Two Aspects of K Ahler Geometry in the G=g Model

The subject of this talk is the role of equivariant KK ahler geometry in the G=G model or, in other words, that of its equivariant supersymmetry. This supersym-metry is somewhat unusual in that it does not model the innnitesimal action of the group of gauge transformations. Nevertheless, this supersymmetry is useful on, at least, two counts. Firstly, it allows one to localise the path integral. A careful evaluation of the xed point contributions leads to an alternative derivation of the Verlinde formula for the G k WZW model. In this part of the story one makes contact with Bismut's theory of equivariant Bott-Chern currents on KK ahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formula. Secondly, the supersymmetry is related to a non-linear generalization (q-deformation) of the ordinary moment map of symplectic geometry in which a representation of the Lie algebra of a group G is replaced by a representation of its group algebra with commutator g; h] = gh ? hg. In the large k limit it reduces to the ordinary moment map of two-dimensional gauge theories.