Classical A_n--W-Geometry

TL;DR

This paper develops a detailed geometric framework for $A_n$--W-geometry, linking it to chiral surfaces in complex projective spaces, and explores its connections with integrable systems, W-transformations, and higher-dimensional generalizations.

Contribution

It provides a comprehensive geometric and algebraic description of $A_n$--W-geometry, including its relation to chiral surfaces, integrable equations, and transformations.

Findings

Frenet-Serret equations relate $CP^n$--W-surfaces to $A_n$-Toda Lax pairs.

W-transformations extend as target-space diffeomorphisms.

Global Pl"ucker formulas connect $A_n$-Toda equations with the Gauss-Bonnet theorem.

Abstract

This is a detailed development for the $A_n$ case, of our previous article entitled "W-Geometries" to be published in Phys. Lett. It is shown that the $A_n$--W-geometry corresponds to chiral surfaces in $CP^n$. This is comes out by discussing 1) the extrinsic geometries of chiral surfaces (Frenet-Serret and Gauss-Codazzi equations) 2) the KP coordinates (W-parametrizations) of the target-manifold, and their fermionic (tau-function) description, 3) the intrinsic geometries of the associated chiral surfaces in the Grassmannians, and the associated higher instanton- numbers of W-surfaces. For regular points, the Frenet-Serret equations for $CP^n$--W-surfaces are shown to give the geometrical meaning of the $A_n$-Toda Lax pair, and of the conformally-reduced WZNW models, and Drinfeld-Sokolov equations. KP coordinates are used to show that W-transformations may be extended as particular diffeomorphisms of the target-space. This leads to higher-dimensional generalizations of the WZNW and DS equations. These are related with the Zakharov- Shabat equations. For singular points, global Pl\"ucker formulae are derived by combining the $A_n$-Toda equations with the Gauss-Bonnet theorem written for each of the associated surfaces.