4.2 Symplectic quotients

 

 

In classical mechanics one deals with a phase-space which is a symplectic manifold X. If a compact Lie group G acts symplectically on X then (under mild assumptions) there is a moment mapis called the reduced phase-space or symplectic quotient. It is a manifold (with singularities) and inherits a natural sym­plectic structure. Its dimension is, in general, given by

 

 

To distinguish it from the ordinary quotient X/G (which is not symplectic) it may be denoted by X//G. The good case (leading to the dimension formula above) is when the generic G-orbit is free (or has only a finite isotropy group).

There are more general constructions of symplectic quotients of the form

 

 

where À is a G-orbit in Lie (G)*. These are related not to the invariant subring but to the part that transforms according to a given irreducible representation of G. In particular, if G is abelian, different integral points À in Lie (G)* correspond to different characters of G.

A basic example is given by taking G=U(\) acting by scalar multiplication on C" = X. Giving C" the symplectic structure from the standard Hermitian metric we find

 

 

It follows that the symplectic quotient /x_1(l)/ t/(l) is the complex projective space Pn-,(C) with the symplectic struc­ture of its standard Kahler metric.

The space

 

Note that P„_1(C) = (C"-0)/C*, the natural complex quotient by the group C* (complexification of U( 1)). Thus P„-,(C) occurs both as a symplectic quotient and as a complexalgebraic quotient. This is in fact a typical story as we shall explain.

Return now to the situation of the preceding section with a compact Lie group G, and its complexification Gc, acting on an algebraic variety X with its ample line-bundle if. We can fix a G-invariant connection on if and its curvature will then be a type (1,1) form corresponding to a G-invariant Kahler metric on X. For example, fix a G-invariant metric on the space of sections of iffc with k large, to give a projective embedding of X. The Kahler metric of X defines a G-invariant symplectic structure. We can therefore form both the Mum- ford quotient of X by Gc and the symplectic quotient X//G. It is a general theorem (see [18]) that these coincide and the symplectic structure of X/G is defined by a Kahler metric on the Mumford quotient. The key step in this identification is to show that every closed Gc-orbit in Xss contains a G-orbit in /*"'(0).

The advantage of the symplectic quotient X // G is that it is obviously compact, and we do not need to worry about stable or semi-stable points. On the other hand the complex structure is not obvious and for this the Mumford quotient is needed.