###
2.6
Large diffeomorphisms

Up to now, I have avoided discussing an important discrete
symmetry of general relativity on topologically nontrivial
spacetimes. The description of a solution of the field equations
in terms of holonomies (Sections
2.2
and
2.3) or moduli (Section
2.4) is invariant under infinitesimal diffeomorphisms, and hence
under “small” diffeomorphisms, those that can be smoothly
deformed to the identity. But if the spacetime manifold is
topologically nontrivial, its group of diffeomorphisms may not be
connected:
may admit “large” diffeomorphisms, which cannot be built up
smoothly from infinitesimal deformations. The group of such large
diffeomorphisms (modulo small diffeomorphisms),
, is called the mapping class group of
; for the torus
, it is also known as the modular group.
The archetype of a large diffeomorphism is a
Dehn twist of a torus, which may be described as the operation of
cutting
along a circumference to obtain a cylinder, twisting one end by
, and regluing. Similar transformations exist for any closed
surface
, and in fact the Dehn twists around generators of
generate
[57, 56]
. It is easy to see that the mapping class group of a spacetime
acts on
, and therefore on the holonomies of Section
2.2
. As diffeomorphisms, elements of the mapping class group also
acts on the constant curvature metrics
, and hence on the moduli of Section
2.4
.

Classically, geometries that differ by actions
of
are exactly equivalent, so the “true” space of vacuum solutions
for a spacetime with the topology
is really
, where
is the moduli space (11). Quantum mechanically, it is not clear whether one should
impose mapping class group invariance on states or whether one
should merely treat
as a symmetry under which states may transform nontrivially
(see, for instance,
[164]). In 2+1 dimensions, though, there seems to be a strong argument
in favor of treating the mapping class group as a genuine
invariance, as follows. Using the Chern-Simons formalism, one can
compute the quantum amplitude for the scattering of a point
particle off another particle
[65], a black hole
[259], or a handle
[67]
. In each case, it is only when one imposes invariance under the
mapping class group that one recovers the correct classical
limit. It may still be that simple enough representations of
lead to sensible physical results, but it is at least clear that
the mapping class group cannot be ignored.