Fusion rules and structure constants of E-series minimal models
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| Format: | Preprint |
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2025
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| _version_ | 1866912383432130560 |
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| author | Nivesvivat, Rongvoram Ribault, Sylvain |
| author_facet | Nivesvivat, Rongvoram Ribault, Sylvain |
| contents | In the ADE classification of Virasoro minimal models, the E-series is the sparsest: their central charges $c=1-6\frac{(p-q)^2}{pq}$ are not dense in the half-line $c\in (-\infty,1)$, due to $q=12,18,30$ taking only 3 values -- the Coxeter numbers of $E_6, E_7, E_8$. The E-series is also the least well understood, with few known results beyond the spectrum.
Here, we use a semi-analytic bootstrap approach for numerically computing 4-point correlation functions. We deduce non-chiral fusion rules, i.e. which 3-point structure constants vanish. These vanishings can be explained by constraints from null vectors, interchiral symmetry, simple currents, extended symmetries, permutations, and parity, except in one case for $q=30$. We conjecture that structure constants are given by a universal expression built from the double Gamma function, times polynomial functions of $\cos(π\frac{p}{q})$ with values in $\mathbb{Q}\big(\cos(\fracπ{q})\big)$, which we work out explicitly for $q=12$.
We speculate on generalizing E-series minimal models to generic integer values of $q$, and recovering loop CFTs as $p,q\to \infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14295 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fusion rules and structure constants of E-series minimal models Nivesvivat, Rongvoram Ribault, Sylvain High Energy Physics - Theory Mathematical Physics In the ADE classification of Virasoro minimal models, the E-series is the sparsest: their central charges $c=1-6\frac{(p-q)^2}{pq}$ are not dense in the half-line $c\in (-\infty,1)$, due to $q=12,18,30$ taking only 3 values -- the Coxeter numbers of $E_6, E_7, E_8$. The E-series is also the least well understood, with few known results beyond the spectrum. Here, we use a semi-analytic bootstrap approach for numerically computing 4-point correlation functions. We deduce non-chiral fusion rules, i.e. which 3-point structure constants vanish. These vanishings can be explained by constraints from null vectors, interchiral symmetry, simple currents, extended symmetries, permutations, and parity, except in one case for $q=30$. We conjecture that structure constants are given by a universal expression built from the double Gamma function, times polynomial functions of $\cos(π\frac{p}{q})$ with values in $\mathbb{Q}\big(\cos(\fracπ{q})\big)$, which we work out explicitly for $q=12$. We speculate on generalizing E-series minimal models to generic integer values of $q$, and recovering loop CFTs as $p,q\to \infty$. |
| title | Fusion rules and structure constants of E-series minimal models |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2502.14295 |