Dichromatic state sum models for four-manifolds from pivotal functors

B�renz, Manuel; Barrett, John W.

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: https://nottingham-repository.worktribe.com/output/943812
• Publisher URL: https://link.springer.com/article/10.1007%2Fs00220-017-3012-9
• Contract Date: Oct 12, 2017