On the relation between 2+1 Einstein gravity and Chern Simons theory

TL;DR

The paper demonstrates that 2+1 dimensional Einstein gravity and Chern-Simons theory with the Poincaré gauge group are not fully equivalent when considering gauge orbits and the reduced phase space. By comparing the action and equations of motion, it shows that equivalence holds for invertible dreibein configurations, but singular metrics in CS theory produce distinct gauge orbits from those in GR. A concrete counterexample on a four-punctured manifold shows two GR states with the same transition data that are not connected by a smooth gauge transformation, implying large translations cannot be gauged in GR. The work highlights how including singular metrics and large gauge transformations affects the phase-space structure, and cautions against naive identifications between CS theory and GR in 2+1 dimensions.

Abstract

A simple example is given to show that the gauge equivalence classes of physical states in Chern Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore not equivalent. It is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects.

Figures (3)

• Figure 1: The gauge orbits (solid lines) and the submanifold of singular metrics (dashed line) in the physical phase space ${\mathcal{P}}$ of Chern Simons theory. The states $\phi_1$ and $\phi_2$ are related by a local Poincaré transformation in Chern Simons theory, but they are not gauge equivalent in Einstein gravity. Instead, $\phi_2$ is related to $\phi_3$ by a large diffeomorphism.
• Figure 2: The space manifold $\mathcal{N}$ is a plane with four punctures. It can be covered by two simply connected regions $\mathcal{N}_\pm$ overlapping along their common boundary, which consists of five lines $\lambda_0$ through $\lambda_4$.
• Figure 3: The images of the maps $\vec{f}_\pm$ on the spatial plane in Minkowski space. The geometry of the space manifold can be read off by gluing the two flat surfaces together along their edges. The space obtained in (a) is a cone with four tips in a row, which is different from that in (b), where the tips form a parallelogram.