A 2-categorical state sum model

TL;DR

This paper constructs a 4-dimensional state sum model using 2-categories and 2-groups, explicitly computing weights that ensure invariance under triangulation moves, advancing the algebraic understanding of 4-manifold invariants.

Contribution

It provides a concrete realization of a 2-categorical state sum model based on a categorified Euclidean group, with explicit simplex weights satisfying invariance conditions.

Findings

Constructed a 4D state sum model using 2-categories.

Explicitly computed simplex weights analogous to 6j-symbols.

Demonstrated invariance under Pachner moves via an hexagon equation.

Abstract

It has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4-manifolds. This idea has been refined recently, by proposing to use 2-groups and their representations as specific examples of 2-categories. The challenge has been to make these proposals fully explicit. Here we give a concrete realization of this program. Building upon our earlier work with Baez and Wise on the representation theory of 2-groups, we construct a four-dimensional state sum model based on a categorified version of the Euclidean group. We define and explicitly compute the simplex weights, which may be viewed a categorified analogue of Racah-Wigner 6$j$-symbols. These weights solve an hexagon equation that encodes the formal invariance of the state sum under the Pachner moves of the triangulation. This result unravels the combinatorial formulation of the Feynman amplitudes of quantum field theory on flat spacetime proposed in [1], which was shown to lead after gauge-fixing to Korepanov's invariant of 4-manifolds.