Exact structure constants of E-series Virasoro minimal models: the polynomiality conjecture, complete $q=18$ and $q=30$ formulae, and the unexplained vanishing of $C^{even}_{13,13,13}$ in $E_{p,30}$

The exceptional E-series of Virasoro minimal models - existing only for $q=12,18,30$, the Coxeter numbers of $E_6,E_7,E_8$ - is the only ADE family whose three-point structure constants are not known in closed form. Nivesvivat and Ribault (SciPost Phys. 18, 163 (2025)) determined its non-chiral fusion rules by numerical conformal bootstrap, conjectured that reduced structure constants (numerical constants divided by a universal double-Gamma reference ansatz) are polynomial functions of $n=-2\cos(\pi p/q)$ with values in $\mathbb{Q}(\cos(\pi/q))$, worked out explicit formulae for the $q=12$ family, and found exactly one vanishing unexplained by all known symmetry and selection mechanisms: $C^{even}_{13,13,13}=0$ in the $E_{p,30}$ family, robust to roughly 38 significant digits. The problem is to complete the exact solution of the E-series: explicit analytic formulae for all independent reduced structure constants of the $q=18$ and $q=30$ families, a proof or disproof of the polynomiality conjecture, and an explanation of the unexplained vanishing.

Background

Virasoro minimal models are two-dimensional conformal field theories (CFTs) whose spectrum of primary fields is finite and built from irreducible representations of the Virasoro algebra, the infinite-dimensional symmetry algebra of two-dimensional conformal invariance. They are parametrized by two coprime integers $p,q\geq 3$ fixing the central charge $c_{p,q}=1-6(p-q)^2/(pq)$, and their modular-invariant torus partition functions fall into the A-D-E classification: the diagonal A-series, the D-series (interpretable as $\mathbb{Z}_2$ orbifolds of the A-series), and the exceptional E-series, which exists only for $q=12,18,30$ - the Coxeter numbers of the Dynkin diagrams $E_6,E_7,E_8$ - with $p$ odd and coprime to $q$. Primary fields are labeled by Kac indices $(r,s)$; a field is called diagonal, $V^d_{\langle r,s\rangle}$, when its left- and right-moving representations coincide, and non-diagonal, $V_{(r,s)}$, otherwise. Beyond the spectrum, the dynamical data of a CFT are its structure constants $C_{i,j,k}$ - the coefficients of three-point functions, equivalently of the operator product expansion (OPE). For the A- and D-series these are known in closed form in terms of the double Gamma function $\Gamma_\beta(x)$, a deformation of the Euler Gamma function characterized by functional equations under shifts of its argument by $\beta$ and $\beta^{-1}$, where $\beta=\sqrt{p/q}$. The E-series is the one ADE family without such an explicit solution; the only partial results are the fusion-matrix computations of Furlan-Ganchev-Petkova (1990), and brute-force case-by-case computation is impractical because the spectra are large and strongly $q$-dependent.

Nivesvivat and Ribault (SciPost Phys. 18, 163 (2025)) attacked this gap with the numerical conformal bootstrap: solving the crossing-symmetry equations of four-point functions, with Virasoro conformal blocks (the chiral building blocks into which four-point functions decompose) computed by Zamolodchikov's recursive representation, and extracting exact structural information from high-precision numerics. They determined the non-chiral fusion rules - i.e. which three-point constants vanish - for all three E-series families. Nearly all vanishings follow from six mechanisms: null vectors (degenerate representations obey differential equations that kill certain couplings), interchiral symmetry (shift equations relating constants under $s\to s+1$), simple currents (fields whose OPE with any field produces a single field, here generating a $\mathbb{Z}_2$ symmetry), extended symmetries (the spectra decompose into blocks forming an Ising-type fusion category for $q=12$ and a Lee-Yang-type category for $q=30$, whose sector fusion constrains couplings), permutation symmetry (odd permutations of the three fields cost the sign of the conformal spin), and left-right parity. The single exception is a purely numerical finding in the $q=30$ family: the even-parity coupling of three diagonal fields $V^d_{\langle 13,s\rangle}$ - the diagonal sector with the largest first Kac index, $r=13$ - vanishes, $C^{even}_{13,13,13}=0$, observed to about 38 significant digits in three independent four-point functions at $p=7$, while its odd-parity partner $C^{odd}_{13,13,13}$ converges to a finite value. Here the even/odd label distinguishes, for fixed first indices $r_1,r_2,r_3$, the two independent constants selected by $s_1+s_2+s_3 \bmod 2$. The vanishing is formally similar to $C^{even}_{7,7,7}=0$ in the $q=12$ family, which is explained by the Ising-type extended symmetry; no analogue explains the $q=30$ case.

