Deformation of the Engle-Livine-Pereira-Rovelli spin foam model by a cosmological constant

In this article, we consider an ad hoc deformation of the Engle-Livine-Pereira-Rovelli model for quantum gravity by a cosmological constant term. This sort of deformation was first introduced by Han for the case of the 4-simplex. In this article, we generalize the deformation to the case of arbitrary vertices, and compute its large-$j$ asymptotics. We show that, if the boundary data correspond to a four-dimensional polyhedron $P$, then the asymptotic formula gives the usual Regge action plus a cosmological constant term. We pay particular attention to the determinant of the Hessian matrix, and show that it can be related to that of the undeformed vertex.

Figure 1

The boundary graph of a 4-simplex.

Figure 2

The two types of crossings $C$ get assigned different numbers $\sigma (C)=\pm 1$.

Figure 3

The boundary graph of a hypercuboid (or a hyperfrustum). This graph has eight nodes, 24 links, and six crossings. The lines going to infinity all meet at node 7.