Large Chiral Diffeomorphisms on Riemann Surfaces and W-algebras

TL;DR

This paper explores large chiral diffeomorphisms on Riemann surfaces, demonstrating how truncated Taylor expansions and Forsyth frames lead to W-algebras, linking symmetry structures to KdV flows.

Contribution

It introduces a novel approach to large diffeomorphisms using Forsyth frames and B.R.S. formulation, establishing explicit connections to W-algebras and KdV flows.

Findings

Linear truncation generates symmetry algebras with structure functions

B.R.S. formulation implements large diffeomorphisms algebraically

Explicit link established between W-algebras and KdV flows

Abstract

The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a B.R.S. formulation (for a given order of truncation) leading to a more algebraic set up. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between KdV flows and W-diffeomorphims