Nonlinear Realisations of W 1+∞

The nonlinear scalar-field realisation of w 1+∞ symmetry in d = 2 dimensions is studied in analogy to the nonlinear realisation of d = 4 conformal symmetry SO(4, 2). The w 1+∞ 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 co-variant 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 ℓ −(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 sub-realisations, including the standard one for a single scalar. We derive the Maurer-Cartan form and discuss the construction of invariant actions.