Rationality of Lorentzian Lattice CFTs And The Associated Modular Tensor Category
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909368695390208 |
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| author | Singh, Ranveer Kumar Sinha, Madhav Tao, Runkai |
| author_facet | Singh, Ranveer Kumar Sinha, Madhav Tao, Runkai |
| contents | We classify the irreducible modules of a rational Lorentzian lattice vertex operator algebra (LLVOA) based on an even, self-dual Lorentzian lattice $Λ\subset\mathbb{R}^{m,n}$ of signature $(m,n)$. We show that the set of isomorphism classes of irreducible modules of the LLVOA are in one-to-one correspondence with the equivalence classes $Λ_0^\circ/Λ_0$ for a certain subset $Λ_0^\circ\subset\mathbb{R}^{m,n}$ and a full rank sublattice $Λ_0\subsetΛ$. We also classify the intertwining operators between the modules and calculate the fusion rules. We then describe the standard construction of modular tensor category (MTC) associated to rational LLCFTs. We explicitly construct the modular data and braiding and fusing matrices for the MTC. As a concrete example, we show that the LLCFT based on a certain even, self-dual Lorentzian lattice of signature $(m, n)$, with $m$ even, realizes the $D(m \bmod 8)$ level 1 Kac-Moody MTC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02744 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rationality of Lorentzian Lattice CFTs And The Associated Modular Tensor Category Singh, Ranveer Kumar Sinha, Madhav Tao, Runkai High Energy Physics - Theory Quantum Algebra We classify the irreducible modules of a rational Lorentzian lattice vertex operator algebra (LLVOA) based on an even, self-dual Lorentzian lattice $Λ\subset\mathbb{R}^{m,n}$ of signature $(m,n)$. We show that the set of isomorphism classes of irreducible modules of the LLVOA are in one-to-one correspondence with the equivalence classes $Λ_0^\circ/Λ_0$ for a certain subset $Λ_0^\circ\subset\mathbb{R}^{m,n}$ and a full rank sublattice $Λ_0\subsetΛ$. We also classify the intertwining operators between the modules and calculate the fusion rules. We then describe the standard construction of modular tensor category (MTC) associated to rational LLCFTs. We explicitly construct the modular data and braiding and fusing matrices for the MTC. As a concrete example, we show that the LLCFT based on a certain even, self-dual Lorentzian lattice of signature $(m, n)$, with $m$ even, realizes the $D(m \bmod 8)$ level 1 Kac-Moody MTC. |
| title | Rationality of Lorentzian Lattice CFTs And The Associated Modular Tensor Category |
| topic | High Energy Physics - Theory Quantum Algebra |
| url | https://arxiv.org/abs/2408.02744 |