Duistermaat-Heckman Theorem

Lutian Zhao UID: 661622198 The Duistermaat-Heckman theorems concern the measure associated to moment map of a torus action of symplectic manifold. Typically, this name refers to two theorems, one is called the ”Duistermaat-Heckman measure", which says that ”the Radon-Nikodym derivative is piecewise polynomial", the definition of each terms will be introduced later. The second one is called ”Duistermaat-Heckman localization formula", which says an oscillating integral could be represented by some summation of values of a certain function, and could be seen as another type of “stationary phase formula" aroused in harmonic analysis. In this article, I’ll discuss the first theorem in greater details. The second theorem is left to N.Zhang for further discussion. We may have a glance at the statement of theorem at the beginning, but to do that, we must first define the ingredients for the theorem. That is Definition 0.1. The Liouville measure for a symplectic manifold (M;!) is the volume form m! = ! n n! , here ! is the n fold wedge product. We define this measure because this measure is preserved by the Hamiltonian vector field X. In fact, recall that the Cartan’s magic formula LX! = {Xd!+d{X!, and note that {X! = d , thus LX! = 0 and so Liouville measure is invariant under symplectomorphism defined by Hamiltonian flow. Thus we could pushforward the Liouville measure to the Lie algebra by the moment map, and this measure is the key in our theorem. Definition 0.2. The Duistermaat-Heckman measure on g associated to Hamiltonian-G space is the pushforward of Liouville measure m!, defined by mDH(U) = R 1(U) m!. So now we have all our ingredient for Duistermaat-Heckman action, we can state the easiest version of the theorem without any difficulty. Theorem 0.3. Let (M;!) be a symplectic manifold with dimension 2n, and be the moment map associated to the T -action : T d ! Symp(M), then the Duistermaat-Heckman measure mDH = m! is a piecewise polynomial multiple of the Lebesgue measure on R, the Lie algebra associated to T action, with the degree of polynomial at most n d. This is the simplest case for Duistermaat-Heckman action. Actually, we could see from a long-time example for Duistermaat-Heckman theorem, e.g., the Archimedean sphere theorem: Example 0.4 (Archimedean sphere). For S = f(sin cos ; sin sin ; cos ) : 2 [0; 2 ]; 2 [0; ]g and symplectic form ! = sin d ^ d , with (sin cos ; sin sin ; cos ) = cos . Now the hamiltonian vector LECTURE 4: THE DUISTERMAAT-HECKMAN MEASURE