Building blocks for $W$-algebras of classical types
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915613844176896 |
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| author | Creutzig, Thomas Kovalchuk, Vladimir Linshaw, Andrew R. |
| author_facet | Creutzig, Thomas Kovalchuk, Vladimir Linshaw, Andrew R. |
| contents | The universal $2$-parameter vertex algebra $W_{\infty}$ of type $W(2,3,4,\dots)$ serves as a classifying object for vertex algebras of type $W(2,3,\dots,N)$ for some $N$ in the sense that under mild hypothesis, all such vertex algebras arise as quotients of $W_{\infty}$. There is an $\mathbb{N} \times \mathbb{N}$ family of such $1$-parameter vertex algebras which, after tensoring with a Heisenberg algebra, are known as $Y$-algebras. They were introduced by Gaiotto and Rapčák and are expected to be the building blocks for all $W$-algebras in type $A$, i.e., every $W$-(super) algebra in type $A$ is an extension of a tensor product of finitely many $Y$-algebras. Similarly, the orthosymplectic $Y$-algebras are $1$-parameter quotients of a universal $2$-parameter vertex algebra $W^{\text{ev}}_{\infty}$ of type $W(2,4,6,\dots)$, which is a classifying object for vertex algebras of type $W(2,4,\dots, 2N)$ for some $N$. Unlike type $A$, these algebras are not all the building blocks for $W$-algebras of types $B$, $C$, and $D$. In this paper, we construct a new universal $2$-parameter vertex algebra of type $W(1^3, 2, 3^3, 4, 5^3,6,\dots)$ which we denote by $W^{\mathfrak{sp}}_{\infty}$ since it contains a copy of the affine vertex algebra $V^k(\mathfrak{sp}_2)$. We identify $8$ infinite families of $1$-parameter quotients of $W^{\mathfrak{sp}}_{\infty}$ which are analogues of the $Y$-algebras. We regard $W^{\mathfrak{sp}}_{\infty}$ as a fundamental object on equal footing with $W_{\infty}$ and $W^{\text{ev}}_{\infty}$, and we give some heuristic reasons for why we expect the $1$-parameter quotients of these three objects to be the building blocks for all $W$-algebras of classical types. Finally, we prove that $W^{\mathfrak{sp}}_{\infty}$ has many quotients which are strongly rational. This yields new examples of strongly rational $W$-superalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Building blocks for $W$-algebras of classical types Creutzig, Thomas Kovalchuk, Vladimir Linshaw, Andrew R. Representation Theory High Energy Physics - Theory Quantum Algebra The universal $2$-parameter vertex algebra $W_{\infty}$ of type $W(2,3,4,\dots)$ serves as a classifying object for vertex algebras of type $W(2,3,\dots,N)$ for some $N$ in the sense that under mild hypothesis, all such vertex algebras arise as quotients of $W_{\infty}$. There is an $\mathbb{N} \times \mathbb{N}$ family of such $1$-parameter vertex algebras which, after tensoring with a Heisenberg algebra, are known as $Y$-algebras. They were introduced by Gaiotto and Rapčák and are expected to be the building blocks for all $W$-algebras in type $A$, i.e., every $W$-(super) algebra in type $A$ is an extension of a tensor product of finitely many $Y$-algebras. Similarly, the orthosymplectic $Y$-algebras are $1$-parameter quotients of a universal $2$-parameter vertex algebra $W^{\text{ev}}_{\infty}$ of type $W(2,4,6,\dots)$, which is a classifying object for vertex algebras of type $W(2,4,\dots, 2N)$ for some $N$. Unlike type $A$, these algebras are not all the building blocks for $W$-algebras of types $B$, $C$, and $D$. In this paper, we construct a new universal $2$-parameter vertex algebra of type $W(1^3, 2, 3^3, 4, 5^3,6,\dots)$ which we denote by $W^{\mathfrak{sp}}_{\infty}$ since it contains a copy of the affine vertex algebra $V^k(\mathfrak{sp}_2)$. We identify $8$ infinite families of $1$-parameter quotients of $W^{\mathfrak{sp}}_{\infty}$ which are analogues of the $Y$-algebras. We regard $W^{\mathfrak{sp}}_{\infty}$ as a fundamental object on equal footing with $W_{\infty}$ and $W^{\text{ev}}_{\infty}$, and we give some heuristic reasons for why we expect the $1$-parameter quotients of these three objects to be the building blocks for all $W$-algebras of classical types. Finally, we prove that $W^{\mathfrak{sp}}_{\infty}$ has many quotients which are strongly rational. This yields new examples of strongly rational $W$-superalgebras. |
| title | Building blocks for $W$-algebras of classical types |
| topic | Representation Theory High Energy Physics - Theory Quantum Algebra |
| url | https://arxiv.org/abs/2409.03465 |