Vertex operator algebra extensions of nonunitary Virasoro minimal models: realizing the remaining D-series and exceptional modular invariants and completing the classification

For coprime integers $p,q\geq 2$ the simple Virasoro vertex operator algebra $L(c_{p,q},0)$ (a minimal model) is rational, and the modular invariant combinations of its characters are classified into A-, D- and exceptional E-types. While the realization problem for the unitary series ($|p-q|=1$) is completely settled — Dong and Lin's classification constructs an extension for every unitary invariant that admits one and proves the $E_7$-type and odd-indexed $D$-type unitary invariants unrealizable — for genuinely nonunitary models ($|p-q|>1$) only four congruence families of exceptional invariants were constructed by Ai, Dong and Lin (2020), together with their uniqueness: the $E_6$- and $E_8$-type invariants whose non-exceptional index is congruent to $+1$ or $-1$ modulo 12, respectively 30. This problem asks to complete the realization program for nonunitary models: construct VOA extensions realizing the $E_7$-type exceptional invariants (the cases $p=18$ or $q=18$, none of which is constructed), the $E_6$- and $E_8$-type invariants at the remaining congruence classes of the non-exceptional index, and the D-series invariants ($p$ or $q$ even), or prove that no such VOA extension exists in those cases; and to classify, up to isomorphism, all VOA extensions of nonunitary rational Virasoro VOAs matching the full list of modular invariants — a necessary step toward classifying rational VOAs of effective central charge less than 1.

Background

A vertex operator algebra (VOA) is the algebraic axiomatization of the chiral algebra of a two-dimensional conformal field theory: a complex vector space graded by conformal weight, carrying vertex operators that satisfy locality, a Jacobi identity, and translation covariance, with a distinguished conformal vector $\omega$ whose modes generate a Virasoro algebra of central charge $c$. The basic building blocks are the Virasoro minimal models: for coprime integers $p,q\geq 2$ with central charge $c_{p,q}=1-6(p-q)^{2}/(pq)$, the simple Virasoro VOA $L(c_{p,q},0)$ is rational (every module is a direct sum of the finitely many irreducibles $M_{r,s}$, $1\leq r\leq q-1$, $1\leq s\leq p-1$, of conformal weight $h_{r,s}=((pr-qs)^2-(p-q)^2)/(4pq)$). The model is unitary exactly when $|p-q|=1$; otherwise it is called nonunitary. The normalized characters $\chi_{r,s}(\tau)$ of the irreducible modules carry an action of the modular group $SL_2(\mathbb{Z})$, and the physically admissible torus partition functions are the nonnegative integer combinations $Z=\sum Z_{r,s}\overline{Z_{r',s'}}$ invariant under this action, called modular invariants. Their classification — the A-D-E classification of Cappelli–Itzykson–Zuber for the unitary series and of Kato for general coprime $(p,q)$ — produces the A-type (diagonal), D-type (arising when $p$ or $q$ is even), and the exceptional $E_6$, $E_7$, $E_8$-type invariants, the latter occurring at $p$ or $q$ equal to 12, 18, 30 respectively. A rational VOA $U$ is an extension of $L(c_{p,q},0)$ when $L(c_{p,q},0)$ embeds as a sub-VOA; throughout the realization problem, extensions are understood in the standard sense in which the correspondence with modular invariants and the classification results quoted below are formulated: $U$ is simple, self-dual and of CFT type (as in Dong–Lin's classification of simple self-dual CFT-type VOAs with $c<1$). As an $L(c_{p,q},0)$-module, $U$ decomposes into a direct sum of irreducibles $M_{r,s}$, and the modular invariant of $L(c_{p,q},0)$ associated to $U$ — built from the characters of the irreducible $L(c_{p,q},0)$-modules appearing in $U$ — is then one of the classified modular invariants. The realization problem asks the converse: which modular invariants arise from VOA extensions, and is each such extension unique? For the unitary series the problem is completely settled: Dong and Lin (2015) classified the simple self-dual CFT-type VOAs with $c<1$ that are preunitary (a preunitary VOA is one that is also a unitary module for its Virasoro algebra) — their classification constructs a unique extension realizing every unitary minimal-model invariant that admits one, and proves that the remaining unitary invariants, of types $(A_{16},E_7)$, $(E_7,A_{18})$ and the odd-indexed $D_{2n+1}$-types, are realized by no VOA extension. Their method, however, does not extend to nonunitary central charges. Ai, Dong and Lin (2020) developed a tensor-product coupling technique for the nonunitary case and constructed VOA structures realizing exactly four exceptional families — $(A_{q-1},E_6)$ ($p=12$, $q\equiv 11 \bmod 12$), $(E_6,A_{p-1})$ ($q=12$, $p\equiv 1 \bmod 12$), $(A_{q-1},E_8)$ ($p=30$, $q\equiv 29 \bmod 30$), and $(E_8,A_{p-1})$ ($q=30$, $p\equiv 1 \bmod 30$) — proving these extensions unique up to isomorphism; since $c_{p,q}=c_{q,p}$ identifies $(p,q)$ with $(q,p)$, these families cover precisely the $E_6$- and $E_8$-type invariants whose non-exceptional index is congruent to $+1$ or $-1$ modulo 12, respectively 30, while the $E_6$- and $E_8$-type invariants at the other residue classes of the non-exceptional index, all $E_7$-type invariants, and all D-series invariants of nonunitary models remain unrealized. A further motivation is the classification program for rational VOAs of small central charge: the effective central charge $c_{\text{eff}}=c-24h_{\min}$ (where $h_{\min}$ is the minimal conformal weight) was introduced by Dong and Mason because results of Dong–Zhang and Mason show that a rational VOA of effective central charge less than 1 is an extension of a rational Virasoro VOA, so classifying the extensions of nonunitary $L(c_{p,q},0)$ is the gateway to classifying all rational VOAs with $c_{\text{eff}}<1$.

