LECTURE 4. THE DUISTERMAAT-HECKMAN THEOREM 317Suppose a symplectic manifold M is equipped with a complex structure com-patible with the symplectic structure (i.e. Mis a Kahler manifold). Then if(£, \7)
is a prequantum line bundle with connection, C naturally acquires a structure of
holomorphic line bundle (we define the a operator on sections of c as the (0, 1) part
\7 (0,l ) of the prequantum connection, in other words a section s is holomorphic if
and only if \7(o,i)s = 0).
Definition 4.4. Suppose M is a symplectic manifold equipped with a prequantum
line bundle with connection (£, \7). The quantization of M is the virtual Hilbert
space
i even i oddRemark 4.5. In many natural situations, only one of the vector spaces Hi(M, C)
is nonzero.
Remark 4.6. If M is compact, all the vector spaces H i (M, C) are finite-
dimensional, and the dimension of the quantization is given by the Riemann-Roch
theorem.
Suppose M is equipped with a prequantum line bundle with connection, and
suppose a group G acts in a Hamiltonian fashion on M, and that the group action
lifts to the total space of C in a way that is compatible with the connection. (The
choice of such a lift is in fact equivalent to the choice of a moment map for the group
action: cf. Remark 3.5.) Then each of the vector spaces H i (M, C) is acted on by
the group G , in other words the quantization of M is a (virtual) representation of
G.
If Mis acted on by a torus T, the multiplicities with which the weights for the
action appear in the representation 1i of T are related to the moment polytope:
all weights that have nonzero multiplicity lie within the moment polytope, and
the asymptotics of the multiplicities of weights are in a natural sense given by the
Duistermaat-Heckman polynomial f from Theorem 2.5. (For a precise statement,
see Section 3.4 of [17].)
4.4. Nonabelian localization
Witten in [41] gave a result (the nonabelian localization principle) that related
intersection pairings on the symplectic quotient Mred of a (compact) manifold M to
data on M itself. Since;;,: H(;(M) -+ H*(Mred) is a surjective ring homomorphism,
all intersection pairings are given in the form JM <ed r;,(ry) for some T/ E H(;(M).
Witten [41] regards the equivariant cohomology parameter X E gas an inte-
gration variable, and seeks to compute the asymptotics in E > 0 of
{ dXe-EIX l^2 / 2 { ry(X)eiwei(μ,X)_
JXEg JMHe finds that this has an asymptotic expansion as E -+ 0 of the form
(4.2) r eE6eiW,ed;;,(ry) + o(p(t:-l/2)e-f.-)
}Mredwhere b is the smallest nonzero critical value of 1μ1^2 , p is a polynomial, and e
is a particular element of H^4 (Mred) (the image r;,({3) of the element f3 E H(;(M)