Topological invariant of 4-manifolds based on a 3-group

TL;DR

This paper develops a topological invariant for 4-manifolds using a generalized 3-group based state sum, extending higher gauge theory and ensuring invariance under Pachner moves.

Contribution

It introduces a triangulation-independent state sum for 4D manifolds based on a 3BF action, generalizing previous 2-group models and verifying topological invariance.

Findings

Constructed a topological state sum Z for 4-manifolds.

Proved invariance of Z under Pachner moves.

Connected the state sum to Porter's TQFT for d=4, n=3.

Abstract

We study a generalization of a 4-dimensional BF-theory in the context of higher gauge theory. We construct a triangulation independent topological state sum Z, based on the classical 3BF action for a general 3-group and a 4-dimensional spacetime manifold. This state sum coincides with Porter's TQFT for d=4 and n=3. In order to verify that the constructed state sum is a topological invariant of the underlying 4-dimensional manifold, its behavior under Pachner moves is analyzed, and it is obtained that the state sum Z remains the same. This paper is a generalization of the work done by Girelli, Pfeiffer, and Popescu for the case of state sum based on the classical 2BF action with the underlying 2-group structure.