$W_{\infty}$--Geometry and Associated Continuous Toda System

TL;DR

This paper explores the geometric structure of an infinite-dimensional Kähler manifold linked to area-preserving diffeomorphisms on a torus, connecting it to a continuous limit of the $A_r$--Toda system and related geometric notions.

Contribution

It introduces a continuous limit of the $A_r$--Toda system and develops $W_{ abla}$--geometry concepts relevant to self--dual Einstein spaces with symmetries.

Findings

Defined an infinite-dimensional Kähler manifold for area-preserving diffeomorphisms.

Established a continuous limit of the $A_r$--Toda system.

Introduced a Pl"ucker type formula for $W_{ abla}$--geometry.

Abstract

We discuss an infinite--dimensional k\"ahlerian manifold associated with the area--preserving diffeomorphisms on two--dimensional torus, and, correspondingly, with a continuous limit of the $A_r$--Toda system. In particular, a continuous limit of the $A_r$--Grassmannians and a related Pl\"ucker type formula are introduced as relevant notions for $W_{\infty}$--geometry of the self--dual Einstein space with the rotational Killing vector.