Twisted geometries, twistors, and conformal transformations

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 γ, and reduces to the identity for γ = ∞. 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 nonlinear 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 [2].

SU(2) flat connection on a Riemann surface and 3D twisted geometry with a cosmological constant

“…The generalized twisted geometry relates naturally to the SU(2) flat connection on Riemann surface. The twist angle ξ in the usual twisted geometry has been interpreted as the extrinsic curvature of the spatial slice, when g is the holonomy of the Ashtekar-Barbero connection along the link [15,16]. A similar interpretation can be obtain for ξ ab from G ab on Riemann surface, which is discussed in the next section.…”

“…The plane where 4D normal rotates is orthogonal to f AB . Θ AB is the boost angle (hyper-dihedral angle) between the 4D-normals of the two tetrahedra [15,16].…”

“…The paths is again not unique. With the pair of base points fixed, the homotopy classes of the paths are again classified by the windings along the meridian 15 . For instance, we may have a path connecting the base points as it is showed in FIG.14, denoted byp 0 .…”

Timelike twisted geometries

“…[42] to investigate the possibility of a null normal vector N I in the simplicity constraints and the subsequent quantization of null hypersurfaces with spacelike 2-surfaces. It also has recently been used to investigate conformal transformations in LQG [43]. Very much in the same spirit of Ref.…”

Octahedron of complex null rays and conformal symmetry breaking

“…Here we are interested instead in a different reduction that takes us directly to SL(2, C), since this is the local gauge group of general relativity. As shown in [14], this reduction can be achieved without using the infinity twistor, but rather requiring conservation of the dilatations between the two sets of twistors. This means that the we preserve not only the pseudo-Hermitian structure Σ, but also γ 5 .…”