N = 2 local and N = 4 non-local reductions of supersymmetric KP hierarchy in N = 2 superspace

Abstract

A N = 4 supersymmetric matrix KP hierarchy is proposed and a wide class of its reductions which are characterized by a finite number of fields are described. This class includes the onedimensional reduction of the two-dimensional N = (2|2) superconformal Toda lattice hierarchy possessing the N = 4 supersymmetry -the N = 4 Toda chain hierarchy -which may be relevant in the construction of supersymmetric matrix models. The Lax pair representations of the bosonic and fermionic flows, corresponding local and nonlocal Hamiltonians, finite and infinite discrete symmetries, the first two Hamiltonian structures and the recursion operator connecting all evolution equations and the Hamiltonian structures of the N = 4 Toda chain hierarchy are constructed in explicit form. Its secondary reduction to the N = 2 supersymmetric α = −2 KdV hierarchy is discussed.

Lax pair formulation of the N=4 Toda chain (KdV) hierarchy in N=4 superspace

“…Our starting point is a manifestly N = 2 supersymmetric Lax pair representation of even flows of the N = 4 Toda chain (KdV) hierarchy [1] …”

“…where the subscript ≥ 0 denotes the differential part of the operator and L T is the operator conjugated Lax operator 1 . The functions v ≡ v(z, θ + , θ − ) and u ≡ u(z, θ + , θ − ) are N = 2 superfields, and D ± are fermionic covariant derivatives All other rules can be derived using these.…”

“…It is instructive to rewrite the Lax operator (1) in another superfield basis [4] J ≡ uv + D − D + ln u, J ≡ −uv,…”

“…which can be derived by rewriting the corresponding formulae of [1] into the superfield basis (5). Second, let us discuss some properties of the Lax operator L T 1 (19) which will be useful in what follows.…”

Symmetries of supersymmetric integrable hierarchies of KP type

“…Subsequently, the subject of supersymmetrization of KP hierarchy [4,5,6,7] and other basic integrable systems (Korteveg-de Vries, nonlinear Schrödinger, Toda lattice etc.) [8,9,10,11,12,13,14,15,16] attracted a lot of interest from purely mathematical point of view, especially, the supersymmetric generalizations of the inverse scattering method, bi-Hamiltonian structures, tau-functions and Sato Grassmannian approach.…”