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Journal of the European Mathematical Society

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Volume 13, Issue 6, 2011, pp. 1737–1768
DOI: 10.4171/JEMS/292

Published online: 2011-09-16

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

Andrew R. Linshaw[1]

(1) Technische Hochschule Darmstadt, Germany

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 $n\geq 1$, 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 $GL_n$, 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 $\mathbb{C}^{n^2+2n}$.

Keywords: Invariant theory, vertex algebra, $\mathcal{W}_{1+\infty}$ algebra, orbifold construction, strong finite generation

Linshaw Andrew: Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge. J. Eur. Math. Soc. 13 (2011), 1737-1768. doi: 10.4171/JEMS/292