For simplicity, let us initially restrict our
attention to the case
. The group
of Section
2.2, or, equivalently, the gauge group in the Chern-Simons formalism
of Section
2.3, is then
. The fundamental group
has two generators,
and
, satisfying a single relation similar to Equation (12
):
It is a bit more convenient to describe the
holonomies as elements of the covering group
[211
]
. Let
denote the two
holonomies corresponding to the curve
. An
matrix
is called hyperbolic, elliptic, or parabolic according to
whether
is greater than, equal to, or less than 2, and the space of
holonomies correspondingly splits into nine sectors. It may be
shown that only the hyperbolic-hyperbolic sector corresponds to a
spacetime in which the
slices are spacelike
[117
, 119, 182
, 209]
. By suitable overall conjugation, the two generators of the
holonomy group in this sector can then be taken to be
To obtain the corresponding geometry, we can
use the quotient space construction of Section
2.2
. Note first that three-dimensional anti-de Sitter space can be
represented as the submanifold of flat
(with coordinates
and metric
) defined by the condition that
It is straightforward to show that the resulting induced metric is
whereTo relate these expressions to the ADM
formalism of Section
2.4, we must first find the slices of constant extrinsic curvature
. For the metric (40
), the extrinsic curvature of a slice of constant
is
, which is independent of
and
. A constant
slice is thus also a slice of constant York time. The standard
flat metric on
, the genus one version of the standard metric (31
), is
To quantize this system, we will need the
classical Poisson brackets, which can be obtained from
Equation (26):
Finally, let us consider the action of the
torus mapping class group. This group is generated by two Dehn
twists, which act on
by
For a torus universe with zero or positive
cosmological constant, similar constructions are possible. I
refer the reader to
[81]
for details.
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