Toward the construction of N=2 supersymmetric integrable hierarchies

TL;DR

This paper proposes a conjecture for Lax operators describing N=2 supersymmetric integrable hierarchies, verifies it in specific cases, and introduces new super-residue definitions and realizations of superconformal algebra.

Contribution

It formulates and tests a conjecture for N=2 supersymmetric Lax operators and introduces novel super-residue and Miura-like realizations for superconformal algebra.

Findings

Verified the conjecture in simple cases

Constructed N=2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy

Developed a new definition of super-residues for degenerate N=2 operators

Abstract

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible $N=2$ supersymmetric integrable hierarchies with $N=2$ super $W_n$ second hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the $N=2$ supersymmetric extensions of the Generalized Non-Linear Schr\"{o}dinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct hamiltonians we are led to a new definition of super-residues for degenerate N=2 supersymmetric pseudodifferential operators. We have found a new non-polinomial Miura-like realization for $N=2$ superconformal algebra in terms of two bosonic chiral--anti--chiral free superfields.