Supersymmetric Bi-Hamiltonian Systems

TL;DR

This paper develops a framework for constructing integrable supersymmetric bi-Hamiltonian systems using SUSY Poisson vertex algebras, introducing the super master formula and applying supersymmetric Drinfeld-Sokolov reduction.

Contribution

It introduces the super master formula for SUSY PVAs and constructs super bi-Hamiltonian hierarchies via a supersymmetric Drinfeld-Sokolov reduction, with explicit examples.

Findings

Construction of super bi-Hamiltonian hierarchies

Introduction of the super master formula for SUSY PVAs

Explicit example from rak{osp}(2|2)

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 $\mathfrak{g}=\mathfrak{osp}(2|2)$.