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
-category
of linear categories. We prove that modular functors in
are equivalent to self-dual balanced braided algebras
in
(a
categorification of the notion of a commutative Frobenius algebra) for which a
condition formulated in terms of factorization homology with coefficients in
is satisfied;
we call such
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
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.
Keywords
modular functors, topological field theories, operads,
mapping class groups
