Nonlinear Realisations of $w_{1+\infty}$

TL;DR

This paper explores a nonlinear realization of the $w_{1+ ablafty}$ symmetry in two dimensions, using coset space methods to identify a simplified set of Goldstone fields and their transformation properties, with implications for invariant actions.

Contribution

It introduces a novel coset construction for $w_{1+ ablafty}$ symmetry in 2D, reducing infinite Goldstone fields to a manageable set with a triangular transformation structure.

Findings

Derived transformation rules with a triangular structure.

Reduced Goldstone fields to a finite set corresponding to Cartan generators.

Discussed invariant action construction based on Maurer-Cartan form.

Abstract

The nonlinear scalar-field realisation of $w_{1+\infty}$ symmetry in $d=2$ dimensions is studied in analogy to the nonlinear realisation of $d=4$ conformal symmetry $SO(4,2)$. The $w_{1+\infty}$ realisation is derived from a coset-space construction in which the divisor group is generated by the non-negative modes of the Virasoro algebra, with subsequent application of an infinite set of covariant constraints. The initial doubly-infinite set of Goldstone fields arising in this construction is reduced by the covariant constraints to a singly-infinite set corresponding to the Cartan-subalgebra generators $v^\ell_{-(\ell+1)}$. We derive the transformation rules of this surviving set of fields, finding a triangular structure in which fields transform into themselves or into lower members of the set only. This triangular structure gives rise to finite-component subrealisations, including the standard one for a single scalar. We derive the Maurer-Cartan form and discuss the construction of invariant actions.