Spin connection of twisted geometry

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.

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.

Figure 2

We “thicken” the triangle in order to smooth out the discontinuity. The path $\gamma$ goes from one tetrahedron to the other through the thickened region.