SL(2,C) Chern-Simons Theory, Flat Connections, and Four-dimensional Quantum Geometry
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).
Cite
@article{arxiv.1512.07690,
title = {SL(2,C) Chern-Simons Theory, Flat Connections, and Four-dimensional Quantum Geometry},
author = {Hal M. Haggard and Muxin Han and Wojciech Kaminski and Aldo Riello},
journal= {arXiv preprint arXiv:1512.07690},
year = {2021}
}
Comments
Presentation substantially revised; 45+5 pages, 6 figures