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Dichromatic state sum models for four-manifolds from pivotal functors

Bärenz, Manuel; Barrett, John W.

Authors

Manuel Bärenz

JOHN BARRETT john.barrett@nottingham.ac.uk
Professor of Mathematical Physics



Abstract

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category.
A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant.
A special case is the four-dimensional untwisted Dijkgraaf-Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models.
Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed.

Citation

Bärenz, M., & Barrett, J. W. (2018). Dichromatic state sum models for four-manifolds from pivotal functors. Communications in Mathematical Physics, 360(2), https://doi.org/10.1007/s00220-017-3012-9

Journal Article Type Article
Acceptance Date Aug 27, 2017
Online Publication Date Nov 24, 2017
Publication Date Jun 30, 2018
Deposit Date Oct 12, 2017
Publicly Available Date Nov 24, 2017
Journal Communications in Mathematical Physics
Print ISSN 0010-3616
Electronic ISSN 1432-0916
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 360
Issue 2
DOI https://doi.org/10.1007/s00220-017-3012-9
Public URL http://eprints.nottingham.ac.uk/id/eprint/47124
Publisher URL https://link.springer.com/article/10.1007%2Fs00220-017-3012-9
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0





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