Tensor category of $\mathbb{Z}_2$-orbifold of Heisenberg vertex operator algebra and its applications
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2026
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| author | Adamovic, Drazen Lin, Xingjun Yang, Jinwei |
| author_facet | Adamovic, Drazen Lin, Xingjun Yang, Jinwei |
| contents | In this paper, we prove the category of finite length modules for the $\mathbb{Z}_2$-orbifold $M(1)^+$ of the Heisenberg vertex operator algebra whose simple composition factors are $M(1)^\pm$ or $M(1,λ)$ for $λ\in \mathbb{C}^\times$ is a vertex and braided tensor category. Our strategy is to show these simple composition factors are $C_1$-cofinite and the category of finite length $M(1)^+$-modules is exactly the category of grading-restricted $C_1$-cofinite modules. We also determine the fusion product decompositions of simple objects and prove the rigidity of this category.
As an application of the tensor category structure of $M(1)^+$-modules, we prove the category $\mathcal{C}_{-1}(sp(2n))$ of grading-restricted generalized modules for the simple affine vertex algebra $L_{-1}(sp(2n))$ is semisimple. For this, we first prove $M(1)^+$ and simple affine vertex algebra $L_{-\frac{1}{2}}(sp(2n))$ form a commutant pair in the simple minimal $W$-algebra $W_{-1}^{min}(sp(2n))$ for $n \geq 2$ and determine $W_{-1}^{min}(sp(2n))$ as well as its irreducible modules obtained from quantum Hamilton reduction as decompositions of $M(1)^+ \otimes L_{-\frac{1}{2}}(sp(2n))$-modules, then we show all the highest weight modules for $L_{-1}(sp(2n))$ in $\mathcal{C}_{-1}(sp(2n))$ are irreducible via the quantum Hamilton reduction.
We also prove a Schur-Weyl duality between $L_{-1}(sp(2n))$ and $M(1)^+$ by showing they form a commutant pair in the $\mathbb{Z}_2$-orbifold of the rank $n$ $βγ$ system, and then establish a braided reversed equivalence between the category $\mathcal{C}_{-1}(sp(2n))$ and the full subcategory of $C_1$-cofinite $M(1)^+$-modules consisting of direct sums of irreducible modules $M(1)^\pm$ and $M\big(1, \frac{s}{\sqrt{-2n}}\big)$ for $s \in \mathbb{Z}_{\geq 0}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12120 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tensor category of $\mathbb{Z}_2$-orbifold of Heisenberg vertex operator algebra and its applications Adamovic, Drazen Lin, Xingjun Yang, Jinwei Quantum Algebra 17B69, 17B65, 18M15, 81R10, 81T40 In this paper, we prove the category of finite length modules for the $\mathbb{Z}_2$-orbifold $M(1)^+$ of the Heisenberg vertex operator algebra whose simple composition factors are $M(1)^\pm$ or $M(1,λ)$ for $λ\in \mathbb{C}^\times$ is a vertex and braided tensor category. Our strategy is to show these simple composition factors are $C_1$-cofinite and the category of finite length $M(1)^+$-modules is exactly the category of grading-restricted $C_1$-cofinite modules. We also determine the fusion product decompositions of simple objects and prove the rigidity of this category. As an application of the tensor category structure of $M(1)^+$-modules, we prove the category $\mathcal{C}_{-1}(sp(2n))$ of grading-restricted generalized modules for the simple affine vertex algebra $L_{-1}(sp(2n))$ is semisimple. For this, we first prove $M(1)^+$ and simple affine vertex algebra $L_{-\frac{1}{2}}(sp(2n))$ form a commutant pair in the simple minimal $W$-algebra $W_{-1}^{min}(sp(2n))$ for $n \geq 2$ and determine $W_{-1}^{min}(sp(2n))$ as well as its irreducible modules obtained from quantum Hamilton reduction as decompositions of $M(1)^+ \otimes L_{-\frac{1}{2}}(sp(2n))$-modules, then we show all the highest weight modules for $L_{-1}(sp(2n))$ in $\mathcal{C}_{-1}(sp(2n))$ are irreducible via the quantum Hamilton reduction. We also prove a Schur-Weyl duality between $L_{-1}(sp(2n))$ and $M(1)^+$ by showing they form a commutant pair in the $\mathbb{Z}_2$-orbifold of the rank $n$ $βγ$ system, and then establish a braided reversed equivalence between the category $\mathcal{C}_{-1}(sp(2n))$ and the full subcategory of $C_1$-cofinite $M(1)^+$-modules consisting of direct sums of irreducible modules $M(1)^\pm$ and $M\big(1, \frac{s}{\sqrt{-2n}}\big)$ for $s \in \mathbb{Z}_{\geq 0}$. |
| title | Tensor category of $\mathbb{Z}_2$-orbifold of Heisenberg vertex operator algebra and its applications |
| topic | Quantum Algebra 17B69, 17B65, 18M15, 81R10, 81T40 |
| url | https://arxiv.org/abs/2604.12120 |