn. L.W. gratefully acknowledges support by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 772960). Acknowledgments

Publisher Copyright:
© The Author(s) 2023. Published by Oxford University Press. All rights reserved.

ID: 389416091