The geometry of W3 algebra: a twofold way for the rebirth of a symmetry

TL;DR

This paper explores the geometric origins of W3 algebras through the interplay of reparametrization symmetry breakings on Riemann surfaces, revealing new symmetries and explicit primary fields within a symplectic framework.

Contribution

It introduces a geometric approach to W3 algebras based on symplectic structures and coordinate transformations, providing explicit calculations of primary fields and new symmetry insights.

Findings

Explicit calculation of primary fields generating W3 algebra

Identification of conditions for well-defined algebraic structures

Emergence of new symmetries from geometric construction

Abstract

The purpose of this note is to show that W3 algebras originate from an unusual interplay between the breakings of the reparametrization invariance under the diffemorphism action on the cotangent bundle of a Riemann surface. It is recalled how a set of smooth changes of local complex coordinates on the base space are collectively related to a background within a symplectic framework. The power of the method allows to calculate explicitly some primary fields whose OPEs generate the algebra as explicit functions in the coordinates: this is achieved only if well defined conditions are satisfied, and new symmetries emerge from the construction. Moreoverer, when primary flelds are introduced outside of a coordinate description the W3 symmetry byproducts acquire a good geometrical definition with respect to holomorphic changes of charts.