A note on the Jeffrey-Kirwan-Witten localisation formula

Abstract

LET A4 be a compact symplectic manifold provided with a Hamiltonian action of a compact Lie group G with Lie algebra g. We note by (M, cr, p) such a data where 0 is the symplectic form of M and ~1: M + g* is the moment map. Let us assume that the action of G on p-‘(O) is free. We can then consider the symplectic manifold Mred = G\p‘(0). It is a symplectic manifold, called the Marsden-Weinstein reduction of M, with symplectic form &d. It is important to be able to compute the integral jm,,d v,,d of a de Rham cohomology class v,,d on Mred . By a theorem of Kirwan [8], any cohomology class v,,d of Mrcd is obtained from an equivariant cohomology class v on M by restriction and reduction. In [12], Witten proposed a formula relating the integral over Mred of v,,d and an integral over M x g of an equivariant cohomology class given in terms of v and the equivariant symplectic form. Witten’s formula has been proven by Kalkman [7], Wu [13] in the case of circle actions and by Jeffrey and Kirwan [6] in the general case. As the localisation formula [l] is an efficient tool to compute integrals over M of equivariant cohomology classes, the formula of Witten can be used to compute H*(Mred) in some cases [7,6]. Let us explain Witten’s statement. Let a be a G-equivariant differential form on M, that is, tl is an equivariant map from g to the space d(M) of differential forms on M. Assume that for X E g, a(X) = ei”~‘X)/?(X) wh ere /I is a closed G-equivariant form on M depending polynomially on the variable X E g and a&X) = p(X) + cr is the value at X E g of the equivariant symplectic form of M. Let a,,d = eiOrcd Bred be the de Rham cohomology class of Mred determined by ~1. We denote by jM a the P-function on g such that its value at X E g is the integral of a(X) over M: