64 4 Geometric structures and Chern-Simons theory
on £. Our construction thus gives us a (6g
—
6)-dimensional family of
spacetimes.
4.2 Geometric structures
The spacetimes of the preceding section are solutions of the vacuum field
equations, but they are physically rather uninteresting. In particular, their
dynamics is trivial; the spatial slices of constant T are nearly isometric,
differing only in overall scale. Intuitively, we have really obtained only half
of the parameters describing the space of solutions - we have specified
initial 'positions', but we have assumed vanishing initial 'momenta'. This
intuition can be made precise in the ADM formalism. The spacetimes
constructed in this manner are precisely the 'static moduli solutions' of
section 4 of the preceding chapter, that is, the solutions for which p
a
= 0.
To generalize this result, it is useful to give a slightly different description
of our construction. Let us return again to two dimensions. We can view
the polygon P(L) as a kind of coordinate patch on the surface S - that
is,
the interior of P(L) is the diffeomorphic image of an open set 0 in Z,
and points in 0 can be identified with their images in P(S). As in any
manifold, coordinate patches are glued together by means of transition
functions, which in this case describe the coordinate changes as we go
from an edge E of P(L) to the edge E
f
identified with E. P(S) has a
constant negative curvature metric, and the metric will remain smooth as
long as the transition functions <j)(E,E
f
) are isometries. In fact, P(L) has
4g edges identified in pairs, and the 2g transition functions <£(£,-,£,') are
precisely the generators of the Fuchsian group F introduced above.
What we have just described is known to mathematicians as a geometric
structure [3, 58, 256]. In general, a manifold M is said to have a geometric
structure (G,X) if M is locally modeled on X with transition functions in
G, much as an ordinary n-dimensional manifold is modeled on R
n
. More
precisely, let G be a Lie group that acts analytically on some manifold X
(the 'model space'), and let M be another manifold of the same dimension
as X. Then a (G,X) structure on M is given by a set of coordinate patches
Ut on M with 'coordinates' & : C7,-
—>
X taking their values in X, and
with transition functions g
t
j =
(j)
i
o
cj>-~
1
\ Ut
n Uj in G. In particular, if we
let X be the hyperbolic plane H
2
and G be the isometry group PSL(2, R),
we obtain a hyperbolic structure on the surface Z.
A fundamental ingredient in the description of a geometric structure is
its holonomy, which can be thought of as measuring the failure of a single
coordinate patch to extend all the way around a closed curve.* Let M
be a (G,X) manifold containing a closed path y. We can cover y with
*
The reader should be cautioned that this is not quite the same 'holonomy' that is found
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