Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps

TL;DR

This work extends Minkowski's tetrahedral reconstruction to homogeneously curved spaces by replacing face normals with Levi-Civita holonomies $O_\ell$ and enforcing a curved closure $O_4 O_3 O_2 O_1=\mathrm{e}$ to recover convex tetrahedra in ${\mathrm S}^3$ or ${\mathrm H}^3$. It develops a full phase-space framework for curved tetrahedra using quasi-Poisson structures on $SU(2)$ and a fusion procedure, showing that the reduced two-dimensional shape space coincides with the moduli space of flat $SU(2)$ connections on a four-punctured sphere and relates to Fenchel–Nielsen-type coordinates. A parallel quasi-Hamiltonian treatment yields explicit leaves, their volumes, and the curvature dependence, with explicit formulas for the reduced symplectic form and Liouville measure that connect to Witten's volume formulas. The results provide a bottom-up bridge between discrete curved geometry and covariant loop quantum gravity with a cosmological constant, suggesting deformed gauge symmetries and quantum-group structures in four-dimensional quantum gravity, and laying groundwork for curved spinfoam models (e.g., with $SL(2,\mathbb{C})$ boundary data) and twisted geometries. Overall, the paper offers a coherent curvature-enabled generalization of Minkowski's theorem, a group-valued phase space for curved tetrahedra, and a path toward quantization compatible with cosmological-constant physics.

Abstract

We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. A concrete realization of this is provided by the relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in 3+1 dimensional covariant loop quantum gravity in the presence of a cosmological constant.

Figures (9)

• Figure 1: A standard numbering of the vertices of the tetrahedron, which also induces a particular topological orientation. The BurntOrangep character of this tetrahedron is simply a device for underlining its spherical nature. However, similar pictures result from stereographic projection of a spherical tetrahedron in ${\mathrm{S}}^3$ onto $\mathbb{R}^3$. This projection sends great two-spheres of ${\mathrm{S}}^3$ into spheres of $\mathbb{R}^3$, though possibly with different radii. Note that the convex or concave aspect of the stereographically projected spherical tetrahedron has no intrinsic meaning.
• Figure 2: The set of simple paths used to interpret the holonomies $\{O_\ell\}$.
• Figure 3: The dihedral angle $\theta_{\ell m}$ spans the arc from outward normal $\ell$ to $m$. Here we illustrate the case $\{\ell,m\}=\{1,3\}$.
• Figure 4: The three vectors involved in the triple product at vertex 4. Given the topological orientation of the tetrahedron, its convexity, and supposing all normals are outward pointing, one finds that $\mathrm{sgn}\left[\left( \hat{n}_1 \times \hat{n}_2\right) . \hat{n}_3\right]>0$.
• Figure 5: A spherical triangle illustrating the notation for Eqs. \ref{['eq_sphcos']} and \ref{['eq_sphcosinv']}, the spherical cosines laws.

Theorems & Definitions (4)

• Theorem 1
• Lemma 1
• proof
• proof