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A modular functor from state sums for finite tensor categories and their bimodules

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Jürgen Fuchs, Gregor Schaumann, and Christoph Schweigert

We construct a modular functor which takes its values in the monoidal
bicategory of finite categories,
left exact functors and natural transformations. The modular functor is defined on bordisms
that are 2-framed. Accordingly we do not need to require that the finite categories
appearing in our construction are semisimple, nor that the finite tensor categories that
are assigned to two-dimensional strata are endowed with a pivotal
structure. Our prescription can be understood as a state-sum construction. The state-sum
variables are assigned to one-dimensional strata and take values in bimodule categories
over finite tensor categories, whereby we also account for the presence of boundaries and
defects. Our construction allows us to explicitly compute functors associated to surfaces and
representations of mapping class groups acting on them.

Keywords:
modular functor, state-sum construction, finite tensor category,
monoidal bicategory, mapping class group, factorization, topological defect

2020 MSC:
18M20, 18M30, 81T45

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 15, pp 436-594.

Published 2022-03-20.

http://www.tac.mta.ca/tac/volumes/38/15/38-15.pdf

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