Meromorphic CFTs have central charges c = 8$\mathbb{N}$: a proof based on the MLDE approach and Rademacher series
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911893241724928 |
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| author | Das, Arpit |
| author_facet | Das, Arpit |
| contents | In this short note, we present a simple and elementary proof that meromorphic conformal field theories (CFTs) have central charges of the form: $c=8N$ with $N\in\mathbb{N}$ (the set of natural numbers) using the modular linear differential equations (MLDEs) approach. We first set up the 1-character MLDE for arbitrary value of the Wronskian index: $\ell$. From this we get the general form of the meromorphic CFT's character. We then study its modular transformations and the asymptotic value of it's Fourier coefficients -- using Rademacher series -- to conclude that odd values of $\ell$ make the character in-admissible implying that the central charge for admissible character has to be a multiple of 8. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02129 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Meromorphic CFTs have central charges c = 8$\mathbb{N}$: a proof based on the MLDE approach and Rademacher series Das, Arpit High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics Number Theory Quantum Algebra In this short note, we present a simple and elementary proof that meromorphic conformal field theories (CFTs) have central charges of the form: $c=8N$ with $N\in\mathbb{N}$ (the set of natural numbers) using the modular linear differential equations (MLDEs) approach. We first set up the 1-character MLDE for arbitrary value of the Wronskian index: $\ell$. From this we get the general form of the meromorphic CFT's character. We then study its modular transformations and the asymptotic value of it's Fourier coefficients -- using Rademacher series -- to conclude that odd values of $\ell$ make the character in-admissible implying that the central charge for admissible character has to be a multiple of 8. |
| title | Meromorphic CFTs have central charges c = 8$\mathbb{N}$: a proof based on the MLDE approach and Rademacher series |
| topic | High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics Number Theory Quantum Algebra |
| url | https://arxiv.org/abs/2312.02129 |