Regge calculus in 2+1 dimensions begins with a triangulated three-manifold, consisting of a collection of flat simplices joined along one-dimensional edges. The curvature of such a manifold is concentrated entirely at the edges. For a simplicial three-manifold with Riemannian signature, composed of simplices with edges of length , Regge’s form of the Einstein-Hilbert action is

where is the conical deficit angle at the edge labeled by the index . A similar expression exists for Lorentzian signature, although the definition of the deficit angle is a bit more complicated [48] .The first hint that (2+1)-dimensional gravity might be exceptional came from the observation by Ponzano and Regge [225] that the Regge action in 2+1 dimensions can be re-expressed in terms of Wigner-Racah -symbols. (See [90] for more about these quantities.) Consider first a single tetrahedron with edge lengths , where the are integers or half-integers. Ponzano and Regge noticed, and Roberts later proved rigorously [234], that in the limit of large ,

where is a -symbol, is the Regge action (65) for the tetrahedron, and is its volume. For a manifold made of a collection of such tetrahedra, the full Regge action will occur in a product of such -symbols. This suggests that the (2+1)-dimensional path integral, which is essentially a sum over geometries of terms of the form , might be expressible as a sum of such products. Ponzano and Regge’s specific proposal, developed by Hasslacher and Perry [153] and modified by Ooguri [219] to account for boundaries, was the following:Consider a three-manifold with boundary , with a given triangulation of . Choose a triangulation of that agrees with the triangulation of the boundary. Label interior edges of tetrahedra by integers or half-integers and exterior (boundary) edges by , and for a given tetrahedron , let denote the spins that color its (interior and exterior) edges. Then

where “int” and “ext” mean “interior” and “exterior” and is a regularization factor that controls divergences in the sum over interior lengths. With this weighting, identities among -symbols may be used to show that the amplitude is invariant under refinement - that is, subdivision of a tetrahedron into four smaller tetrahedra - suggesting that we are dealing with a “topological” theory that does not depend on the choice of triangulation. This is, of course, what one would hope for, based on the classical characteristics of (2+1)-dimensional gravity.The “topological” feature of the Ponzano-Regge model was made more precise by Turaev and Viro [262], who discovered an improved regularization, based on the technology of quantum groups. The “spins” in Equation (67) can be viewed as labeling representations of . If these are replaced by representations of the quantum group (“quantum ”), with , , the number of such representations is finite, and the sum over interior edge lengths is automatically cut off. With appropriate substitutions (e.g., “quantum” -symbols [90]), the Ponzano-Regge amplitude (67) becomes well-defined without any regularization.

The construction of physical states as appropriate functions of boundary edge lengths is described in Section 11.2 of [81] . The resulting amplitudes can be computed for simple topologies [162, 161], and have several key features:

- For large but finite , the Turaev-Viro quantum group regularization introduces a cosmological constant to the Regge action [204, 203], Correspondingly, the quantum -symbols are related to spherical tetrahedra rather than flat tetrahedra [255] .
- In the large limit, the Turaev-Viro Hilbert space is isomorphic to the space of gauge-invariant functions of flat connections [219, 220, 242] . This establishes a direct link to Chern-Simons quantization: Just as (2+1)-dimensional Lorentzian gravity can be written as an Chern-Simons theory with a configuration space of flat connections, three-dimensional Euclidean gravity can be written as an Chern-Simons theory with a configuration space of flat connections.
- For a closed three-manifold , the Turaev-Viro amplitude - now interpreted as a partition function - is equal to the absolute square of the partition function of an Chern-Simons theory with coupling constant [228, 260, 233], This again establishes an equivalence with Euclidean gravity in first-order form: The first-order Euclidean action with can be written as a difference of Chern-Simons actions, so in agreement with Equation (70).
- A candidate for a discrete version of the Wheeler-DeWitt equation in three dimensions has been found [47], for which the Ponzano-Regge wave functions are solutions.

