R. P. Malik — Page 9

arXiv1993

N=2 Super - $W_{3}$ Algebra and N=2 Super Boussinesq Equations

We study classical $N=2$ super-$W_3$ algebra and its interplay with $N=2$ supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs - covariant reduction approach. These techniques have been previously applied by us in the bosonic $W_3$ case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general $N=2$ super Boussinesq equation and two kinds of the modified $N=2$ super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear $N=2$ super-$W_3^{\infty}$ symmetry associated with $N=2$ super-$W_3$. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second hamiltonian structure.

hep-th↗

arXiv1992

Boussinesq-type equations from nonlinear realizations of $W_3$

We construct new coset realizations of infinite-dimensional linear $W_3^{\infty}$ symmetry associated with Zamolodchikov's $W_3$ algebra which are different from the previously explored $sl_3$ Toda realization of $W_3^{\infty}$. We deduce the Boussinesq and modified Boussinesq equations as constraints on the geometry of the corresponding coset manifolds.The main characteristic features of these realizations are:i. Among the coset parameters there are the space and time coordinates $x$ and $t$ which enter the Boussinesq equations, all other coset parameters are regarded as fields depending on these coordinates;ii. The spin 2 and 3 currents of $W_3$ and two spin 1 $U(1)$ Kac- Moody currents as well as two spin 0 fields related to the $W_3$currents via Miura maps, come out as the only essential parameters-fields of these cosets. The remaining coset fields are covariantly expressed through them;iii.The Miura maps get a new geometric interpretation as $W_3^{\infty}$ covariant constraints which relate the above fields while passing from one coset manifold to another; iv. The Boussinesq equation and two kinds of the modified Boussinesq equations appear geometrically as the dynamical constraints accomplishing $W_3^{\infty}$ covariant reductions of original coset manifolds to their two-dimensional geodesic submanifolds;v. The zero-curvature representations for these equations arise automatically as a consequence of the covariant reduction. The approach proposed could provide a universal geometric description of the relationship between $W$-type algebras and integrable hierarchies.

hep-th↗