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

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 diffeomorphism 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. Moreover, when primary fields are introduced outside of a coordinate description, the W3 symmetry byproducts acquire a good geometrical definition with respect to holomorphic changes of charts.