Jorma Louko is an Associate Professor in Applied Mathematics at the School of Mathematical Sciences, University of Nottingham

2+1 gravity is topological even without spacetime orientability, and quantisable for selected couplings

General relativity in four and more spacetime dimensions has local dynamical degrees of  freedom, as manifested for example in gravitational waves. In three spacetime dimensions,  by contrast, Einstein’s equations preclude local dynamics but allow still dynamics in the  global properties. This makes (2+1)-dimensional general relativity a dynamically simple but geometrically interesting arena for quantising gravity.

A significant development due to Achucarro and Townsend in 1986 and to Witten in 1988 was to formulate (2+1)-dimensional general relativity as a Chern-Simons theory,  underscoring the topological character of the dynamics and opening the door for new quantisation techniques, on which the current status of the field is largely based. The standard Chern-Simons formulation assumes however at the outset that the spacetime is orientable, and the vast majority of the current literature is subject to this restriction.

The present paper by Chen, Witt and Plotkin gives (2+1)-dimensional general relativity a Chern-Simons formulation that applies to both orientable and nonorientable spacetimes, using as the fundamental variables densities rather than differential forms, and showing that the nonorientable theory is still topological. The work lays the foundations for the study of 2+1 Chern-Simons gravity and related Chern-Simons theories on nonorientable spacetimes. As a first application, a simplified theory with an Abelian gauge group is quantised on selected nonorientable manifolds, showing that the values of the coupling
constants are constrained much more stringently than in the orientable case.

Read the full article: Class. Quantum Grav. 31 055008
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