Geometric parametrization of $SO(D+1)$ phase space of all dimensional loop quantum gravity

TL;DR

This paper extends the twisted-geometry parametrization from 4D to all-dimensional loop quantum gravity, providing a geometric understanding and a new gauge reduction method for the phase space.

Contribution

It generalizes the twisted-geometry parametrization to all dimensions and introduces a gauge reduction procedure based on the geometric interpretation of the phase space.

Findings

Provides a geometric parametrization of the $SO(D+1)$ phase space.

Introduces a new gauge reduction procedure for simplicity constraints.

Identifies the classical state space of discrete ADM data.

Abstract

To clarify the geometric information encoded in the $SO(D+1)$ spin-network states for the higher dimensional loop quantum gravity, we generalize the twisted-geometry parametrization of the $SU(2)$ phase space for $(1+3)$ dimensional loop quantum gravity to that of the $SO(D+1)$ phase space for the all-dimensional case. The Poisson structure in terms of the twisted geometric variables suggests a new gauge reduction procedure, with respect to the discretized Gaussian and simplicity constraints governing the kinematics of the theory. Endowed with the geometric meaning via the parametrization, our reduction procedure serves to identify proper gauge freedom associated with the anomalous discretized simplicity constraints and subsequently leads to the desired classical state space of the (twisted) discrete ADM data.