The Plebanski sectors of the EPRL vertex

Understand

Modern spin-foam models of four dimensional gravity are based on a discrete version of the $Spin(4)$ Plebanski formulation.

• Beyond what is already in the literature, we clarify the meaning of different Plebanski sectors in this classical discrete model.
• We show that the linearized simplicity constraints used in the EPRL and FK models are not sufficient to impose a restriction to a single Plebanski sector, but rather, three Plebanski sectors are mixed.
• We propose this as the reason for certain extra `undesired' terms in the asymptotics of the EPRL vertex analyzed by Barrett et al.