# Invariant chiral differential operators and the W_3 algebra

Andrew R. Linshaw

Arxiv ID: 0710.0194•Last updated: 8/10/2020

Attached to a vector space V is a vertex algebra S(V) known as the beta-gamma
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 W_3 algebra with c=-2 plays a fundamental role in the structure
of S(V)^{g[t]}.

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