EQUIVARIANT KAHLER GEOMETRY AND LOCALIZATION IN G-G MODEL

We analyze in detail the equivariant supersymmetry of the G/G model. I n spite of the fact that this supersymmetry does not model the infinit esimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reduci ble connections and a careful evaluation of the fixed point contributi ons leads to an alternative derivation of the Verlinde formula for the G(k) WZW model. We show that the supersymmetry of the G/G model can b e regarded as an infinite dimensional realization of Bismut's theory o f equivariant Bott-Chem currents on Kahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formu la. We also show that the supersymmetry is related to a non-linear gen eralization (q-deformation) of the ordinary moment map of symplectic g eometry in which a representation of the Lie algebra of a group G is r eplaced by a representation of its group algebra with commutator [g, h ] = gh - hg. In the large k limit it reduces to the ordinary moment ma p of two-dimensional gauge theories.