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Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

Communications in Mathematical Physics volume 360, pages 663–714 (2018)Cite this article

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.

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Affiliations

  1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Manuel Bärenz

  2. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    John Barrett

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  1. Manuel Bärenz
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Correspondence to Manuel Bärenz.

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Communicated by C. Schweigert

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Bärenz, M., Barrett, J. Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors. Commun. Math. Phys. 360, 663–714 (2018). https://doi.org/10.1007/s00220-017-3012-9

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