34 2
Classical general relativity
in 2hi
dimensions
It is clear from equation (2.70) that these 'wouldbe diffeomorphism'
degrees of freedom are related to the 'wouldbe gauge' degrees of freedom
in the ChernSimons formulation. Unfortunately, in the absence of a local
decomposition of degrees of freedom in the metric formalism analogous
to (2.81), it is not clear how to write down the metric equivalent of the
boundary action (2.92). This means that for now, at least, the precise form
of the boundary action for general relativity can only be given in 2+1
dimensions, since only then is a ChernSimons formulation possible.
2.7 Comparing generators of invariances
In section 4, we computed the generators of diffeomorphisms in the
ADM formulation and analyzed their Poisson algebra. In particular, we
found that the algebra (2.47)(2.48) was not a true Lie algebra  the
commutators were characterized by fielddependent structure functions
rather than structure constants. On the other hand, we have now seen
that the firstorder formalism allows us to express the diffeomorphisms as
ordinary gauge transformations, whose generators should satisfy a genuine
Lie algebra. By studying the relationship between these two formulations,
we can gain further insight into the structure of (2+l)dimensional gravity.
We begin by considering a decomposition of the firstorder action
(2.61) into space and time components. On a manifold with the topology
M « [0,1] x 2, it is easy to check that up to boundary terms, the action
can be written as
/ =2
where the constraints are
[dt[
d
2
x
{eVef&ja

co
t
a
%
a

e
t
a
%
a
],
(2.97)
,e] = ^ [d
t
ej
a
 djef +
e
abc
(co
ib
e
jc

co
jb
e
ic
)}
,e] = \& \dicof  djcof +
e
abc
(co
ib
co
jc
 Ae
ib
e
jc
)] .
1
(2.98)
These constraints have a straightforward intepretation. When A = 0, the
# constraint tells us that cof, the induced 50(2,1) connection on Z, is flat.
The # constraint then implies that e? is a cotangent vector to the space
of flat connections. Indeed, let
cot
a
(s)
be a oneparameter family of flat
connections  solutions of the ^ constraint  on 2. Then the derivative
of
$>
a
[co(s)]
with respect to s is
dco(s)
~ds~