Bi-Hamiltonian Sturcture of Super KP Hierarchy

TL;DR

This paper develops the bi-Hamiltonian structure of the super KP hierarchy, extending classical integrable systems into supersymmetric frameworks, and introduces a super BKP hierarchy through specific constraints.

Contribution

It provides the first supersymmetric extension of the KP bi-Hamiltonian structure and constructs a related super BKP hierarchy via anti-self-dual constraints.

Findings

Derived the supersymmetric bi-Hamiltonian structure of super KP hierarchy.

Proposed a universal super W-algebra from the hierarchy.

Constructed the super BKP hierarchy with anti-self-dual constraints.

Abstract

We obtain the bi-Hamiltonian structure of the super KP hierarchy based on the even super KP operator $\Lambda = \theta^{2} + \sum^{\infty}_{i=-2}U_{i} \theta^{-i-1}$, as a supersymmetric extension of the ordinary KP bi-Hamiltonian structure. It is expected to give rise to a universal super $W$-algebra incorporating all known extended superconformal $W_{N}$ algebras by reduction. We also construct the super BKP hierarchy by imposing a set of anti-self-dual constraints on the super KP hierarchy.