Cosmological constant in loop quantum gravity vertex amplitude

A new $q$ deformation of the Euclidean EPRL/FK (Engle-Pereira-Rovelli-Livine/Freidel-Krasnov) vertex amplitude is proposed by using the evaluation of the Vassiliev invariant associated with a 4-simplex graph [related to two copies of quantum SU(2) group at different roots of unity] embedded in a 3-sphere. We show that the large-$j$ asymptotics of the $q$-deformed vertex amplitude gives the Regge action with a cosmological constant. In the end we also discuss its relation with a Chern-Simons theory on the boundary of 4-simplex.

Figure 1

The ${\Gamma }_5^{+}$ graph with one crossing between $l_{31}$ and $l_{42}$.

Figure 2

The building blocks for Vassiliev invariant.

Figure 3

Evaluation of Vassiliev invariant.

Figure 4

The evaluation of 4-simplex graph via Vassiliev invariant.

Figure 5

The ${\Gamma }_5^{-}$ graph.