The first-order formalism takes as its fundamental variables an orthonormal frame (“triad” or “dreibein”) , which determines a metric , and a spin connection . As in the Palatini formalism, and are treated as independent quantities. In terms of the one-forms

the first-order action takes the form with Euler-Lagrange equations The first of these implies that the connection is torsion-free, and, if is invertible, that has the standard expression in terms of the triad. Given such a spin connection, Equation (16) is then equivalent to the standard Einstein field equations.The action (14) has two sets of invariances, the local Lorentz transformations

and the “local translations” Provided the triad is invertible, the latter are equivalent to diffeomorphisms on a shell; more precisely, the combination of transformations with parameters and is equivalent to the diffeomorphism generated by the vector field . The invertibility condition for is important; if it is dropped, the first-order formalism is no longer quite equivalent to the metric formalism [194] .As first noted by Achúcarro and Townsend [2] and further developed by Witten [277, 279], the first-order action (14) is equivalent to that of a Chern-Simons theory. Consider first the case of a vanishing cosmological constant. The relevant gauge group - the group of the geometric structure - is then the Poincaré group , with standard generators and and commutation relations

The corresponding gauge potential is If one defines a bilinear form (or “trace”) it is straightforward to show that the action (14) can be written as with . Equation (22) may be recognized as the standard Chern-Simons action [278] for the group .A similar construction is possible when . For , the pair of one-forms together constitute an gauge potential, with a Chern-Simons action

that is again equivalent to Equation (14), provided we set . If , the complex one-form may be viewed as an gauge potential, whose Chern-Simons action is again equivalent to the first-order gravitational action. For any value of , it is easily checked that the transformations (17) are just the gauge transformations of . Vacuum general relativity in 2+1 dimensions is thus equivalent - again up to considerations of the invertibility of - to a gauge theory. We can now connect the first-order formalism to the earlier description of geometric structures. The field equations coming from the action (22) are simply implying that the field strength of the gauge potential vanishes, i.e., that is a flat connection. Such a connection is completely determined by its holonomies, that is, by the Wilson loops around closed noncontractible curves . This use of the term “holonomy” is somewhat different from that of Section 2.2, but the two are equivalent. Indeed, any structure on a manifold determines a corresponding flat bundle [147] : We simply form the product in each patch, giving the local structure of a bundle, and use the transition functions of the geometric structure to glue the fibers on the overlaps. The holonomy group of this flat bundle can be shown to be isomorphic to the holonomy group of the geometric structure, and for (2+ 1)-dimensional gravity, the flat connection constructed from the geometric structure is that of the Chern-Simons theory. An explicit construction may be found in Section 4.6 of [81] ; see also [7, 263] .The first-order action allows us an additional step that was unavailable in the geometric structure formalism - we can compute the symplectic structure on the space of solutions. The basic Poisson brackets follow immediately from the action:

The resulting brackets among the holonomies have been evaluated by Nelson, Regge, and Zertuche [210, 211] for , for which the two factors in the gauge group may be taken to be independent. The brackets are nonzero only for holonomies of curves that intersect, and can be written in terms of holonomies of “rerouted” curves; symbolically, where is the oriented intersection number at the point that the curves cross. The composition of loops implicit in the brackets (27) makes it difficult to find small closed subalgebras of the sort needed for quantization. However, Nelson and Regge have succeeded in constructing a small but complete (actually overcomplete) set of holonomies on a surface of arbitrary genus that form a closed algebra [213, 212], and Loll has found a complete set of “configuration space” variables [178] .By generalizing a discrete combinatorial approach to Chern-Simons theory due to Fock and Rosly [122] and Alekseev et al. [3, 4, 5], several authors have further explored the quantum group structure of these brackets, which can be expressed in terms of the quantum double of the Lorentz group [37, 36, 61, 201] . It is also interesting that the symplectic structure obtained in this way is closely related to the symplectic structure on the abstract space of loops on first discovered by Goldman [145, 146] .

http://www.livingreviews.org/lrr-2005-1 | © Max Planck Society and
the author(s)
Problems/comments to |