Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 4

$\operatorname{SL}(2,\mathbb{C})$ Chern-Simons theory, flat connections, and four-dimensional quantum geometry

Pages: 1067 – 1158



Hal M. Haggard (Physics Program, Bard College, Annandale-on-Hudson, New York, U.S.A.; and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Muxin Han (Institut für Quantengravitation, Universität Erlangen-Nürnberg, Erlangen, Germany; and Department of Physics, Florida Atlantic University, Boca Raton, Fl., U.S.A.)

Wojciech Kaminski (Faculty of Physics, University of Warsaw, Poland)

Aldo Riello (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)


A correspondence between three-dimensional flat connections and constant curvature four-dimensional simplices is used to give a novel quantization of geometry via complex $\operatorname{SL}(2,\mathbb{C})$ 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 threedimensional 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 [1].

Published 16 January 2020