Intersection Pairings on Quotients and Moduli Spaces, and Witten’s Nonabelian Localization

Many moduli spaces in complex algebraix geometry can be expressed as quotients, in the sense of Mumford’s geometric invariant theory [18], of nonsingular complex projective varieties X by actions of complex reductive groups G . Any such quotient can also be identified with a symplectic quotient (or Marsden-Weinstein reduction) of the variety X by a maximal compact subgroup K of the reductive group G [14], [18], [19]. This symplectic quotient is µ -1 (0)/ K , where µ : X → k * is a moment map for the action of K on X equipped with a suitable symplectic form.