Twistorial phase space for complex Ashtekar variables

TL;DR

This paper extends the twistorial phase space framework from SU(2) to SL(2,C) for complex Ashtekar variables, offering new insights into loop quantum gravity and the EPRL spinfoam model.

Contribution

It generalizes the twistorial phase space to Lorentzian variables, providing a new derivation of the simplicity constraints in loop quantum gravity.

Findings

Phase space decomposed into twistorial variables for SL(2,C)

Solution space of simplicity constraints derived

Key properties of EPRL spinfoam model recovered

Abstract

We generalise the SU(2) spinor framework of twisted geometries developed by Dupuis, Freidel, Livine, Speziale and Tambornino to the Lorentzian case, that is the group SL(2,C). We show that the phase space for complex valued Ashtekar variables on a spinnetwork graph can be decomposed in terms of twistorial variables. To every link there are two twistors---one to each boundary point---attached. The formalism provides a new derivation of the solution space of the simplicity constraints of loop quantum gravity. Key properties of the EPRL spinfoam model are perfectly recovered.