Super loop groups, Hamiltonian actions and super Virasoro algebras

The quotient LG/G of a super loop group LG by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding JLand:LG/G→Lgand* with a coadjoint orbit of the centrally extended super loop algebra Lgand. The algebra diffc S1 of superconformal vector fields on the circle is shown to have a natural representation as Hamiltonian vector fields of LG/G generated by an equivariant moment map J: LG/G→diffc S1*. This map is obtained by composition of JLand with a super Poisson map JS:Lgand*→diffc S1* 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 ρ:u(1)→G, an associated 'twisted' moment map Jρ:LG/G→diffCS1* is also derived generating a super Poisson bracket realization of a super Virasoro subalgebra Vir of the semi-direct sum diffCcS1∝Lgand. The corresponding super Poisson map JSρ:Lgand*→Vir* 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 Lgand*. (orig.)