Invariant theory and the ${\mathcal{W}}_{1+\infty }$ algebra with negative integral central charge

Abstract

The vertex algebra ${\mathcal{W}}_{1+\infty ,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that ${\mathcal{W}}_{1+\infty ,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of ${\mathcal{W}}_{1+\infty ,-n}$ due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of $G{L}_n$, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of ${\mathcal{W}}_{1+\infty ,-n}$ are parametrized by a closed subvariety of ${\mathbb{C}}^{n^2+2n}$.