The spin connection of twisted geometry

TL;DR

The paper addresses how to define a torsionless spin connection for twisted geometry, a discrete generalization of 3D geometry used in loop gravity. It builds a torsionless connection by thickening faces and interpolating the triad via polar decomposition, yielding a holonomy $U(e)=\exp A$ and a distributional connection $\Gamma = -A\,d\tau$; in Regge geometries, the bone holonomy reduces to a rotation by the Regge deficit angle $\delta_l$, recovering the standard Cartan spin connection. An explicit expression for the connection in terms of face normals is derived, and the curvature is shown to coincide with Regge curvature in the Regge limit while allowing richer structures beyond Regge for general twisted geometries. The work clarifies that twisting does not imply torsion and that twisted geometry provides a valid discretization compatible with the classical limit of GR, potentially enabling a broader exploration of curvature in discrete quantum gravity frameworks.

Abstract

Twisted geometry is a piecewise-flat geometry less rigid than Regge geometry. In Loop Gravity, it provides the classical limit for each step of the truncation utilized in the definition of the quantum theory. We define the torsionless spin-connection of a twisted geometry. The difficulty given by the discontinuity of the triad is addressed by interpolating between triads. The curvature of the resulting spin connection reduces to the Regge curvature in the case of a Regge geometry.

Figures (1)

• Figure 1: In a twisted geometry two adjacent triangles have the same area and the same normal, but the angles and the edge lengths can differ. The two triangles can be identified, at the price of a discontinuity of the metric across the triangle.