SL(2,C) Chern-Simons Theory, Flat Connections, and Four-dimensional Quantum Geometry

Understand

A correspondence between three-dimensional flat connections and constant curvature four-dimensional simplices is used to give a novel quantization of geometry via complex SL(2,C) Chern-Simons theory.

• The resulting quantum geometrical states are hence represented by the 3d blocks of analytically continued Chern-Simons theory.
• In the semiclassical limit of this quantization the three-dimensional Chern-Simons action, remarkably, becomes the discrete Einstein-Hilbert action of a 4-simplex, featuring the appropriate boundary terms as well as the essential cosmological term proportional to the simplex's curved 4-volume.
• Both signs of the curvature and associated cosmological constant are present in the class of flat connections that give rise to this correspondence.