#### marcus

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So I suspect (having spent time looking at other potentially influential developments) that anyone who wants to follow Loop gravity research would be well advised to be reading

Speziale Wieland 1207.6348 and getting prepared to understand Speziale's ILQGS talk on 13 November.

So I'll list some titles and links in this thread. But also it would be very interesting if someone disagrees and thinks that some other reformulation of LQG that is currently being actively pursued has a better chance and could make a stronger showing at the upcoming Loops 2013 conference.

First of all here's the main paper.

http://arxiv.org/abs/1207.6348

**The twistorial structure of loop-gravity transition amplitudes**

Simone Speziale, Wolfgang Wieland

The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings 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 an 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 non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schroedinger 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 behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space.

40 pages, 3 figures