Asymptotics of lowest unitary SL(2,C) invariants on graphs
Pietro Donà, Simone Speziale
Abstract
We describe a technique to study the asymptotics of $\mathrm{SL}(2,\mathbb{C})$ invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian 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.