Localization theorems by symplectic cuts

Abstract

Given a compact symplectic manifold M with the Hamiltonian action of a torus T, let zero be a regular value of the moment map, and M0 the symplectic reduction at zero. Denote by κ0 the Kirwan map H*T (M) → H*(M0). For an equivariant cohomology class η ∈H*T(M) we present new localization formulas which express ∫M0κ0(η) as sums of certain integrals over the connected components of the fixed point set MT. To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T) we give a new proof of the Jeffrey-Kirwan localization formula [JK1].