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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: M may admit “large” diffeomorphisms, which cannot be built up smoothly from infinitesimal deformations. The group of such large diffeomorphisms (modulo small diffeomorphisms), D(M ), is called the mapping class group of M ; for the torus T 2, 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 T2 along a circumference to obtain a cylinder, twisting one end by 2p, and regluing. Similar transformations exist for any closed surface S, and in fact the Dehn twists around generators of p1(S) generate D(S)   [5756] . It is easy to see that the mapping class group of a spacetime M acts on p1(M ), and therefore on the holonomies of Section  2.2 . As diffeomorphisms, elements of the mapping class group also acts on the constant curvature metrics gij, and hence on the moduli of Section  2.4 .

Classically, geometries that differ by actions of D(M ) are exactly equivalent, so the “true” space of vacuum solutions for a spacetime with the topology R × S is really M/D(M ), where M is the moduli space (11View Equation). Quantum mechanically, it is not clear whether one should impose mapping class group invariance on states or whether one should merely treat D(M ) 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 D(M ) lead to sensible physical results, but it is at least clear that the mapping class group cannot be ignored.


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