Communications in Mathematical Physics

, Volume 360, Issue 2, pp 663–714 | Cite as

Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

Open Access


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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Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.School of Mathematical SciencesUniversity of NottinghamUniversity Park, NottinghamUK

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