Super loop groups, Hamiltonian actions and super Virasoro algebras (Q921103)

The authors give to the quotient of a super loop group by the group of constant loops a super symplectic structure. This is identified via a momentum map with a coadjoint orbit of the centrally extended super loop algebra. The algebra of superconformal vector fields on the circle is represented as Hamiltonian vector fields on this quotient; this representation is generated also by a momentum map which in turn is induced by a super Poisson map defining the super symmetric extension of the Sugawara formula. Quantization of the bracket is discussed and the Kac-Todorov formula on unitary highest weight representations is deduced. A super Poisson bracket realization of the super Virasoro algebra is also given. The associated momentum map is shown to be a non-abelian generalization of the super Miura map. Applications to super integrable systems are also given.