Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity

This paper focuses on the semiclassical behavior of spinfoam quantum gravity in four dimensions. There has been long-standing confusion, known as flatness problem, about whether curved geometry exists regime amplitude. The confusion is resolved by present work. By numerical computations, we explicitly find Regge geometries that contribute dominantly to large-$j$ Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) amplitudes triangulations. These are with small deficit angles and relate complex critical points dominant contribution from amplitude proportional ${e}^{i\mathcal{I}}$, where $\mathcal{I}$ action plus corrections higher order curvature. As a result, regime, reduces an integral over weighted by-product, our result also provides mechanism relax cosine problem model. Our results provide important evidence supporting consistency gravity.