Conformal-Net Classification: Invariants and Limits

Classification is always relative to a class of nets and an equivalence relation. Below central charge one, irreducible local conformal nets are classified by finite-index extensions of discrete-series Virasoro nets and an ADE list of Q-systems. Outside that regime, central charge, fusion rules, and modular $S,T$ data are important invariants but do not generally determine a net up to isomorphism.

Required background. DHR Sectors and Modular Tensor Categories of Nets supplies sector invariants. Helpful background. Extensions, Orbifolds, Cosets, and Alpha-Induction supplies local Q-systems, Full Two-Dimensional CFT from Chiral Nets supplies full centers, Nonrational, Infinite-Index, and Logarithmic Net Problems marks the finiteness boundary, and Irrational CFT, Fusion Kernels, and Crossing shows why continuous data require different invariants.

Equivalence and layers of invariant data

Two vacuum conformal nets $\mathcal A$ and $\mathcal B$ are isomorphic when there is a unitary $V:\mathcal H_{\mathcal A}\to\mathcal H_{\mathcal B}$ such that, for every interval $I$ and covariance transformation $g$ ,

V\mathcal A(I)V^*=\mathcal B(I), \qquad V\Omega_{\mathcal A}=\Omega_{\mathcal B}, \qquad VU_{\mathcal A}(g)V^*=U_{\mathcal B}(g).

This is stronger than equivalence of representation categories. Useful invariants include central charge, the Virasoro subnet, the DHR category with associator and braiding, $\mu$ , alpha-induction coupling matrices, and the isomorphism class of the extension Q-system. Fusion coefficients forget associators and braiding. Modular $S,T$ forget the local interval algebras and may not distinguish inequivalent categories. Even a braided category does not specify which commutative Q-system, if any, realizes a particular extension.

The logical hierarchy is therefore

\text{net isomorphism} \Longrightarrow \text{equivalent local extension data} \Longrightarrow \text{equivalent braided sector data},

with neither converse valid without a classification theorem for a restricted class.

Classification below central charge one

Assume an irreducible local diffeomorphism-covariant net on $S^1$ with central charge $c<1$ . Positivity of the Virasoro representation forces the discrete values

c=1-\frac{6}{m(m+1)}, \qquad m=3,4,\ldots .

The subnet generated by the stress tensor is the completely rational Virasoro net $\operatorname{Vir}_c$ . The inclusion $\operatorname{Vir}_c(I)\subset\mathcal A(I)$ has finite index, so $\mathcal A$ is controlled by a local Q-system in the Virasoro DHR category. Kawahigashi and Longo first classify the local irreducible extensions of these Virasoro nets in Kawahigashi and Longo 2004, Theorem 4.1, pp. 509–510. They then prove that local conformal nets with $c<1$ , up to isomorphism, are classified by pairs of $A$ , $D_{2n}$ , $E_6$ , or $E_8$ Dynkin diagrams whose Coxeter numbers differ by one Kawahigashi and Longo 2004, Theorem 5.1, p. 513.

The theorem excludes $E_7$ from the local type-I extension list and does not classify $c=1$ , nonunitary minimal models, or arbitrary nets sharing the same fusion ring. Its proof uses far more than modular matrices: complete rationality of the Virasoro subnet, the Cappelli–Itzykson–Zuber modular-invariant classification, alpha-induction, subfactor principal graphs, existence of the relevant Q-systems, and uniqueness information for their local realizations. The local algebra and positivity conditions close the gap left by numerical modular data.

The Ising entry

For $m=3$ ,

c=1-\frac6{3\cdot4}=\frac12.

The Coxeter numbers are $h(A_2)=3$ and $h(A_3)=4$ , so the pair $(A_2,A_3)$ lies in the theorem’s list. Its Virasoro net has sectors $\mathbf1,\sigma,\varepsilon$ , global dimension $4$ , and no additional proper bosonic local extension. Thus the classification identifies the Ising net itself, not merely its three fusion rules. This is the exact operator-algebraic result returned to Minimal Models and Fusion Rules.

An independent consistency check uses the Coxeter formula $c=1-6/(hh')$ for adjacent $h=3,h'=4$ , again giving $1/2$ . The sector invariant alone would not perform the final step: the same fusion ring can occur with different twists, and even matching $S,T$ does not exhibit interval algebras or a vacuum-preserving unitary.

Why a complete invariant must be faithful

A proposed classification invariant is complete only if equality of the invariant forces the chosen equivalence of nets. Central charge cannot see extensions at fixed $c$ ; a fusion ring omits associators and braiding; modular matrices omit the concrete sector category; and the sector category omits which local Q-system embeds the Virasoro subnet. Even adding that Q-system does not finish the argument until its realization is proved and intervalwise covariance is reconstructed. The classification below central charge one succeeds because each of these lifts is established in the restricted discrete-series setting. Elsewhere the same list remains a hierarchy of obstructions and comparison data, not a completeness theorem.

This distinction also separates classification from enumeration. Producing many nets with pairwise different invariants proves those examples inequivalent, but it does not show that every net in the class appears. Surjectivity of the list is a theorem obligation, just as injectivity requires proving that repeated data give isomorphic nets.

Adversarial nonconverse

Suppose two candidate nets have identical fusion coefficients and identical numerical $S,T$ matrices. Declaring them isomorphic omits at least the associator realization, the local extension Q-system, the embeddings of the Virasoro subnet, and the unitary equivalence of every interval algebra. The strongest justified statement is equality of the displayed invariants. The $c<1$ theorem upgrades such data only because its restricted hypotheses and Q-system classification prove existence and uniqueness; it cannot be exported to nonrational or $c\ge1$ theories.

Exercises

Use Coxeter numbers to identify the ADE pair at $c=1/2$ , and state which extra theorem turns that pair into a net classification rather than a numerical label.

Solution

$c=1/2$ gives $m=3$ , so the adjacent Coxeter numbers are $3$ and $4$ . These are $h(A_2)$ and $h(A_3)$ . The upgrade comes from the Kawahigashi–Longo $c<1$ classification: finite-index Virasoro inclusion plus classification and realization of local Q-systems, with isomorphism of vacuum conformal nets as the equivalence relation.