Problem Statement

Let $p,q\geq 2$ be coprime with $|p-q|>1$ (nonunitary), and let $L(c_{p,q},0)$ be the rational Virasoro vertex operator algebra of central charge $c_{p,q}=1-6(p-q)^2/(pq)$ with irreducible modules $M_{r,s}$ of weights $h_{r,s}=((pr-qs)^2-(p-q)^2)/(4pq)$. Against the classified list of modular invariants of Cappelli–Itzykson–Zuber and Kato — of which the A-type diagonal case and the four $E_6$- and $E_8$-type congruence families with non-exceptional index congruent to $+1$ or $-1$ modulo 12, respectively 30, are realized by unique VOA extensions (Ai–Dong–Lin 2020) — determine the remaining cases and complete the classification: (1) Realize the exceptional invariants of types $(A_{q-1},E_7)$ and $(E_7,A_{p-1})$ (occurring when $p=18$ or $q=18$; the unitary instances at $|p-q|=1$ are settled — Dong–Lin's classification proves no extension realizes them — so every coprime nonunitary value of the other index is open, with no residue restriction beyond coprimality), and likewise the $E_6$-type invariants at $p=12$ or $q=12$ and the $E_8$-type invariants at $p=30$ or $q=30$ for coprime values of the other index outside the constructed congruence families (other index not congruent to $+1$ or $-1$ modulo 12, respectively 30), by explicit vertex operator algebra structures on direct sums $L(c_{p,q},0)\oplus\bigoplus_i M_{r_i,s_i}$ whose associated modular invariant equals the given invariant, and prove uniqueness up to isomorphism, or prove that no such VOA extension exists. (2) Likewise realize the D-series invariants $(A_{q-1},D_{p/2+1})$ and $(D_{q/2+1},A_{p-1})$ (with $p$ or $q$ even, $|p-q|>1$) by VOA extensions — noting that these correspond to $\mathbb{Z}_2$ simple-current extensions (a simple current being an irreducible module whose fusion product with every irreducible module is again irreducible) whose vacuum orbit contains modules whose conformal weights must be integral for an ordinary VOA structure — and prove uniqueness up to isomorphism, or prove nonexistence in the obstructed cases. (3) Combine (1) and (2) with the A-type diagonal case and the constructed $E_6$/$E_8$ congruence families into a complete classification: for every modular invariant in the classified list of every nonunitary rational Virasoro VOA, either exhibit a VOA extension realizing it and prove its uniqueness up to isomorphism, or prove that none exists; and identify the resulting consequences for the classification of rational vertex operator algebras of effective central charge less than 1. These three parts are one coherent objective: matching the full A-D-E list of modular invariants of nonunitary Virasoro minimal models with VOA extensions, existence and uniqueness included.

References

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