REALIZATIONS OF CLASSICAL AND QUANTUM-W3 SYMMETRY

We consider realisations of Zamolodchikov's nonlinear W3 algebra at the classical and quantum level. Recent work has produced gaugings of the classical W3 algebra starting from a theory of n scalar fields phi-i, given the existence of a set of coefficients d(ijk) satisfying a certain algebraic identity. We note that a solution exists for each Jordan algebra determined by a cubic norm form, leading to an infinite family of "generic" models for all n, plus four special cases with n = 5, 8, 14 and 26. Taking free-field ansatze for the spin-two and spin-three currents, we then formulate the conditions for the quantum W3 algebra to be satisfied. We show how the generic classical models may be extended to the quantum case for every n, reducing to the construction of Fateev and Zamolodchikov for n = 2. These models are seen to be examples of a completely general construction, which produces a realisation of W3 from an arbitrary realisation of the Virasoro algebra and an additional scalar field.