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
[108, 277
, 279
, 2
], 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”)
, 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]
.