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

TL;DR

This paper introduces a novel approach to quantum geometry using complex SL(2,C) Chern-Simons theory, establishing a correspondence between flat connections and 4D simplices, and deriving a discrete Einstein-Hilbert action in the semiclassical limit.

Contribution

It presents a new quantization method of geometry via complex Chern-Simons theory and links 3D flat connections to 4D quantum geometrical states, including boundary and cosmological terms.

Findings

Quantum states represented by 3D Chern-Simons blocks

Semiclassical limit yields discrete Einstein-Hilbert action

Framework accommodates both signs of curvature and cosmological constant

Abstract

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. We provide a Wilson graph operator that picks out this class of connections. We discuss how to promote these results to a model of Lorentzian covariant quantum gravity encompassing both signs of the cosmological constant. This paper presents the details for the results reported in reference [1] (arXiv:1509.00458).