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Deformed phase space for 3d loop gravity and hyperbolic discrete geometries

Valentin Bonzom, Maité Dupuis, Florian Girelli, Etera R. Livine

TL;DR

The paper develops a classical phase-space framework for 3D loop quantum gravity with negative cosmological constant by embedding ISO$(3)$ into a Heisenberg double and deforming it to SL$(2,\mathbb{C})$, yielding a lattice phase space with deformed translations and SU$(2)$ rotations. It introduces two Iwasawa decompositions to realize SL$(2,\mathbb{C})$ as a phase space and shows that the deformation parameter $\kappa$ governs curvature, recovering ISO$(3)$ in the limit $\kappa\to 0$ and producing a hyperbolic discrete geometry otherwise. The lattice construction uses ribbon graphs with Gauss and flatness constraints that form a first-class algebra, and geometrical interpretation ties the constraints to hyperbolic cosine laws and dihedral angles, validating a consistent hyperbolic discretization of 3D gravity. The model is shown to be topological, with solutions dependent only on the topology (genus) of the underlying surface, and the authors discuss the path toward quantization and the expected connection to quantum groups and Turaev-Viro-type dynamics. Overall, the work provides a concrete classical bridge between discrete hyperbolic geometry and quantum group-inspired LQG dynamics for $\Lambda<0$, opening avenues for quantization and continuum-limit links to BF theory with cosmological constant.

Abstract

We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing the phase space $T^*\mathrm{SU}(2)\sim\mathrm{ISO}(3)$ as the Heisenberg double of the Lie group $\mathrm{SO}(3)$ provided with the trivial cocyle. Tackling the issue of accounting for a non-vanishing cosmological constraint $Λ\ne 0$ in the canonical framework of 3D loop quantum gravity, $\mathrm{SL}(2,\mathbb{C})$ viewed as the Heisenberg double of $\mathrm{SU}(2)$ provided with a non-trivial cocyle is introduced as a phase space. It is a deformation of the flat phase space $\mathrm{ISO}(3)$ and reproduces the latter in a suitable limit. The $\mathrm{SL}(2,\mathbb{C})$ phase space is then used to build a new, deformed LQG phase space associated to graphs. It can be equipped with a set of Gauss constraints and flatness constraints, which form a first class system and Poisson-generate local 3D rotations and deformed translations. We provide a geometrical interpretation for this lattice phase space with constraints in terms of consistently glued hyperbolic triangles, i.e. hyperbolic discrete geometries, thus validating our construction as accounting for a constant curvature $Λ<0$. Finally, using ribbon diagrams, we show that our new model is topological.

Deformed phase space for 3d loop gravity and hyperbolic discrete geometries

TL;DR

The paper develops a classical phase-space framework for 3D loop quantum gravity with negative cosmological constant by embedding ISO into a Heisenberg double and deforming it to SL, yielding a lattice phase space with deformed translations and SU rotations. It introduces two Iwasawa decompositions to realize SL as a phase space and shows that the deformation parameter governs curvature, recovering ISO in the limit and producing a hyperbolic discrete geometry otherwise. The lattice construction uses ribbon graphs with Gauss and flatness constraints that form a first-class algebra, and geometrical interpretation ties the constraints to hyperbolic cosine laws and dihedral angles, validating a consistent hyperbolic discretization of 3D gravity. The model is shown to be topological, with solutions dependent only on the topology (genus) of the underlying surface, and the authors discuss the path toward quantization and the expected connection to quantum groups and Turaev-Viro-type dynamics. Overall, the work provides a concrete classical bridge between discrete hyperbolic geometry and quantum group-inspired LQG dynamics for , opening avenues for quantization and continuum-limit links to BF theory with cosmological constant.

Abstract

We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing the phase space as the Heisenberg double of the Lie group provided with the trivial cocyle. Tackling the issue of accounting for a non-vanishing cosmological constraint in the canonical framework of 3D loop quantum gravity, viewed as the Heisenberg double of provided with a non-trivial cocyle is introduced as a phase space. It is a deformation of the flat phase space and reproduces the latter in a suitable limit. The phase space is then used to build a new, deformed LQG phase space associated to graphs. It can be equipped with a set of Gauss constraints and flatness constraints, which form a first class system and Poisson-generate local 3D rotations and deformed translations. We provide a geometrical interpretation for this lattice phase space with constraints in terms of consistently glued hyperbolic triangles, i.e. hyperbolic discrete geometries, thus validating our construction as accounting for a constant curvature . Finally, using ribbon diagrams, we show that our new model is topological.

Paper Structure

This paper contains 23 sections, 7 theorems, 162 equations, 8 figures.

Table of Contents

  1. LQG phase space and its deformation
  2. Definitions
  3. $\mathrm{ISO}(3)$ as phase space
  4. $\mathrm{SL}(2,{\mathbb C})$ as a phase space
  5. Left Iwasawa decomposition
  6. Right Iwasawa decomposition
  7. $\mathrm{SL}(2,{\mathbb C})$ phase space as a deformation of the $\mathrm{ISO}(3)$ phase space
  8. Poisson Lie symmetries
  9. Lattice with Heisenberg double and constraints
  10. Graphs, ribbons and phase spaces
  11. The constraints
  12. The Gauss law ${\mathcal{G}}_v$ as generator of $\mathrm{SU}(2)$ transformations
  13. The flatness constraints as generator of the (deformed) translations
  14. A first class constraint algebra
  15. Geometric meaning of the constraints
  16. ...and 8 more sections

Key Result

Theorem 2.1

Drinfeld double as symmetry structure The multiplication maps are Poisson maps, so they define left and right Poisson actions of respectively ${\mathcal{D}}_{\pi_-}$ and ${\mathcal{D}}_{-\pi_-}$ on ${\mathcal{D}}_{\pi_+}$, lu. Explicitly, we have the symmetry transformations $U$ and $V$ that do Poisson commute with $D$ an element of the Heisenberg double and

Figures (8)

  • Figure 1: The ribbon is orientated such that $\ell u = {\tilde{u}} \tilde{\ell}$. The solid lines are the strands and the dashed lines represent the thickened source and target points of each edge. Flipping the box is not allowed, only rotations in the plane are authorized.
  • Figure 2: The four possible ways to glue two ribbon edges together. The product of the $\mathrm{SU}(2)$ elements along the solid lines always only contains some $u$ and/or ${\tilde{u}}^{-1}$. The product of elements along the dashed lines always only contains some $\ell$ and/or $\tilde{\ell}^{-1}$.
  • Figure 3: Part of a ribbon graph. There is a closed face in bold lines, with holonomy ${\tilde{u}}_4^{-1} {\tilde{u}}_5^{-1} {\tilde{u}}_1^{-1} u_2 u_3$. There are two closed ribbon vertices with holonomies $\ell_1\ell_2\tilde{\ell}_7^{{-1}}$ and $\ell_5\tilde{\ell}_1^{{-1}} \tilde{\ell}_6^{{-1}}$ in dashed lines.
  • Figure 4: A 3-valent ribbon vertex with its incident edges inward. The Gauss law is ${\mathcal{G}} = \ell_1 \ell_2 \ell_3 = {\bf 1}$.
  • Figure 5: This represents the way the vectors $\hat{\mathfrak{b}}_e, \hat{\mathfrak{b}}_e^{op}$ are assigned to the dashed lines of the ribbon vertex.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 3.1
  • Proposition 3.2