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

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

1992-06-29

arxiv.org/abs/hep-th/9206108v1

Class.Quant.Grav.10:19-36,1993

Physics

High Energy Physics

High Energy Physics - Theory

20 pages, plain TeX, IC/92/122

Scientific paper

10.1088/0264-9381/10/1/006

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.

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