We study a generalization of BF-theories in the context of higher gauge theory. We construct a topological state sum $Z$, based on the classical 3BF action for a general semistrict Lie 3-group and a triangulation of a 4-dimensional spacetime manifold. The 3BF action is constructed using a 2-crossed module which encodes a 3-group (as introduced by Picken and Faria Martins [1]), while the state sum $Z$ is an example of Porter’s TQFT [2] for $d=4$ and $n=3$. In order to verify that the constructed state sum is a topological invariant of the underlying manifold, its behavior under Pachner moves is analyzed, and it is obtained that the state sum $Z$ remains the same. Our results are a generalization of the work done by Girelli, Pfeiffer, and Popescu [3] for the case of state sum based on the classical 2BF action with the underlying 2-group structure.

1. J. Faria Martins and R. Picken, Diff. Geom. Appl. 29, 179 (2011), arXiv:0907.2566.
2. T. Porter, J. Lond. Math. Soc. (2)58, No. 3, 723 (1998), MR 1678163.
3. F. Girelli, H. Pfeiffer and E. M. Popescu, Jour. Math. Phys. 49, 032503 (2008), arXiv:0708.3051.