N = 4 super NLS-mKdV hierarchies

Abstract

N = 2 extension of affine algebra sl(2) ⊕ u(1) possesses a hidden global N = 4 supersymmetry and provides a second hamiltonian structure for a new N = 4 supersymmetric integrable hierarchy defined on N = 2 affine supercurrents. This system is an N = 4 extension of at once two hierarchies, N = 2 NLS and N = 2 mKdV ones. It is related to N = 4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks construction which relates N = 2 sl(2) ⊕ u(1) algebra to "small" N = 4 SCA. We also find the underlying affine hierarchy for another integrable system with the N = 4 SCA second hamiltonian structure, "quasi" N = 4 KdV hierarchy. It respects only N = 2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N = 2 affine algebra admitting a quaternionic structure possesses N = 4 supersymmetry and so can be used to produce N = 4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies. 1. Introduction. The integrable hierarchies of differential equations in 1 + 1 dimensions have been widely studied in the last several years both in the physical and in the mathematical contexts. There is an ever-growing evidence that their relevance should not be confined to the realm of pure mathematics, but rather their beautiful mathematical structures show themselves naturally when investigating physical problems. At first they appeared in connection with the discretized versions, via the matrix-models approach, of 2-dimensional gravity (see [1] and references therein). Recently they sprang up rather unexpectedly in the domain of 4-dimensional field theories as an underlying structure of the N = 2 super Yang-Mills theories in the Seiberg-Witten approach [2]. The remarkable relation of the KdV-type hierarchies, via their second hamiltonian structure, to the conformal algebras (Virasoro algebra and its extensions) establishes a link between such hierarchies and the string theories which has still to be fully appreciated.One is naturally led to investigate supersymmetric extensions of the above bosonic hierarchies. Besides N = 1 supersymmetric systems which were studied in a lot of papers starting from refs.[3], it is very interesting to consider the larger N cases. One of the main sources of interest in extended supersymmetric hierarchies with N > 1 is that they could lead to a sort of "unification or grand-unification" of known hierarchies. It could happen that seemingly unrelated bosonic or lower supersymmetric (N = 1, 2) hierarchies are different manifestations of a single "unifying" larger N supersymmetric hierarchy.During the last years, N = 2 supersymmetric hierarchies were a subject of many exhaustive studies. Much less was known about systems with higher N. In a series of papers [4][5][6][7] the first example of hierarchy with N = 4 supersymmetry, N = 4 super KdV system, was discovered and studied. It is related, through its second hamiltonian structure, to "small" N = 4 superconformal algebra, is bi-hamiltonian and admits a Lax-pair formulation. It encom...

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