Twistorial structure of loop-gravity transition amplitudes

The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many pressing questions remain. In this paper we address some of them using a twistorial description, which sheds new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form a canonical pair. The primary simplicity constraints are solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct a SU(2) holonomy and prove it to correspond to the (lattice version of the) Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a nontrivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schr\"odinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is a spinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behavior of the model and our construction, clarifying the relation of the Regge geometries with the original phase space.