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arXiv · 1608.06255

The $\mathbb{Z}_2$-orbifold of the $\mathcal{W}_3$-algebra

Masoumah Al-Ali·Andrew R. Linshaw
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Abstract

The Zamolodchikov $\mathcal{W}_3$-algebra $\mathcal{W}^c_3$ with central charge $c$ has full automorphism group $\mathbb{Z}_2$. It was conjectured in the physics literature over 20 years ago that the orbifold $(\mathcal{W}^c_3)^{\mathbb{Z}_2}$ is of type $\mathcal{W}(2,6,8,10,12)$ for generic values of $c$. We prove this conjecture for all $c \neq \frac{559 \pm 7 \sqrt{76657}}{95}$, and we show that for these two values, the orbifold is of type $\mathcal{W}(2,6,8,10,12,14)$. This paper is part of a larger program of studying orbifolds and cosets of vertex algebras that depend continuously on a parameter. Minimal strong generating sets for orbifolds and cosets are often easy to find for generic values of the parameter, but determining which values are generic is a difficult problem. In the example of $(\mathcal{W}^c_3)^{\mathbb{Z}_2}$, we solve this problem using tools from algebraic geometry.

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BibTeXRIS

Masoumah Al-Ali, Andrew R. Linshaw. 2016-08-22. The $\mathbb{Z}_2$-orbifold of the $\mathcal{W}_3$-algebra. https://doi.org/10.1007/s00220-016-2812-7

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