The Hamiltonian Structures of the super KP hierarchy Associated with an Even Parity SuperLax Operator

TL;DR

This paper explores the Hamiltonian structures of the supersymmetric KP hierarchy with an even parity superLax operator, revealing local and non-local structures and their relation to super $w_{}$ algebra.

Contribution

It derives two Hamiltonian structures for the super KP hierarchy using Gelfand-Dikii methods, highlighting their local/non-local nature and connection to super $w_{}$ algebra.

Findings

First Hamiltonian structure is local and linear.

Second Hamiltonian structure is non-local and nonlinear.

Connections with super $w_{}$ algebra are discussed.

Abstract

We consider the even parity superLax operator for the supersymmetric KP hierarchy of the form $L~=~D^2 + \sum_{i=0}^\infty u_{i-2} D^{-i+1}$ and obtain the two Hamiltonian structures following the standard method of Gelfand and Dikii. We observe that the first Hamiltonian structure is local and linear whereas the second Hamiltonian structure is non-local and nonlinear among the superfields appearing in the Lax operator. We discuss briefly on their connections with the super $w_{\infty}$ algebra.