The spin connection of twisted geometry

TL;DR

This paper defines a torsionless spin-connection for twisted geometries in Loop Gravity, addressing discontinuities in triads and connecting to Regge curvature in special cases.

Contribution

It introduces a method to define a torsionless spin-connection for twisted geometries, handling triad discontinuities and linking to Regge geometry.

Findings

The curvature of the new spin connection reduces to Regge curvature for Regge geometries.

A method to interpolate between triads to define the spin connection.

Provides a classical limit framework for Loop Gravity truncations.

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.