Nonlinear Realisations of w₁₊∞

الأصل · EN

The nonlinear scalar-field realisation of w₁₊∞ symmetry in d=2 dimensions is studied in analogy to the nonlinear realisation of d=4 conformal symmetry SO(4,2). The w₁₊∞ 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 vℓ₋₍ℓ₊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.