For the non-vanishing constants, the authors imported loop-CFT technology: in the exactly solvable loop CFTs (continuum limits of statistical loop models such as the $O(n)$ model, where $n=-2\cos(\pi p/q)$ is the statistical loop weight), numerical structure constants divided by a universal reference ansatz built from $\Gamma_\beta$ leave residuals that are polynomials in $n$. Defining the reduced structure constants $c_{i,j,k}=C_{i,j,k}/C^{ref}_{i,j,k}$ with the same double-Gamma reference ansatz, they conjectured that in the E-series the reduced constants are polynomial functions of $n$ with values in the number field $\mathbb{Q}(\cos(\pi/q))$, i.e. polynomials in $n$ whose coefficients lie in the field generated over the rationals by $\cos(\pi/q)$. They worked out explicit formulae for the whole $q=12$ family, and verified the conjecture only on selected structure constants in a few four-point functions for $q=18$ and $q=30$, leaving complete formulae open. The E-series are also related to Pasquier's exactly solvable restricted solid-on-solid (RSOS) lattice models labeled by Dynkin diagrams, a possible route to both selection rules and the $n$-dependence. Finally, the same paper speculates on extensions $\widetilde{E}_{p,q}$ of the E-series to generic integer $q$ - necessarily non-rational and/or logarithmic CFTs whose $p,q\to\infty$ limit might recover the loop CFTs with $PSU(n)$ symmetry (the projective special unitary group) - and reports unsuccessful bootstrap tests of candidate spectra at $q=9,15$; the authors themselves describe this as speculation.

Problem Statement

Complete the exact solution of the E-series Virasoro minimal models, i.e. of the families $E_{p,q}$ with $q=18$ and $q=30$ ($p$ odd and coprime to $q$), by determining all three-point structure constants analytically. In the source's formulation:

1. Provide explicit analytic formulae - in the sense of the double-Gamma reference ansatz, i.e. explicit reduced structure constants $c_{i,j,k}=C_{i,j,k}/C^{ref}_{i,j,k}$ - for all independent three-point structure constants of the $E_{p,18}$ and $E_{p,30}$ families, valid for all admissible $p$, extending the complete $q=12$ results of Nivesvivat-Ribault. The reduced constants are to be established by derivation or certification from first principles (for example from the double-Gamma reference ansatz, the RSOS lattice realization, or fusion-matrix/graph-algebra technology), not merely transcribed or reconstructed from numerical bootstrap data: numerical tables of coefficients, however precise, do not by themselves constitute the required formulae.

2. Prove, or disprove, the polynomiality conjecture: reduced structure constants of the E-series are polynomial functions of $n=-2\cos(\pi p/q)$ with values in $\mathbb{Q}(\cos(\pi/q))$. This must be settled by an argument that does not presuppose the form of the item-1 formulae: exhibiting those formulae in polynomial form, or reading polynomiality off them, does not discharge this item, which requires an independent proof or disproof (for example from the RSOS lattice realization or the double-Gamma functional equations).

3. Explain the vanishing $C^{even}_{13,13,13}=0$ in the $E_{p,30}$ family: derive it from an identified mechanism - an additional extended symmetry, a hidden simple-current or graph-algebra selection rule, a selection rule following from the lattice (RSOS) realization, or an identity among double-Gamma factors - or show that it follows from the analytic formulae of item 1.

These three deliverables jointly constitute one objective, making the E-series exactly solvable on the same footing as the A- and D-series: item 1 supplies the explicit constants, item 2 requires an independent proof (or disproof) of the polynomiality that those constants are merely conjectured to exhibit, and item 3 requires an identified mechanism for the only vanishing that the standard toolkit leaves unexplained (or, in the alternative branch stated in item 3, a derivation of that vanishing from the item-1 formulae).

The source additionally speculates on extensions $\widetilde{E}_{p,q}$ of the E-series to generic integer $q$ (possibly non-rational or logarithmic CFTs whose $p,q\to\infty$ limit recovers loop/PSU(n) CFT spectra). Extensions to generic integer $q$ are not part of the required completion.

References

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