Twistorial phase space for complex Ashtekar variables

TL;DR

The paper addresses how to formulate a Lorentzian twistorial description of the loop quantum gravity phase space using complex Ashtekar variables. It develops a truncation on a spinnetwork into copies of $T^*SL(2,\mathbb{C})$ and then provides a twistorial decomposition by attaching two twistors to each link, along with a rigorous area-matching constraint. Classically, it rewrites the linear simplicity constraints in twistorial language and demonstrates a clean separation into a first-class piece and a master constraint; quantum analysis then yields the EPRL relations $\rho=\beta j_0$ and $j=j_0$, with a master constraint enabling weak imposition of the second constraint. The results establish a geometric bridge between the canonical $SL(2,\mathbb{C})$ phase space and spinfoam amplitudes, offering a spinorial derivation of the EPRL vertex and a framework for analyzing gauge reductions and inner products in this setting.

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.