Invariant chiral differential operators and the W3 algebra

AbstractAttached to a vector space V is a vertex algebra S(V) known as the βγ-system or algebra of chiral differential operators on V. It is analogous to the Weyl algebra D(V), and is related to D(V) via the Zhu functor. If G is a connected Lie group with Lie algebra g, and V is a linear G-representation, there is an action of the corresponding affine algebra on S(V). The invariant space S(V)g[t] is a commutant subalgebra of S(V), and plays the role of the classical invariant ring D(V)G. When G is an abelian Lie group acting diagonally on V, we find a finite set of generators for S(V)g[t], and show that S(V)g[t] is a simple vertex algebra and a member of a Howe pair. The Zamolodchikov W3 algebra with c=−2 plays a fundamental role in the structure of S(V)g[t]