Super loop groups,Hamiltonian actions and super Virasoro algebras

The quotient of a super loop group by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding with a coadjoint orbit of the centrally extended super loop algebra. The algebra of super-conformal vector fields on the circle is shown to have a natural representation as Hamiltonian vector fields on generated by an equivariant moment map. This map is obtained by composition of315-8 with a super Poisson map defining a supersymmetric extension of the classical Sugawara formula. Upon quantization, this yields the corresponding formula of Kac and Todorov on unitary highest weight representations. For any homomorphism :u(1)G, an associated twisted moment map is also derived, generating a super Poisson bracket realization of a super Virasoro subalgebra of the semi-direct sum. The corresponding super Poisson map is interpreted as a nonabelian generalization of the super Miura map and applied to two super KdV hierarchies to derive corresponding integrable generalized super MKdV hierarchies in.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation (USA)