Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912844442763264 |
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| author | McRae, Robert Yang, Jinwei |
| author_facet | McRae, Robert Yang, Jinwei |
| contents | Let $\mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $\mathcal{O}_c$ admits vertex algebraic tensor category structure for any $c\in\mathbb{C}$. Here, we determine the structure of this tensor category when $c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that $\mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory $\mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $\mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$. Next, as a straightforward consequence of the braided tensor category structure on $\mathcal{O}_c$ together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras $\mathcal{W}(p)$, including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that $\mathcal{O}_c^0$ is braided tensor equivalent to the $PSL(2,\mathbb{C})$-equivariantization of the category of $\mathcal{W}(p)$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_02170 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$ McRae, Robert Yang, Jinwei Quantum Algebra Mathematical Physics Category Theory Representation Theory 17B68, 17B69, 18D10, 81R10 Let $\mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $\mathcal{O}_c$ admits vertex algebraic tensor category structure for any $c\in\mathbb{C}$. Here, we determine the structure of this tensor category when $c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that $\mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory $\mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $\mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$. Next, as a straightforward consequence of the braided tensor category structure on $\mathcal{O}_c$ together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras $\mathcal{W}(p)$, including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that $\mathcal{O}_c^0$ is braided tensor equivalent to the $PSL(2,\mathbb{C})$-equivariantization of the category of $\mathcal{W}(p)$-modules. |
| title | Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$ |
| topic | Quantum Algebra Mathematical Physics Category Theory Representation Theory 17B68, 17B69, 18D10, 81R10 |
| url | https://arxiv.org/abs/2011.02170 |