Deformed spinor networks for loop gravity: towards hyperbolic twisted geometries

In the context of a canonical quantization of general relativity, one can deform the loop gravity phase space on a graph by replacing the $$T^*\mathrm{SU}(2)$$T∗SU(2) phase space attached to each edge by $$\mathrm{SL}(2,{\mathbb C})$$SL(2,C) seen as a phase space. This deformation is supposed to encode the presence of a non-zero cosmological constant.Here we show how to parametrize this phase space in terms of spinor variables, thus obtaining deformed spinor networks for loop gravity, with a deformed action of the gauge group $$\mathrm{SU}(2)$$SU(2) at the vertices. These are to be formally interpreted as the generalization of loop gravity twisted geometries to a hyperbolic curvature.