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1 Introduction

The task of quantizing general relativity is one of the outstanding problems of modern theoretical physics. Attempts to reconcile quantum theory and general relativity date back to the 1930s (see  [240] for a historical review), and decades of hard work have yielded an abundance of insights into quantum field theory, from the discovery of DeWitt-Faddeev-Popov ghosts to the development of effective action and background field methods to the detailed analysis of the quantization of constrained systems. But despite this enormous effort, no one has yet succeeded in formulating a complete, self-consistent quantum theory of gravity  [83] .

The obstacles to quantizing gravity are in part technical. General relativity is a complicated nonlinear theory, and one should expect it to be more difficult than, say, electrodynamics. Moreover, viewed as an ordinary field theory, general relativity has a coupling constant 1/2 G with dimensions of an inverse mass, and standard power-counting arguments - finally confirmed in 1986 by explicit computations  [149] - indicate that the theory is nonrenormalizable. But the problem of finding a consistent quantum theory of gravity goes deeper. General relativity is a geometric theory of spacetime, and quantizing gravity means quantizing spacetime itself. In a very basic sense, we do not know what this means. For example:

Faced with such problems, it is natural to look for simpler models that share the important conceptual features of general relativity while avoiding some of the computational difficulties. General relativity in 2+1 dimensions - two dimensions of space plus one of time - is one such model. As a generally covariant theory of spacetime geometry, (2+1)-dimensional gravity has the same conceptual foundation as realistic (3+ 1)-dimensional general relativity, and many of the fundamental issues of quantum gravity carry over to the lower dimensional setting. At the same time, however, the (2+1)-dimensional model is vastly simpler, mathematically and physically, and one can actually write down viable candidates for a quantum theory. With a few exceptions, (2+1)-dimensional solutions are physically quite different from those in 3+1 dimensions, and the (2+1)-dimensional model is not very helpful for understanding the dynamics of realistic quantum gravity. In particular, the theory does not have a good Newtonian limit  [107Jump To The Next Citation Point49Jump To The Next Citation Point94Jump To The Next Citation Point] . But for understanding conceptual problems - the nature of time, the construction of states and observables, the role of topology and topology change, the relationships among different approaches to quantization - the model has proven highly instructive.

Work on (2+1)-dimensional gravity dates back to 1963, when Staruszkiewicz first described the behavior of static solutions with point sources  [246Jump To The Next Citation Point] . Progress continued sporadically over the next twenty years, but the modern rebirth of the subject can be traced to the seminal work of Deser, Jackiw, ’t Hooft, and Witten in the mid-1980s  [107Jump To The Next Citation Point105Jump To The Next Citation Point106Jump To The Next Citation Point249Jump To The Next Citation Point103Jump To The Next Citation Point277Jump To The Next Citation Point279Jump To The Next Citation Point] . Over the past twenty years, (2+ 1)-dimensional gravity has become an active field of research, drawing insights from general relativity, differential geometry and topology, high energy particle theory, topological field theory, and string theory.

As I will explain below, general relativity in 2+1 dimensions has no local dynamical degrees of freedom. Classical solutions to the vacuum field equations are all locally diffeomorphic to spacetimes of constant curvature, that is, Minkowski, de Sitter, or anti-de Sitter space. Broadly speaking, three ways to introduce dynamics have been considered:

  1. Point particles can be added, appearing as conical “defects” in an otherwise constant curvature spacetime. Most of the earliest papers in the field  [246105106107249103] were investigations of the dynamics of such conical singularities.
  2. If a negative cosmological constant is present, black hole solutions can be found  [4140] . For such solutions, dynamics at either the horizon or the boundary at infinity can lead to local degrees of freedom  [782475996512491], although these are certainly not yet completely understood  [82Jump To The Next Citation Point] .
  3. One can consider nontrivial spatial or spacetime topologies  [277Jump To The Next Citation Point279Jump To The Next Citation Point] . Such “cosmological” solutions have moduli - a finite number of parameters that distinguish among geometrically inequivalent constant curvature manifolds - and these can become dynamical.

In this paper, I will limit myself to the third case, (2+1)-dimensional vacuum “quantum cosmology.” This review is based in part on a series of lectures in  [76Jump To The Next Citation Point] and an earlier review  [74], and much of the material can be found in more detail in a book  [81Jump To The Next Citation Point] . There is not yet a comprehensive review of gravitating point particles in 2+1 dimensions, although  [65Jump To The Next Citation Point197Jump To The Next Citation Point19537Jump To The Next Citation Point36Jump To The Next Citation Point19963Jump To The Next Citation Point183] will give an overview of some results. Several good general reviews of the (2+1)-dimensional black hole exist  [7539], although a great deal of the quantum mechanics is not yet understood  [82] .

Although string theory is perhaps the most popular current approach to quantum gravity, I will have little to say about it here: While some interesting results exist in 2+1 dimensions, almost all of them are in the context of black holes (see, for example,  [157170187188189]). I will also have little to say about (2+1)-dimensional supergravity, although many of the results described below can be generalized fairly easily, and I will not address the coupling of matter except for a brief discussion in Section  5 .

Throughout, I will use units 16pG = 1 and h = 1 unless otherwise noted.


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