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Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries

Maité Dupuis, Florian Girelli, Etera R. Livine

TL;DR

The paper develops a spinor-based parametrization of the deformed loop gravity phase space with a nonzero cosmological constant by promoting edge data from $T^*\mathrm{SU}(2)$ to $\mathrm{SL}(2,\mathbb{C})$ on graphs. It introduces $\kappa$-deformed spinors and braided-covariant spinors, derives explicit maps to reconstruct $\mathrm{SB}(2,\mathbb{C})$ and $\mathrm{SU}(2)$ holonomies, and shows how a full $\mathrm{SL}(2,\mathbb{C})$ phase space arises as a symplectic reduction with a classical $r$-matrix. The framework yields a deformed action of $\mathrm{SU}(2)$ at vertices and a spinor-network picture that generalizes twisted geometries to hyperbolic (cosmological-constant) settings, paving the way toward hyperbolic twisted geometries and quantum-group deformations in loop quantum gravity. It also outlines a program to interpret the resulting geometry, relate to $q$-deformed amplitudes, and extend to spherical cases and higher-rank structures. Overall, the work provides a classical foundation for incorporating a cosmological constant into LQG via spinor methods and sets the stage for further geometrical and quantum developments.

Abstract

In the context of a canonical quantization of general relativity, one can deform the loop gravity phase space on a graph by replacing the T*SU(2) phase space attached to each edge by SL(2,C) seen as a phase space. This deformation is supposed to encode the presence of a non-zero cosmological constant. Here we show how to parametrize this phase space in terms of spinor variables, thus obtaining deformed spinor networks for loop gravity, with a deformed action of the gauge group SU(2) at the vertices. These are to be formally interpreted as the generalization of loop gravity twisted geometries to a hyperbolic curvature.

Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries

TL;DR

The paper develops a spinor-based parametrization of the deformed loop gravity phase space with a nonzero cosmological constant by promoting edge data from to on graphs. It introduces -deformed spinors and braided-covariant spinors, derives explicit maps to reconstruct and holonomies, and shows how a full phase space arises as a symplectic reduction with a classical -matrix. The framework yields a deformed action of at vertices and a spinor-network picture that generalizes twisted geometries to hyperbolic (cosmological-constant) settings, paving the way toward hyperbolic twisted geometries and quantum-group deformations in loop quantum gravity. It also outlines a program to interpret the resulting geometry, relate to -deformed amplitudes, and extend to spherical cases and higher-rank structures. Overall, the work provides a classical foundation for incorporating a cosmological constant into LQG via spinor methods and sets the stage for further geometrical and quantum developments.

Abstract

In the context of a canonical quantization of general relativity, one can deform the loop gravity phase space on a graph by replacing the T*SU(2) phase space attached to each edge by SL(2,C) seen as a phase space. This deformation is supposed to encode the presence of a non-zero cosmological constant. Here we show how to parametrize this phase space in terms of spinor variables, thus obtaining deformed spinor networks for loop gravity, with a deformed action of the gauge group SU(2) at the vertices. These are to be formally interpreted as the generalization of loop gravity twisted geometries to a hyperbolic curvature.

Paper Structure

This paper contains 6 sections, 1 theorem, 88 equations, 4 figures.

Table of Contents

  1. Standard spinor phase space and holonomy reconstruction
  2. $\mathrm{SL}(2,{\mathbb C})$ phase space
  3. New spinor variables and deformed action of rotations
  4. The tilded sector - An exact copy of the straight sector
  5. q-Deformed holonomy reconstruction
  6. Towards hyperbolic twisted geometries

Key Result

Theorem 5.1

Consider ${\mathcal{S}}_\kappa\sim {\mathbb C}^2$ the phase space given in terms of the following (non-canonical) Poisson brackets ($t_A\in {\mathbb C}^2$) and its subspace ${\mathcal{S}}_{\kappa*}\equiv {\mathbb C}^2/ \{t_a\in{\mathbb C}^2, \, \langle t|t\rangle=0\}$. The symplectic reduction ${\mathcal{P}}_\kappa= ({\mathcal{S}}_{\kappa*}\times{\mathcal{S}}_{\kappa*})//{\mathcal{M}}$ of the pha

Figures (4)

  • Figure 1: We fatten the edges of a graph into a ribbon. The ribbon can be read clockwise as a plaquette encoding the constraint $(\tilde{\ell} \tilde{u}) (\ell u)= D^{{-1}} D= \mathbb{I}$, which encodes the equivalence of the two Iwasawa decompositions.
  • Figure 2: The spinors live at the vertices of the ribbon. We can read off the mappings between spinors, $|t\rangle\,=\,\ell\,|\tau\rangle$ and $|\tt\rangle\,=\,\tilde{\ell}\,|\tilde{\tau}\rangle$ respectively for the straight and tilded sectors separately, and $|\tau\rangle\,\propto\,u\,|\tt]$ and $|\tilde{\tau}\rangle\,\propto\,\tilde{u}\,|t]$ for the $\mathrm{SU}(2)$ holonomies relating the two sectors. Notice that we have to take the dual spinors, which we haven't represented in the diagram. This graphical representation is consistent with the realization of the symmetries (left rotation in this case) as described in Fig. \ref{['leftgauge']}.
  • Figure 3: The different realizations of the left $\mathrm{SU}(2)$ rotations in terms of ribbons.
  • Figure 4: On the left side, we have a standard spinor network. We flatten the graph to encode $T^*\mathrm{SU}(2)$ as $\mathrm{ISO}(3)$. In this case, $\tilde{u}_i=u^{{-1}}_i$, and we have the same spinors standing at the extremities of each ribbon. On the right side, we deform $\mathrm{ISO}(3)$ into $\mathrm{SL}(2,{\mathbb C})$. Hence the braided covariant spinors appear: the extremities of the ribbon have both a covariant and a braided covariant spinor.

Theorems & Definitions (1)

  • Theorem 5.1