Dichromatic state sum models for four-manifolds from pivotal functors
Manuel Bärenz, John W. Barrett
TL;DR
This work develops a versatile framework for 4-manifold invariants built from Kirby calculus by introducing a generalized dichromatic invariant I_F, defined via a pivotal functor $F:\mathcal C\to\mathcal D$ between a spherical fusion category and a ribbon fusion category. The construction yields a chain-mail state-sum that encompasses the Crane–Yetter model (modular and nonmodular targets) and extends it to nonmodular settings, with 1- and 2-handles labeled by different Kirby colours, thereby capturing fundamental-group data beyond mere signature and Euler characteristic. Central technical tools include a generalized sliding property, insertion lemmas, and a robust invariance proof under handle moves, along with simplifications in unitary and modularisable scenarios that connect to Petit’s and Broda’s invariants. The paper also presents a state-sum formulation via chain-mail and triangulations, clarifies connections to Dijkgraaf–Witten theory, Walker–Wang TQFTs, and potential links to quantum gravity approaches, and discusses how cutting strands or modularisation affect computational tractability and interpretability. Together, these results provide a comprehensive, flexible platform for constructing and analyzing 4-manifold invariants from categorical data, with explicit applications to topological quantum field theories and related physical 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.