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Derived Modular Functors

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Speaker: 
Lukas Woike
Affiliation: 
Universität Hamburg
Date: 
Fri, 2019-12-13 15:30 - 16:30
Location: 
MPIM Lecture Hall
Parent event: 
Extra talk

I will present an approach to a class of non-semisimple representation
categories, more specifically non-semisimple modular tensor
categories, via homotopy theory and low-dimensional topology. This
will lead to so-called derived modular functors that provide a
consistent family of higher algebraic invariants for a modular tensor
category that are constructed in a way reminiscent of factorization
homology.

It is already well-known that for a semisimple modular tensor
category, the Reshetikhin-Turaev construction yields an extended
three-dimensional topological field theory and hence by restriction a
modular functor. By work of Lyubashenko the construction of a modular
functor from a modular tensor category remains possible in the
non-semisimple case. We explain that the latter construction is the
shadow of a derived modular functor featuring homotopy coherent
mapping class group actions on chain complex valued conformal blocks
and a version of factorization and self-sewing via homotopy coends. On
the torus, we find a derived version of the Verlinde algebra, an
algebra over the little disk operad. The talk is based on joint work
with Christoph Schweigert (Hamburg).

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