Dichromatic state sum models for four-manifolds from pivotal functors

TL;DR

This paper introduces a new family of four-manifold invariants using state sum models derived from pivotal functors, generalizing the Crane-Yetter invariant and connecting to quantum gravity approaches.

Contribution

It develops a general state sum formula for four-manifold invariants parameterized by pivotal functors, extending existing models like Crane-Yetter to nonmodular categories.

Findings

Crane-Yetter invariant recovered as a special case

Nonmodular categories yield stronger invariants than signature and Euler

State space dimensions match ground state degeneracies in Walker-Wang models

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 parameterised 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.