Coset approach to the N = 2 supersymmetric matrix GNLS hierarchies

Abstract

We discuss a large class of coset constructions of the N=2 sl(n|n − 1) affine superalgebra. We select admissible subalgebras, i.e. subalgebras that induce linear chiral/antichiral constraints on the coset supercurrents. We show that all the corresponding coset constructions lead to N=2 matrix GNLS hierarchies. We develop an algorithm to compute the relative Hamiltonians and flows. We spell out completely the case of the N=2 sl(3|2), which possesses four admissible subalgebras. 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.

Extended N = 2 supersymmetric matrix (1, s)-KdV hierarchies

“…The relation of the chirality preserving scalar Lax operators of Ref. [5] with the former ones (which have been introduced in [10,4,6]) was observed recently [11]. For the Lax operator L s (3) the N = 2 and N = 1 residues 1 vanish since it does not contain the N = 2 fermionic derivatives acting as operators.…”

“…are characterized by a finite number of fields and contain two out of three families of N = 2 supersymmetric hierarchies with N = 2 super W s algebras as their second Hamiltonian structure in the scalar case at F = F = 0 (see [6] for details). It appears that besides reductions (2), there exist other reductions of the Lax operator (1) which in the scalar case correspond to the last remaining family of N = 2 hierarchies with the N = 2 super W s algebras as their second Hamiltonian structure, i.e.…”

“…Let us remark that the Lax-pair representation (6) (with a generic matrix Lax operator of the type [D, L] = 0) being multiplied by the projector −D∂ −1 D from the right, can identically be rewritten in the same form but with the operators L and A p replaced by the operators L ≡ −DL∂ −1 D and A p ≡ −DA p ∂ −1 D, respectively, obeying the chirality preserving condition DL = LD = 0 (as opposed to the former condition [D, L] = 0 we started with). In the scalar (generic matrix) case, Lax operators of the last type have been considered in [5] ( [6]). The relation of the chirality preserving scalar Lax operators of Ref.…”

“…Their bosonic limits have been analyzed in [4], and three different families of the corresponding bosonic hierarchies and their Lax operators have been selected. Then, a complete description in terms of super Lax operators for two out of three families has been proposed in [5,6], and the generalization to the matrix case has been derived in [6].…”

“…The extension of this system by the superfields F and F can be straightforwardly derived from the Lax-pair representation (3), (6), and we do not present it here.…”