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

Abstract

We study a generalization of 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 M4. 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 the 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.

Henneaux-Teitelboim Gauge Symmetry and its Applications to Higher Gauge Theories

“…But whenever one needs to discuss the gauge transformations off-shell, the HT subgroup simply cannot be ignored anymore. Typical situations include the Batalin-Vilkovisky formalism [1], various generalizations of gauge symmetry in the context of higher gauge theories and quantum gravity [28], and even the traditional contexts such as the Coleman-Mandula theorem [29]. The situations in which HT transformations play a significant role may be rare, but nevertheless they tend to be important.…”

“…After discussing the Chern-Simons theory as a toy example, we move to the more important case of 3BF theory. This theory is relevant for building models of quantum gravity, see [3,15,16,28,30]. Therefore, it is important to study its gauge symmetry, and in particular the role of HT transformations.…”

“…where B ∈ A 2 (M 4 , g), C ∈ A 1 (M 4 , h), and D ∈ A 0 (M 4 , l) are Lagrange multipliers, and F ∈ A 2 (M 4 , g), G ∈ A 3 (M 4 , h), and H ∈ A 4 (M 4 , l) represent the fake 3-curvature given by the equation (28). The forms , g , , h , and , l are G-invariant symmetric nondegenerate bilinear forms on g, h, and l, respectively.…”

Gauge symmetry of the 3BF theory for a generic semistrict Lie three-group

“…that it is triangulation independent. This construction was recently carried out in [14], where the 3BF state sum for a general two-crossed module and a closed and orientable four-dimensional manifold M 4 is defined. Unfortunately, in order to rigorously define this state sum, one needs the higher category generalizations of the Peter-Weyl and Plancherel theorems, from ordinary groups to the cases of two-groups and three-groups.…”

“…the underlying two-crossed module. This construction has been recently carried out in [14], where a triangulation independent state sum Z of a topological HGT for an arbitrary two-crossed module and a four-dimensional closed and orientable spacetime manifold M 4 is defined. Once the topological state sum is formulated, one could proceed to modify the amplitudes of the state sum in order to impose the simplicity constraints and obtain the state sum describing the full theory.…”