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arXiv · 1512.07690

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

Hal M. Haggard·Muxin Han·Wojciech Kaminski·Aldo Riello
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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).

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Hal M. Haggard, Muxin Han, Wojciech Kaminski, Aldo Riello. 2015-12-24. SL(2,C) Chern-Simons Theory, Flat Connections, and Four-dimensional Quantum Geometry. https://doi.org/10.4310/atmp.2019.v23.n4.a3

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