Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe

TL;DR

This review shows that quantum gravity in $2+1$ dimensions, despite lacking local degrees of freedom, remains richly structured, with global data encoded in holonomies and moduli. It surveys a spectrum of quantization approaches—reduced phase space, Chern-Simons, covariant canonical, loop, and lattice/spin-foam methods—highlighting both their successes and fundamental nonuniqueness, especially in the presence of large diffeomorphisms and topology changes. The torus universe serves as a concrete testing ground where explicit quantizations reveal connections between moduli-space dynamics, Maass operators, and modular invariance, while general genus cases and path-integral treatments expose challenges such as divergences in topology sums and the problem of time. Overall, the work positions $2+1$ gravity as a crucial, though nontrivial, laboratory for exploring how different quantum gravity frameworks relate to each other and what lessons might carry over to the physical $3+1$ theory.

Abstract

In three spacetime dimensions, general relativity drastically simplifies, becoming a ``topological'' theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)-dimensional vacuum gravity in the setting of a spatially closed universe.

Figures (4)

• Figure 1: The curve $\gamma$ is covered by coordinate patches $U_i$, with transition functions $g_i \in G$. The composition $g_1\circ \dots\circ g_n$ is the holonomy of the curve.
• Figure 2: The ADM decomposition is based on the Lorentzian version of the Pythagoras theorem.
• Figure 3: Three tetrahedra can occur in Lorentzian dynamical triangulation.
• Figure 4: A manifold $M$ with a single boundary $\Sigma$ describes the birth of a universe in the Hartle--Hawking approach to quantum cosmology.