$W$--geometry of the Toda systems associated with non-exceptional simple Lie algebras

Abstract

The present paper describes the $W$--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the $B,C$ and $D$ series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Plücker embedding of the $A$-case to the flag manifolds associated with the fundamental representations of $B_n$, $C_n$ and $D_n$, and a direct proof that the corresponding Kähler potentials satisfy the system of two--dimensional finite non-periodic (conformal) Toda equations. It is shown that the $W$--geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of $C{P}^N$ with appropriate choices of $N$. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Plücker embedding. These conditions are automatically fulfiled when Toda equations hold.

30 pages, no figure

References in corpus (1)

• W-Geometries

Cited by in corpus (1)

• ODE/IM correspondence and modified affine Toda field equations