Fusion rules and structure constants of E-series minimal models

Abstract

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 $\cos (\pi \frac{p}{q})$ with values in $\mathbb{Q}(\cos (\frac{\pi }{q}))$, 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 $p,q\to \infty$.

22 pages, v2, many clarifications according to reviewers' comments

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• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders