From twistors to twisted geometries

TL;DR

This work establishes a deep link between twistors and the edge phase space of loop quantum gravity by showing that twisted geometries emerge as a symplectic reduction of twistor space under an area-matching constraint. It identifies null twistors (modulo an overall phase) with edge degrees of freedom, interpreting them as ruled null rays in Minkowski space and showing that the area-matching constraint H=0 enforces consistent face geometry. The authors present two equivalent reductions—one to holonomy–flux variables and one to twisted-geometries data—thus embedding twisted geometries within a broader twistor framework and clarifying their geometric meaning. The results position twisted geometries within a hierarchy that connects twistors to Regge geometries and open avenues for a possible twistor-based gravity formulation.

Abstract

In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors.