Hamiltonian structure and coset construction of the supersymmetric extensions of N=2 KdV hierarchy

A manifestly N=2 supersymmetric coset formalism is applied to analyse the "fermionic" extensions of N=2 $a=4$ and $a=-2$ KdV hierarchies. Both these hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but non-linear superalgebra by a $\hat{U(1)}$ subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.Comment: 11 pages, Latex, a few modifications in the tex

Matrix models without scaling limit

In the context of hermitean one--matrix models we show that the emergence of the NLS hierarchy and of its reduction, the KdV hierarchy, is an exact result of the lattice characterizing the matrix model. Said otherwise, we are not obliged to take a continuum limit to find these hierarchies. We interpret this result as an indication of the topological nature of them. We discuss the topological field theories associated with both and discuss the connection with topological field theories coupled to topological gravity already studied in the literature.Comment: Latex, SISSA-ISAS 161/92/E

The (N,M)-th KdV hierarchy and the associated W algebra

We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding$M$pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi--field representation of KP hierarchy as sub--systems and naturally appears in multi--matrix models. The N+2M-1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are {\it local} and {\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} and W_\infty algebra, respectively. We call W(N, M) the generating algebra of the extended W_\infty algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual$W_N$ algebra. We show that there exist M distinct reductions of the (N, M)--th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)--th KdV hierarchy. Correspondingly the W(N, M) algebra is reduced to the W_{N+M} algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions.Comment: 40 pages, LaTeX, SISSA-171/93/EP, BONN-HE-46/93, AS-IPT-49/9

Hamiltonian Structures of the Multi-Boson KP Hierarchies, Abelianization and Lattice Formulation

We present a new form of the multi-boson reduction of KP hierarchy with Lax operator written in terms of boson fields abelianizing the second Hamiltonian structure. This extends the classical Miura transformation and the Kupershmidt-Wilson theorem from the (m)KdV to the KP case. A remarkable relationship is uncovered between the higher Hamiltonian structures and the corresponding Miura transformations of KP hierarchy, on one hand, and the discrete integrable models living on {\em refinements} of the original lattice connected with the underlying multi-matrix models, on the other hand. For the second KP Hamiltonian structure, worked out in details, this amounts to finding a series of representations of the nonlinear \hWinf algebra in terms of arbitrary finite number of canonical pairs of free fields.Comment: 12 pgs, (changes in abstract, intro and outlook+1 ref added). LaTeX, BGU-94 / 1 / January- PH, UICHEP-TH/94-