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2.3 The Chern-Simons formulation

The formalism of geometric structures provides an elegant description of vacuum spacetimes in 2+1 dimensions, but it is rather remote from the usual physicist’s approach. In particular, the Einstein-Hilbert action is nowhere in sight, and even the metric makes only a limited appearance. Fortunately, the description is closely related to the more familiar first-order Chern-Simons formalism  [108277Jump To The Next Citation Point279Jump To The Next Citation Point2Jump To The Next Citation Point], which, in turn, can connect us back to the standard metric formalism.

The first-order formalism takes as its fundamental variables an orthonormal frame (“triad” or “dreibein”) e a m, which determines a metric g = j e ae b mn ab m n, and a spin connection w ab m . As in the Palatini formalism, e and w are treated as independent quantities. In terms of the one-forms

1 ea = emadxm, wa = -eabcwmbcdxm, (13) 2
the first-order action takes the form
integral [ ( ) ] I = 2 ea /\ dwa + 1-eabcwb /\ wc + /\eabcea /\ eb /\ ec , (14) M 2 6
with Euler-Lagrange equations
b c Ta = dea + eabcw /\ e = 0, (15) 1- b c /\- b c Ra = dwa + 2 eabcw /\ w = - 2 eabce /\ e. (16)
The first of these implies that the connection is torsion-free, and, if e is invertible, that w has the standard expression in terms of the triad. Given such a spin connection, Equation (16View Equation) is then equivalent to the standard Einstein field equations.

The action (14View Equation) has two sets of invariances, the local Lorentz transformations

a abc a a abc de = e ebtc, dw = dt + e wbtc, (17)
and the “local translations”
a a abc a abc de = dr + e wbrc, dw = - /\e ebrc. (18)
Provided the triad e is invertible, the latter are equivalent to diffeomorphisms on a shell; more precisely, the combination of transformations with parameters ra = q .ea and t a = q .wa is equivalent to the diffeomorphism generated by the vector field q . The invertibility condition for e 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  [277Jump To The Next Citation Point279Jump To The Next Citation Point], the first-order action (14View Equation) is equivalent to that of a Chern-Simons theory. Consider first the case of a vanishing cosmological constant. The relevant gauge group - the group G of the geometric structure - is then the Poincaré group ISO(2, 1), with standard generators J a and Pa and commutation relations

[ ] [ ] [ ] J a,J b = eabcJc, J a, Pb = eabcPc, Pa, Pb = 0. (19)
The corresponding gauge potential is
A = eaPa + waJa. (20)
If one defines a bilinear form (or “trace”)
tr(J aPb) = jab, tr(J aJ b) = tr(PaPb) = 0, (21)
it is straightforward to show that the action (14View Equation) can be written as
( ) k integral 2 ICS[A] = --- tr A /\ dA + --A /\ A /\ A , (22) 4p M 3
with k = 1/(4G) . Equation (22View Equation) may be recognized as the standard Chern-Simons action  [278Jump To The Next Citation Point] for the group ISO(2, 1) .

A similar construction is possible when /\ /= 0 . For 2 /\ = - 1/l < 0, the pair of one-forms (±)a a a A = w ± e /l together constitute an SO(2, 1)× SO(2, 1) gauge potential, with a Chern-Simons action

(+) (-) (+) (-) I[A ,A ] = ICS[A ]- ICS[A ] (23)
that is again equivalent to Equation (14View Equation), provided we set k = l/(4G) . If /\ > 0, the complex one-form V~ -- Aa = wa + i /\ea may be viewed as an SL(2,C) 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 (17View Equation) are just the gauge transformations of A . Vacuum general relativity in 2+1 dimensions is thus equivalent - again up to considerations of the invertibility of e - 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 (22View Equation) are simply
F [A] = dA + A /\ A = 0, (24)
implying that the field strength of the gauge potential A vanishes, i.e., that A is a flat connection. Such a connection is completely determined by its holonomies, that is, by the Wilson loops
( integral ) U = P exp - A (25) g g
around closed noncontractible curves g . This use of the term “holonomy” is somewhat different from that of Section  2.2, but the two are equivalent. Indeed, any (G, X) structure on a manifold M determines a corresponding flat G bundle  [147] : We simply form the product G × Ui in each patch, giving the local structure of a G bundle, and use the transition functions gij 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  [81Jump To The Next Citation Point] ; see also  [7263] .

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:

{ } 1 eia(x),wjb(x') = --jabeijd2(x- x'). (26) 2
The resulting brackets among the holonomies have been evaluated by Nelson, Regge, and Zertuche  [210211Jump To The Next Citation Point] for /\ < 0, for which the two SL(2,R) factors in the gauge group G 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,
{ \/ } 1 (\/ > < ) = ± --e(p) - 2 , (27) , 4l
where e(p) is the oriented intersection number at the point p that the curves cross. The composition of loops implicit in the brackets (27View Equation) 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  [213212], 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.  [345], 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  [37Jump To The Next Citation Point36Jump To The Next Citation Point61Jump To The Next Citation Point201Jump To The Next Citation Point] . 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 S first discovered by Goldman  [145146] .

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