Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-07-18
Int.J.Mod.Phys. A10 (1995) 2537-2578
Physics
High Energy Physics
High Energy Physics - Theory
31 pgs., LaTeX, BGU-94 / 16 / June- PH, UICHEP-TH/94-7 (extended version, new paragraphs and references added)
Scientific paper
10.1142/S0217751X95001212
Toda lattice hierarchy and the associated matrix formulation of the $2M$-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which abelianize the second KP Hamiltonian structure, we are able to obtain an unified formalism for the reduced $SL(M+1,M-k)$-KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded $SL (M+1,M-k)$ matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free-field representations of the associated $W(M,M-k)$ Poisson bracket algebras generalizing the familiar nonlinear $W_{M+1}$-algebra. Discrete B\"{a}cklund transformations for $SL(M+1,M-k)$-KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the $SL (M+1,1)$-KdV hierarchy.
Aratyn Henrik
Nissimov Emil
Pacheva Svetlana
Zimerman A. H.
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