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Origin of the quantum group symmetry in 3d quantum gravity

Maïté Dupuis, Laurent Freidel, Florian Girelli, Abdulmajid Osumanu, Julian Rennert

TL;DR

The paper derives the classical origin of quantum group deformations observed in 3d gravity by introducing a boundary-term–induced canonical transformation parameterized by a fiducial vector n with n^2 = -Λ. This deformation modifies the boundary symmetry algebra, yielding a Λ-dependent 𝔡_{σs} (Drinfeld double) structure that discretizes into a Heisenberg double phase space per link via subdivision and truncation. Upon quantization, the boundary/edge data realize U_q(su(2)) (with q = e^{ħ γ/2}, γ ∝ √|Λ|), providing a concrete bridge between continuum gravity, boundary degrees of freedom, and TV-type quantum group amplitudes. The work thus explains how quantum group symmetries emerge from a principled first-principles continuum approach, clarifying their role as corner/edge data and their relation to Chern-Simons quantization and TV models.

Abstract

It is well-known that quantum groups are relevant to describe the quantum regime of 3d gravity. They encode a deformation of the gauge symmetries parametrized by the value of the cosmological constant. They appear as a form of regularization either through the quantization of the Chern-Simons formulation or the state sum approach of Turaev-Viro. Such deformations are perplexing from a continuum and classical picture since the action is defined in terms of undeformed gauge invariance. We present here a novel way to derive from first principle and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deformed the gauge invariance and the boundary symmetries, and makes them depend on the cosmological constant. Second we implement a discretization procedure relying on a truncation of the degrees of freedom from the continuum.

Origin of the quantum group symmetry in 3d quantum gravity

TL;DR

The paper derives the classical origin of quantum group deformations observed in 3d gravity by introducing a boundary-term–induced canonical transformation parameterized by a fiducial vector n with n^2 = -Λ. This deformation modifies the boundary symmetry algebra, yielding a Λ-dependent 𝔡_{σs} (Drinfeld double) structure that discretizes into a Heisenberg double phase space per link via subdivision and truncation. Upon quantization, the boundary/edge data realize U_q(su(2)) (with q = e^{ħ γ/2}, γ ∝ √|Λ|), providing a concrete bridge between continuum gravity, boundary degrees of freedom, and TV-type quantum group amplitudes. The work thus explains how quantum group symmetries emerge from a principled first-principles continuum approach, clarifying their role as corner/edge data and their relation to Chern-Simons quantization and TV models.

Abstract

It is well-known that quantum groups are relevant to describe the quantum regime of 3d gravity. They encode a deformation of the gauge symmetries parametrized by the value of the cosmological constant. They appear as a form of regularization either through the quantization of the Chern-Simons formulation or the state sum approach of Turaev-Viro. Such deformations are perplexing from a continuum and classical picture since the action is defined in terms of undeformed gauge invariance. We present here a novel way to derive from first principle and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deformed the gauge invariance and the boundary symmetries, and makes them depend on the cosmological constant. Second we implement a discretization procedure relying on a truncation of the degrees of freedom from the continuum.

Paper Structure

This paper contains 54 sections, 7 theorems, 209 equations, 3 figures.

Table of Contents

  1. Canonical analysis of the 3d gravity action with a cosmological constant
  2. From metric formulation to first order formulation.
  3. Equations of motion.
  4. Action symmetries
  5. Symplectic form and Poisson brackets.
  6. Charges algebra
  7. Quantum algebra of observables.
  8. Gauge theory for ${\mathfrak{d}}_{\sigma s}$.
  9. New variables and new action
  10. Gravity Action and canonical transformation
  11. Boundary term and canonical transformation.
  12. Nature of the vector $n$.
  13. Symmetries of the action.
  14. Deformed boundary symmetry algebra and Manin pairs
  15. New charges algebra.
  16. ...and 39 more sections

Key Result

Proposition 1

In the component connected to the identity, where ${\cal D}_{\sigma s}=\mathop{\mathrm{SU}}\nolimits \bowtie \mathop{\mathrm{AN}}\nolimits\ni G =\ell_c h_c$, there exist a boundary symplectic potential $\vartheta$ and a boundary Lagrangian $L_\partial$ given by such that $\Theta_c$ decomposes as a sum of a total derivative and a total variation As a corollary we have that $\Omega_c = \delta \The

Figures (3)

  • Figure 1: The two subregions/triangles $c^*$ and $c'^*$ with their respective reference point/center $c$ and $c'$. $\Gamma$ is the dual 2-complex. The segment $[cc']$ forms a link, dual to the edge $[vv']$ shared by $c^*$ and $c'^*$. The $\mathop{\mathrm{AN}}\nolimits$ and $\mathop{\mathrm{SU}}\nolimits$ holonomies $\ell_{cx}$ and $h_{cx}$ are based at $c$ and go to a point $x$ in the cell $c^*$. These holonomies can be put together as a single ${\mathfrak{D}}_{\sigma s}$ holonomy $G_c(x)=\ell_{cx}h_{cx}$.
  • Figure 2: The constraint \ref{['iwatop']} provides the natural way to define a ribbon structure associated to each link $[cc']$. It encodes that the holonomy around the ribbon is trivial.
  • Figure 3: The ribbon data encodes all the geometric data. In particular, when the ribbons meet at a node, the Gauss constraint $\ell_1\ell_2\ell_3=L^c_{v"v}L^c_{v'v"} L^c_{vv'}=1$ encodes the gauge invariance at the node and is the generalization of the flat case $X_1+X_2+X_3=0$.

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6