Supersymmetric Bi-Hamiltonian Systems

Abstract

We construct super Hamiltonian integrable systems within the theory of supersymmetric Poisson vertex algebras (SUSY PVAs). We provide a powerful tool for the understanding of SUSY PVAs called the super master formula. We attach some Lie superalgebraic data to a generalized SUSY W-algebra and show that it is equipped with two compatible SUSY PVA brackets. We reformulate these brackets in terms of odd differential operators and obtain super bi-Hamiltonian hierarchies after performing a supersymmetric analog of the Drinfeld–Sokolov reduction on these operators. As an example, an integrable system is constructed from \({\mathfrak {g}}=\mathfrak {osp}(2|2)\).