Asymptotics of lowest unitary SL(2,C) invariants on graphs

TL;DR

This paper develops a technique to analyze the asymptotic behavior of SL(2,C) invariant tensors in quantum gravity models, revealing a broader set of critical configurations beyond classical geometries.

Contribution

It introduces a new algorithm for studying asymptotics of SL(2,C) invariants on graphs and extends understanding of critical points in quantum gravity models.

Findings

Critical configurations include conformal twisted geometries, not just Regge geometries.

Multiple critical points can exist for modular graphs, with mixed Euclidean and Lorentzian signatures.

The approach offers a new perspective on geometric variables in quantum gravity asymptotics.

Abstract

We describe a technique to study the asymptotics of SL(2,C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible.