Deformations of Lorentzian Polyhedra: Kapovich-Millson phase space and SU(1,1) Intertwiners

We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with $N$ faces. Starting with the Schwinger representation of the $\mathfrak{su}(1,1)$ Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the three-dimensional Minkowski space $\mathbb{R}^{1,2}$. Considering $N$ copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with $N$ faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of $SU(1,1)$-invariant observables, whose flow by the Poisson bracket generate both area-preserving and area-changing deformations. We further show that the area-preserving observables form a $\mathfrak{gl}_{N}(\mathbb{R})$ Lie algebra and that they generate a $GL_{N}(\mathbb{R})$ action on Lorentzian polyhedra at fixed total area. That action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a $GL_{N}(\mathbb{R})$ transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as $SU(1,1)$ intertwiners between unitary $SU(1,1)$-representations from the principal continuous series. Those $SU(1,1)$-intertwiners are the building blocks of spin network states in loop quantum gravity in 3+1 dimensions for time-like slicing and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.