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

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.

40 Pages + 5 Appendices. 11 Figures. v2: Refined presentation of the general algorithm, additional minor amendments. v3: paragraph added in section 5 about curved embeddings

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Cited by in corpus (31)

• The End of a Black Hole's Evaporation -- Part I
• Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity
• Spinfoam on Lefschetz Thimble: Markov Chain Monte-Carlo Computation of Lorentzian Spinfoam Propagator
• Numerical analysis of spin foam dynamics and the flatness problem
• The End of a Black Hole's Evaporation -- Part II
• How-To compute EPRL spin foam amplitudes
• Asymptotic analysis of spin-foams with time-like faces in a new parameterisation
• Twisted Geometries Coherent States for Loop Quantum Gravity
• Complex critical points in Lorentzian spinfoam quantum gravity: 4-simplex amplitude and effective dynamics on double-${\Delta }_3$ complex
• Numerical analysis of the self-energy in covariant Loop Quantum Gravity
• Spin-foams as semi-classical vertices: gluing constraints and a hybrid algorithm
• Markov Chain Monte Carlo methods for graph refinement in Spinfoam Cosmology
• Renormalization of group field theories for quantum gravity: new scaling results and some suggestions
• Summing bulk quantum numbers with Monte Carlo in spin foam theories
• Spinfoams and high performance computing
• Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
• Numerical evaluation of spin foam amplitudes beyond simplices
• Radiative corrections to the Lorentzian EPRL spin foam propagator
• Hopf link volume simplicity constraints in spin foam models
• Spinfoams: Foundations
• Finiteness of spinfoam vertex amplitude with timelike polyhedra, and the full amplitude
• Classical dynamics for Loop Gravity: The 2-vertex model
• Numerical approach to the black-to-white hole transition
• Toward matter dynamics in spin foam quantum gravity
• Curvature effects in the spectral dimension of spin foams
• Causal structure in spin-foams
• Tunneling of quantum geometries in spinfoams
• Geometry from local flatness in Lorentzian spin foam theories
• Studying the EPRL spinfoam self-energy
• Spinfoam tunneling of quantum geometries in angle variables
• (2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality