Fusion rules and structure constants of E-series minimal models
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 $c∈ (-∞,1)$, 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 under...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SciPost
2025-05-01
|
| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.18.5.163 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850270514397839360 |
|---|---|
| author | Rongvoram Nivesvivat, Sylvain Ribault |
| author_facet | Rongvoram Nivesvivat, Sylvain Ribault |
| author_sort | Rongvoram Nivesvivat, Sylvain Ribault |
| collection | DOAJ |
| description | 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 $c∈ (-∞,1)$, 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 $\cos(\pi\frac{p}{q})$ with values in $\mathbb{Q}\big(\cos(\frac{\pi}{q})\big)$, 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 $p,q\to ∞$. |
| format | Article |
| id | doaj-art-04708e1d3d144e048aaa5dc34268b8ab |
| institution | OA Journals |
| issn | 2542-4653 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj-art-04708e1d3d144e048aaa5dc34268b8ab2025-08-20T01:52:38ZengSciPostSciPost Physics2542-46532025-05-0118516310.21468/SciPostPhys.18.5.163Fusion rules and structure constants of E-series minimal modelsRongvoram Nivesvivat, Sylvain RibaultIn 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 $c∈ (-∞,1)$, 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 $\cos(\pi\frac{p}{q})$ with values in $\mathbb{Q}\big(\cos(\frac{\pi}{q})\big)$, 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 $p,q\to ∞$.https://scipost.org/SciPostPhys.18.5.163 |
| spellingShingle | Rongvoram Nivesvivat, Sylvain Ribault Fusion rules and structure constants of E-series minimal models SciPost Physics |
| title | Fusion rules and structure constants of E-series minimal models |
| title_full | Fusion rules and structure constants of E-series minimal models |
| title_fullStr | Fusion rules and structure constants of E-series minimal models |
| title_full_unstemmed | Fusion rules and structure constants of E-series minimal models |
| title_short | Fusion rules and structure constants of E-series minimal models |
| title_sort | fusion rules and structure constants of e series minimal models |
| url | https://scipost.org/SciPostPhys.18.5.163 |
| work_keys_str_mv | AT rongvoramnivesvivatsylvainribault fusionrulesandstructureconstantsofeseriesminimalmodels |