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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.20545 |
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| _version_ | 1866914416332636160 |
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| author | King, Alastair Hardiman, Leonard |
| author_facet | King, Alastair Hardiman, Leonard |
| contents | Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra module. When applied to the $\mathcal{TM}$ realisation of the modular invariant partition function (arXiv:1911.09024), this yields an identification of the diagonal entries of the modular invariant with the NIM-rep multiplicities, providing a categorical generalisation of Böckenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on $\mathcal{TM}$ required for modular invariance is automatically satisfied, and that $\mathcal{TM}$ recovers the full centre construction of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306, arXiv:0807.3356). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20545 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modular invariants and NIM-reps King, Alastair Hardiman, Leonard Quantum Algebra Mathematical Physics 18M20, 81T40 Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra module. When applied to the $\mathcal{TM}$ realisation of the modular invariant partition function (arXiv:1911.09024), this yields an identification of the diagonal entries of the modular invariant with the NIM-rep multiplicities, providing a categorical generalisation of Böckenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on $\mathcal{TM}$ required for modular invariance is automatically satisfied, and that $\mathcal{TM}$ recovers the full centre construction of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306, arXiv:0807.3356). |
| title | Modular invariants and NIM-reps |
| topic | Quantum Algebra Mathematical Physics 18M20, 81T40 |
| url | https://arxiv.org/abs/2603.20545 |