Fusion rules and structure constants of E-series minimal models

TL;DR

This paper investigates the E-series Virasoro minimal models, using a semi-analytic bootstrap to compute correlation functions, deduce fusion rules, and propose a universal formula for structure constants, advancing understanding of these sparse models.

Contribution

It introduces a semi-analytic bootstrap method to compute 4-point functions and proposes a universal formula for structure constants in E-series minimal models.

Findings

Non-chiral fusion rules are deduced and explained by symmetry constraints.

A conjectured universal expression for structure constants involving double Gamma functions.

Explicit results for the case q=12, with insights into general q extensions.

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