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.
Highlights
The Crane-Yetter model [CYK97] is a state sum invariant of four-dimensional manifolds that determines a topological quantum field theory (TQFT)
There is interest in four-dimensional TQFTs from solidstate physics, where they allow the study of topological insulators, for example in the framework of Walker and Wang [WW12], which is expected to be the Hamiltonian formulation of the Crane-Yetter TQFT
Petit recovers [Pet08, Remark 4.4], up to a factor depending on the Euler characteristic, a generalised Broda invariant for a ribbon fusion category D satisfying the conditions of Definition 3.6
Summary
The Crane-Yetter model [CYK97] is a state sum invariant of four-dimensional manifolds that determines a topological quantum field theory (TQFT). No topological state sum has modelled four-dimensional quantum gravity in a satisfactory way. Petit’s “dichromatic invariant” [Pet08] does exactly this: in addition to the ribbon fusion category, one chooses a full fusion subcategory and labels the 2handles with the Kirby colour of the subcategory Whether this change improves the invariant remained unstudied at the time. If the target category of the functor is modularisable, which is often the case, the generalised invariant can be cast in the form of a state sum. The Crane-Yetter state sum is recovered as a special case, both for modular and nonmodular ribbon fusion categories. A handy overview of the different known special cases of the generalised dichromatic invariant is given as a Table in Sect. 8, together with some comments on the results
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