Gauging of Geometric Actions and Integrable Hierarchies of KP Type
arXiv:hep-th/0009194 · doi:10.1142/S0217751X01003573
Abstract
This work consist of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups with central extensions, with gauge group being certain (infinite-dimensional) subgroup of . We show that there exist generalized ``zero-curvature'' representation of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of being Kac-Moody group the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld-Sokolov integrable hierarchies based on the loop algebra ${\hat \cG}$. For ${\hat \cG} = {\hat {SL}}(M+R)$ the latter hiearchies are equivalent to a class of constrained (reduced) KP hierarchies called ${\sl cKP}_{R,M}$, which contain as special cases a series of well-known integrable systems (mKdV, AKNS, Fordy-Kulish, Yajima-Oikawa etc.). We describe in some detail the loop algebras of additional (non-isospectral) symmetries of ${\sl cKP}_{R,M}$ hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey-Stewartson and -wave resonant systems are also identified as additional symmetry flows of ${\sl cKP}_{R,M}$ hierarchies. Along the way we exhibit explicitly the interrelation between the Sato pseudo-differential operator formulation and the algebraic (generalized) Drinfeld-Sokolov formulation of ${\sl cKP}_{R,M}$ hierarchies. Also we present the explicit derivation of the general Darboux-Bäcklund solutions of ${\sl cKP}_{R,M}$ preserving their additional (non-isospectral) symmetries, which for R=1 contain among themselves solutions to the gauged WZNW field equations.
LaTeX209, 47 pages