Spinning geometry = Twisted geometry

TL;DR

The paper shows that the SU(2) holonomy-flux phase space used in loop gravity, previously represented by twisted geometries, can also be captured by continuous, piecewise-flat spinning geometries with edge torsion. By decomposing edge flux into angular momenta and minimally deforming link shapes, edges become helices whose parameters encode all flux–holonomy data, enabling a continuous geometric realization of the truncated phase space $P_\Gamma$. The authors prove isomorphisms between spinning geometries and twisted geometries, and extend the construction to the entire one-skeleton, providing a unified picture that preserves discrete observables while enabling a continuous spacetime interpretation. This framework suggests new avenues for dynamics by working on the continuous two-skeleton and clarifies the role of torsion in discrete gravity, with potential implications for implementing the scalar constraint in loop gravity.

Abstract

It is well known that the SU(2)-gauge invariant phase space of loop gravity can be represented in terms of twisted geometries. These are piecewise-linear-flat geometries obtained by gluing together polyhedra, but the resulting geometries are not continuous across the faces. Here we show that this phase space can also be represented by continuous, piecewise-flat three-geometries called spinning geometries. These are composed of metric-flat three-cells glued together consistently. The geometry of each cell and the manner in which they are glued is compatible with the choice of fluxes and holonomies. We first remark that the fluxes provide each edge with an angular momentum. By studying the piecewise-flat geometries which minimize edge lengths, we show that these angular momenta can be literally interpreted as the spin of the edges: the geometries of all edges are necessarily helices. We also show that the compatibility of the gluing maps with the holonomy data results in the same conclusion. This shows that a spinning geometry represents a way to glue together the three-cells of a twisted geometry to form a continuous geometry which represents a point in the loop gravity phase space.

Figures (5)

• Figure 1: A helix (in blue) wrapping around a cylinder.
• Figure 2: The flag $(\hat{\bm{\omega}}, \hat{\bm{r}}_0$) (in blue) with respect to a fixed basis $\bm{\tau}_i$. To measure $\theta^2$, one projects $\hat{\bm{\omega}}$ down the direction of $\bm{\tau}_0$ into a plane perpendicular to $\hat{\bm{\omega}}$. $\theta^2$ is the angle from this projection to $\hat{\bm{r}}_0$, measured within the plane perpendicular to $\hat{\bm{\omega}}$.
• Figure 3: A plot of $\Delta_{\varphi}=(f_\varphi^2 - 4 g_\varphi^2 \sin^2 \varphi)$, which is positive for all $\varphi>0$.
• Figure 4: A plot of $f_\varphi + 2 g_\varphi \sin \varphi$, which is positive for all $\varphi>0$.
• Figure 5: A typical plot of $2(r^{2}_{\varphi}K_{\varphi}\varphi)(f_{\varphi}+ 2 g_\varphi \sin\varphi)- \bm{S}\cdot {\bm{D}}$ where $s\cos\delta = s\sin\delta = 1$. Notice there are seven solutions for $\varphi$.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3