Fusion rules and structure constants of E-series minimal models

TL;DR

The paper advances the understanding of E-series minimal models by using a semi-analytic bootstrap to compute 4-point functions, from which non-chiral fusion rules and structure constants are extracted. It systematically applies constraints from null vectors, interchiral symmetry, simple currents, extended symmetries, permutations, and parity to restrict OPE coefficients, and then proposes a universal, n-dependent form for reduced structure constants with explicit q=12 results. The authors implement numerical bootstrap to test this conjecture in q=18 and q=30, finding strong agreement across several 4-point functions and identifying a notable unexplained vanishing in the q=30 case. The work points toward a broader generalization to generic integer q and hints at connections to loop CFTs in the large-p, large-q limit, potentially enriching the landscape of non-rational CFTs.

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 $\cos(π\frac{p}{q})$ 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 $p,q\to \infty$.