Although it has not been universally appreciated, the existence of a divergence in the sum (67) - regulated either by an explicit cut-off or by quantum group tricks - is rather mysterious, given the absence of local excitations and the general well-behavedness of gravity in three dimensions. This mystery may have recently been solved by Freidel and Louapre [131], who show that a residual piece of the diffeomorphism symmetry has not been factored out of the Ponzano-Regge action. Because of this symmetry, the sum (67) overcounts physical configurations, and the regulator is simply the remaining gauge volume. Freidel and Louapre further show that the symmetry can instead be gauge-fixed, leading to a sum over a restricted and considerably simplified class of “collapsed” triangulations.

While the mathematics of Ponzano-Regge and Turaev-Viro models has been studied extensively, so far only a bit of attention has been given to the “traditional” issues of quantum gravity. A few numerical investigations of the Ponzano-Regge path integral have been undertaken [151], but the evidence of a continuum limit is thus far inconclusive. The model has been used to study conditional probabilities and the emergence of quasiclassical behavior in quantum gravity [223], but the cut-off dependence of these results makes their physical significance unclear. In an interesting recent paper, Colosi et al. have investigated the dynamics of a single tetrahedron [241], showing that a quantum description of the evolution can be given in terms of a boundary amplitude.

A number of observables, whose expectation values generally give topological information about the spacetime or about knots within spacetime, have been discussed in [23, 54, 261, 139] . With a few exceptions, though, work in this area has remained largely mathematical in nature; fairly little is understood about the physics of these observables, although some are probably related to length spectra [46] and perhaps volumes [127, 64], and others are almost certainly connected to scattering amplitudes for test particles.

The Ponzano-Regge and Turaev-Viro models are examples of “spin foam” models [35, 222], that is, a model based on simplicial complexes with faces, edges, and vertices labeled by group representations and intertwiners. A key question is whether one can extend such models to Lorentzian signature. It has been known for several years how to generalize the Ponzano-Regge action for a single tetrahedron [48, 101, 191], and recently considerable progress has been made in constructing Lorentzian spin foam models [222, 125, 102] .

Probably the most elegant derivation of a Lorentzian spin foam description starts with the first-order action (14), with , for a triangulated manifold [125, 128] . One can rewrite the action in terms of a set of discrete variables: a Lie algebra element corresponding to the integral of along the edge of a tetrahedron in the triangulation, and a holonomy of the connection around the edge. The path integral then becomes an integral over these variables. As in the continuum path integral of Section 3.10, the integral over the produces a delta function for each edge. This translates back to the geometric statement that the constraints require the connection to be flat, and thus to have trivial holonomy around a contractible curve surrounding an edge.

For the Euclidean Ponzano-Regge action, , and the key trick is now to use the Plancherel formula to express each as a sum over the characters of finite-dimensional representations of . Fairly straightforward arguments then permit an exact evaluation of the remaining integrals over the , reproducing the symbols in the Ponzano-Regge action. To obtain a Lorentzian version, one must replace by . The corresponding Plancherel formula involves a sum over both the (continuous) principle series of representations of and the discrete series. Consequently, edges may now be labeled either by discrete or continuous spins. Similar methods may be used for supergravity [177] .

The resulting rather complicated expression for the partition function may be found in [125] . The appearance of both continuous and discrete labels has a nice physical interpretation [130] : Continuous representations describe spacelike edges, and seem to imply a continuous length spectrum, while discrete representations label timelike edges, and suggest discrete time. These results should probably not yet be considered conclusive, since they require operators that do not commute with all of the constraints, but they are certainly suggestive.

While spin foam models ordinarily assume a fixed spacetime topology, recent work has suggested a method for summing over all topologies as well, thus allowing quantum fluctuations of spacetime topology [132] . These results will be discussed in Section 3.11 . Methods from 2+1 dimensions have also been generalized to higher dimensions, leading to new insights into the construction of spin foams.

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