Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866915035192754176 |
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| author | Nakano, Hiromu Hunziker, Florencia Orosz Camacho, Ana Ros Wood, Simon |
| author_facet | Nakano, Hiromu Hunziker, Florencia Orosz Camacho, Ana Ros Wood, Simon |
| contents | We prove that the categories of weight modules over the simple $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator algebras at fractional admissible levels and central charges are rigid (and hence the categories of weight modules are braided ribbon categories) and that the decomposition formulae of fusion products of simple projective modules conjectured by Thomas Creutzig, David Ridout and collaborators hold (including when the decomposition involves summands that are indecomposable yet not simple). In addition to solving this old open problem, we develop new techniques for the construction of intertwining operators by means of integrating screening currents over certain cycles, which are expected to be of independent interest, due to their applicability to many other algebras. In the example of $\mathfrak{sl}(2)$ these new techniques allow us to give explicit formulae for a logarithmic intertwining operator from a pair of simple projective modules to the projective cover of the tensor unit, namely, the vertex operator algebra as a module over itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras Nakano, Hiromu Hunziker, Florencia Orosz Camacho, Ana Ros Wood, Simon Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 17B69, 17B10, 17B67 (Primary) 81T40 (Secondary) We prove that the categories of weight modules over the simple $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator algebras at fractional admissible levels and central charges are rigid (and hence the categories of weight modules are braided ribbon categories) and that the decomposition formulae of fusion products of simple projective modules conjectured by Thomas Creutzig, David Ridout and collaborators hold (including when the decomposition involves summands that are indecomposable yet not simple). In addition to solving this old open problem, we develop new techniques for the construction of intertwining operators by means of integrating screening currents over certain cycles, which are expected to be of independent interest, due to their applicability to many other algebras. In the example of $\mathfrak{sl}(2)$ these new techniques allow us to give explicit formulae for a logarithmic intertwining operator from a pair of simple projective modules to the projective cover of the tensor unit, namely, the vertex operator algebra as a module over itself. |
| title | Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 17B69, 17B10, 17B67 (Primary) 81T40 (Secondary) |
| url | https://arxiv.org/abs/2411.11387 |