Finite W_3 Transformations in a Multi-time Approach

TL;DR

This paper explores finite W_3 transformations within a multi-time framework, utilizing Riemannian geometry to formulate extended generators and analyze their properties, including global transformations and W-Schwarzians.

Contribution

It introduces a geometric formulation of finite W_3 transformations using Christoffel symbols and discusses their global and Schwarzian aspects in a multi-time setting.

Findings

Extended W_3 generators expressed via Christoffel symbols

Finite transformations depend explicitly on these generators

Analysis of global SL(3) transformations and W-Schwarzians

Abstract

Classical {\W}$_3$ transformations are discussed as restricted diffeomorphism transformations (\W-Diff) in two-dimensional space. We formulate them by using Riemannian geometry as a basic ingredient. The extended {\W}$_3$ generators are given as particular combinations of Christoffel symbols. The defining equations of \W-Diff are shown to depend on these generators explicitly. We also consider the issues of finite transformations, global $SL(3)$ transformations and \W-Schwarzians.