The corresponding decomposition of the
conjugate momentum is described in
[206]
: Up to a diffeomorphism, the trace-free part of
can be written as a holomorphic quadratic differential
, that is, a transverse traceless tensor with respect to the
covariant derivative compatible with
. The space of such quadratic differentials parametrizes the
cotangent space of the moduli space
[1], and the reduced phase space becomes, essentially, the cotangent
bundle of the moduli space.
With the decomposition of
[206], the momentum constraints
become trivial, while the Hamiltonian constraint becomes an
elliptic differential equation that determines the scale factor
in Equation (31
) as a function of
and
,
Three-dimensional gravity again reduces to a
finite-dimensional system, albeit one with a complicated
time-dependent Hamiltonian. The physical phase space is
parametrized by
, which may be viewed as coordinates for the cotangent bundle of
the moduli space of
. For a surface of genus
, this gives us
degrees of freedom, matching the results of Section
2.2
.
If
, this correspondence can be made more explicit: For
and
, the space (11
) of geometric structures is itself a cotangent bundle, whose
base space is the space of hyperbolic structures on
. This follows from the fact that the group
is the cotangent bundle of
. Concretely, in the first-order formalism of Section
2.3, the curvature equation (16
) with
implies that
is a flat
connection; and if
is a curve in the space of such flat connections, the tangent
vector
satisfies the torsion equation (17
). For
, I know of no such direct correspondence, and the general
relationship between the ADM and first-order solutions seems less
transparent.
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