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

TL;DR

This paper explores the W-geometry of finite non-periodic Toda systems linked to non-exceptional simple Lie algebras, revealing their geometric interpretation as special holomorphic surfaces in complex projective spaces.

Contribution

It generalizes the Plücker embedding to B, C, D series Lie algebras and connects Toda systems with the differential geometry of W-surfaces in projective spaces.

Findings

W-geometry coincides with the geometry of special holomorphic surfaces in CP^N.

The Plücker embedding generalization satisfies Toda equations.

W-surfaces meet quadratic holomorphic differential conditions.

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 Pl\"ucker 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 K\"ahler 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 $CP^N$ with appropriate choices of $N$. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Pl\"ucker embedding. These conditions are automatically fulfiled when Toda equations hold.