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A correspondence between three-dimensional flat connections and constantcurvature four-dimensional simplices is used to give a novel quantization ofgeometry via complex SL(2,C) Chern-Simons theory. The resulting quantumgeometrical states are hence represented by the 3d blocks of analyticallycontinued Chern-Simons theory. In the semiclassical limit of this quantizationthe three-dimensional Chern-Simons action, remarkably, becomes the discreteEinstein-Hilbert action of a 4-simplex, featuring the appropriate boundaryterms as well as the essential cosmological term proportional to the simplex'scurved 4-volume. Both signs of the curvature and associated cosmologicalconstant are present in the class of flat connections that give rise to thiscorrespondence. We provide a Wilson graph operator that picks out this class ofconnections. We discuss how to promote these results to a model of Lorentziancovariant quantum gravity encompassing both signs of the cosmological constant.This paper presents the details for the results reported in reference [1](arXiv:1509.00458).

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