Twisted geometries, twistors and conformal transformations

TL;DR

This paper explores the twistorial representation of twisted geometries in loop quantum gravity, analyzing the action of conformal transformations, especially dilatations, on the geometric and extrinsic structures of spin network states.

Contribution

It introduces a twistorial framework for twisted geometries, studies the conformal group action, and relates extrinsic geometry to gauge-invariant parametrizations, revealing new insights into their structure.

Findings

Dilatation preserves intrinsic geometry but alters extrinsic geometry.

Translation and conformal boost generators do not preserve the geometric structure.

The continuum limit reproduces the transformation of extrinsic geometry and rescaling of areas and volumes.

Abstract

The twisted geometries of spin network states are described by simple twistors, isomorphic to null twistors with a time-like direction singled out. The isomorphism depends on the Immirzi parameter, and reduces to the identity when the parameter goes to infinity. Using this twistorial representation we study the action of the conformal group SU(2,2) on the classical phase space of loop quantum gravity, described by twisted geometry. The generators of translations and conformal boosts do not preserve the geometric structure, whereas the dilatation generator does. It corresponds to a 1-parameter family of embeddings of T*SL(2,C) in twistor space, and its action preserves the intrinsic geometry while changing the extrinsic one - that is the boosts among polyhedra. We discuss the implication of this action from a dynamical point of view, and compare it with a discretisation of the dilatation generator of the continuum phase space, given by the Lie derivative of the group character. At leading order in the continuum limit, the latter reproduces the same transformation of the extrinsic geometry, while also rescaling the areas and volumes and preserving the angles associated with the intrinsic geometry. Away from the continuum limit its action has an interesting non-linear structure, but is in general incompatible with the closure constraint needed for the geometric interpretation. As a side result, we compute the precise relation between the extrinsic geometry used in twisted geometries and the one defined in the gauge-invariant parametrization by Dittrich and Ryan, and show that the secondary simplicity constraints they posited coincide with those dynamically derived in the toy model of [1409.0836].