The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy

Abstract

The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N = 2 supersymmetric (n, m)-GNLS hierarchy are constructed in terms of N = 2 superfields in two different superfield bases with local evolution equations. Their bosonic limits are studied in detail. New local and nonlocal bosonic and fermionic integrals both for the N = 2 supersymmetric (n, m)-GNLS hierarchy and its bosonic counterparts are derived. As an example, in the n=1, m=1 case, the algebra and the symmetry transformations for some of them are worked out and a rich N=4 supersymmetry structure is uncovered.Recently there has been an intense research activity on N=2 supersymmetric integrable hierarchy. Basically this stems from the interest spurred by the ordinary integrable hierarchies and their relation to the 2D gravity and N=2 supersymmetric models representing superstring vacua. Although an analysis of this connection for the N=2 supersymmetric integrable hierarchies is still lacking, it is nevertheless a fact that the latter hierarchies have extremely rich and interesting structures. In [1] a large class of such hierarchies was introduced: the N = 2 supersymmetric (n, m) Generalized Nonlinear Schrödinger (GNLS) hierarchies. They were subsequently studied in a number of other papers, [2,3,4,5,6].The goal of the present letter is to fill a gap in our knowledge of these hierarchies by analyzing their Hamiltonian structures. In particular we produce here the first two Hamiltonian structures and the relevant recursion operators, as well as related local and non-local conserved charges, in two different superfield representations which possess local flow equations.Let us start with a short summary of the main facts concerning the N = 2 supersymmetric (n, m)-GNLS hierarchy [1, 6] which will be useful in what follows.The Lax operator of the N = 2 supersymmetric (n, m)-GNLS hierarchy has the following form

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

“…which leads to the supersymmetric Toda chain. Then, in order to derive the Lax operator we are looking for, we follow a trick proposed in [32] and express each lattice function entering the spectral equation (19) in terms of lattice functions defined at the single lattice point i using eqs. (13) and (21).…”

Hamiltonian Structure and Coset Construction of the Supersymmetric Extensions of N=2 KdV Hierarchy

“…This approach allows for a simpler analysis than the Dirac constraint procedure, because the constraints H = H = 0 can be imposed in the equations of motion. In this Section we will show how to apply the constraints H = H = 0 to the algebra (2.12) and will demonstrate that the Dirac brackets for the remaining supercurrents J, Ψ, Ψ close on a non-local algebra [1,2].…”

Coset approach to the N = 2 supersymmetric matrix GNLS hierarchies

“…The non-local second Hamiltonian structure of the N=2 matrix GNLS hierarchies is obtained via Dirac procedure from the local N=2 sl(n|n − 1) affine superalgebra. We observe that to any second Hamiltonian structure with pure bosonic or pure fermionic superfield content there correspond two different N=2 matrix GNLS hierarchies.In the last ten years there has been a great progress in the construction of new integrable hierarchies with extended N ≥ 2 supersymmetry [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. One of the main reasons of this success lies in the fact that the second Hamiltonian structures of the KdV-type hierarchies are extended superconformal algebras.…”

“…Finally the two different Lax operators for two (out of three) integrable hierarchies associated with N=2 W S have been proposed in [9] and [13].At the same time another series of integrable hierarchies with N=2 supersymmetry, the so called (n, m)-GNLS ones [14], have been constructed. These hierarchies include only spin 1/2 chiral/antichiral superfields and possess non-local N=2 superalgebras as their second Hamiltonian structures [16]. One can try to construct new hierarchies by 'joining' Lax operators of this class with Lax operators of the previous (KdV) type.…”

“…The aim of this letter is to show that all (k|l, m)-MGNLS hierarchies (and (l, m)-GNLS as a particular case for k = 1) can be reproduced via the coset approach, starting from the local N=2 sl(n|n − 1) superalgebra. The corresponding non-local superalgebras [16,13] naturally appear in this scheme after applying Dirac reduction procedure. Thus, the classification problem reduces to the classification of admissible (see below) cosets of N=2 sl(n|n − 1) superalgebra.The letter is organized as follows.…”