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

TL;DR

This paper generalizes the reconstruction of tetrahedra to curved spaces using face holonomies, unifying spherical and hyperbolic geometries, and connects these structures to quantum gravity with a cosmological constant.

Contribution

It introduces a framework for curved tetrahedra using group-valued moment maps, extending phase space concepts and linking to loop quantum gravity with a cosmological constant.

Findings

Unified description of spherical and hyperbolic tetrahedra

Introduction of hyperbolic simplices for complete coverage

Connection to quantum groups in loop quantum gravity

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.