A classification of modular functors via factorization homology
- 1Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, Université Paris Cité, Sorbonne Université, CNRS, Paris, France
- 2Université Bourgogne Europe, CNRS, IMB UMR 5584, Dijon, France
Abstract
Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing.They can formally be described as modular algebras over certain extensions of the modular surface operad, with the values of the algebra lying in a suitable symmetric monoidal .2;1/-category S of linear categories.We prove that modular functors in S are equivalent to self-dual balanced braided algebras A in S (a categorification of the notion of a commutative Frobenius algebra) for which a condition formulated in terms of factorization homology with coefficients in A is satisfied; we call such A connected.The equivalence in one direction is afforded by genus-zero restriction.Our construction of the inverse equivalence is entirely topological and can be thought of as a far reaching generalization of the construction of modular functors from skein theory.In order to verify the connectedness condition in practice, we prove that it can be reduced to a single condition in genus one.Moreover, we show that cofactorizability of A, a condition known to be satisfied for modular categories, is sufficient.We recover in particular Lyubashenko's construction of a modular functor from a (not necessarily semisimple) modular category and show that it is determined by its genus-zero part.Additionally, we exhibit modular functors that do not come from modular categories and outline applications to the theory of vertex operator algebras.
Related
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BibTeX
@article{brochier2026classification,
title = {{A classification of modular functors via factorization homology}},
author = {Adrien Brochier and Lukas Woike},
journal = {Geometry \& Topology},
year = {2026},
month = jul,
doi = {10.2140/gt.2026.30.2157},
url = {https://doi.org/10.2140/gt.2026.30.2157},
note = {CC BY},
}
RIS
TY - JOUR
TI - A classification of modular functors via factorization homology
AU - Adrien Brochier
AU - Lukas Woike
PY - 2026
DA - 2026/07/31
JO - Geometry & Topology
DO - 10.2140/gt.2026.30.2157
UR - https://doi.org/10.2140/gt.2026.30.2157
AB - Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing.They can formally be described as modular algebras over certain extensions of the modular surface operad, with the values of the algebra lying in a suitable symmetric monoidal .2;1/-category S of linear categories.We prove that modular functors in S are equivalent to self-dual balanced braided algebras A in S (a categorification of the notion of a commutative Frobenius algebra) for which a condition formulated in terms of factorization homology with coefficients in A is satisfied; we call such A connected.The equivalence in one direction is afforded by genus-zero restriction.Our construction of the inverse equivalence is entirely topological and can be thought of as a far reaching generalization of the construction of modular functors from skein theory.In order to verify the connectedness condition in practice, we prove that it can be reduced to a single condition in genus one.Moreover, we show that cofactorizability of A, a condition known to be satisfied for modular categories, is sufficient.We recover in particular Lyubashenko's construction of a modular functor from a (not necessarily semisimple) modular category and show that it is determined by its genus-zero part.Additionally, we exhibit modular functors that do not come from modular categories and outline applications to the theory of vertex operator algebras.
SN - 1364-0380
SN - 1465-3060
ER -