Corrigendum: The Plebanski sectors of the EPRL vertex

TL;DR

This paper corrects a sign error in the characterization of Plebanski sectors in the EPRL vertex asymptotics, clarifying the relation between sectors, orientations, and the Regge action.

Contribution

It provides a corrected analysis of the Plebanski sectors in the EPRL model, refining the understanding of the asymptotic behavior for Regge-like boundary data.

Findings

The Plebanski sectors (II+) and (II-) are associated with signs of the product of sector and orientation.

The correction aligns the asymptotics with classical expectations relating BF and Einstein-Hilbert actions.

The sign error correction clarifies the interpretation of the EPRL vertex asymptotics.

Abstract

We correct what amounts to a sign error in the proof of part (i.) of theorem 3 in Class.Quant.Grav.28 225003 (arXiv:1107.0709). The Plebanski sectors isolated by the linear simplicity constraints do not change --- they are still the three sectors (deg), (II+), and (II-). What changes is the characterization of the continuum Plebanski two-form corresponding to the first two terms in the asymptotics of the EPRL vertex amplitude for Regge-like boundary data. These two terms do not correspond to Plebanski sectors (II+) and (II-), but rather to the two possible signs of the product of the sign of the sector --- +1 for (II+) and -1 for (II-) --- and the sign of the orientation $\epsilon_{IJKL}B^{IJ} \wedge B^{KL}$ determined by $B^{IJ}$. This is consistent with what one would expect, as this is exactly the sign which classically relates the BF action, in Plebanski sectors (II+) and (II-), to the Einstein-Hilbert action, whose discretization is the Regge action appearing in the asymptotics.