TY - JOUR

T1 - Classification of Consistent Systems of Handlebody Group Representations

AU - Müller, Lukas

AU - Woike, Lukas

N1 - Publisher Copyright: © The Author(s) 2023. Published by Oxford University Press. All rights reserved.

PY - 2024

Y1 - 2024

N2 - 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 M (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in M. 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.

AB - 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 M (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in M. 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.

U2 - 10.1093/imrn/rnad178

DO - 10.1093/imrn/rnad178

M3 - Journal article

AN - SCOPUS:85188614322

VL - 2024

SP - 4767

EP - 4803

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 6

ER -