Invariant chiral differential operators and the W3W3 algebra

Attached to a vector space VV is a vertex algebra S(V)S(V) known as the βγβγ-system or algebra of chiral differential operators on VV. It is analogous to the Weyl algebra D(V)D(V), and is related to D(V)D(V) via the Zhu functor. If GG is a connected Lie group with Lie algebra gg, and VV is a linear GG-representation, there is an action of the corresponding affine algebra on S(V)S(V). The invariant space S(V)g[t]S(V)g[t] is a commutant subalgebra of S(V)S(V), and plays the role of the classical invariant ring D(V)GD(V)G. When GG is an abelian Lie group acting diagonally on VV, we find a finite set of generators for S(V)g[t]S(V)g[t], and show that S(V)g[t]S(V)g[t] is a simple vertex algebra and a member of a Howe pair. The Zamolodchikov W3W3 algebra with c=−2c=−2 plays a fundamental role in the structure of S(V)g[t]S(V)g[t].