Split Property and Complete Rationality

Complete rationality is a theorem-ready package, not a synonym for having some rational characters. For a chiral conformal net it combines the split property, strong additivity, and finite two-interval $\mu$ -index. The package forces finitely many irreducible DHR sectors of finite statistical dimension and a nondegenerate braided sector category; removing any one finiteness hypothesis invalidates that conclusion.

Required background. Conformal Nets and Covariance Axioms supplies the vacuum net, Modular Theory, Nuclearity, and the Split Property supplies the operator-algebraic split theorem, and Strong Additivity, Haag Duality, and the μ-Index supplies the disconnected-region index. Helpful background. The Split Property and Approximate Tensor Products gives the subsystem interpretation.

Statistical independence of separated intervals

For intervals with proper closure inclusion $\overline{I_1}\subset I_2$ , the split property requires a type-I factor $F$ such that

\mathcal A(I_1)\subset F\subset\mathcal A(I_2).

Equivalently, for separated intervals $I_1$ and $I_3\subset I_2'$ , the multiplication map extends to a normal spatial isomorphism

\mathcal A(I_1)\,\overline\otimes\,\mathcal A(I_3) \simeq \mathcal A(I_1)\vee\mathcal A(I_3), \qquad A\otimes B\mapsto AB.

The collar between the intervals is essential. Type-III local algebras do not themselves factorize as $B(\mathcal H_1)\otimes B(\mathcal H_2)$ , and the split inclusion is not canonical: different intermediate type-I factors can exist. Nuclearity or sufficiently controlled growth of conformal-energy eigenspaces is a common sufficient mechanism, but split is the property used in the rationality theorem.

Complete rationality and its theorem

For an irreducible conformal net on $S^1$ , complete rationality means:

the split property for separated intervals;
strong additivity after deletion of one point; and
finite $\mu_{\mathcal A}=[\mathcal A(E')':\mathcal A(E)]$ for a two-interval region $E$ .

The original formulation and its immediate sector bound are Definition 8, Theorem 9, and Corollary 10 of Kawahigashi, Longo, and Müger 2001, pp. 639–642. Under these hypotheses every DHR sector is a finite direct sum of irreducibles, there are finitely many irreducibles, every statistical dimension is finite, and the braiding is nondegenerate. Moreover,

\mu_{\mathcal A}=\sum_{i\in\operatorname{Irr}(\mathcal A)}d_i^2.

The proof mechanism starts with the finite-index two-interval subfactor. Its canonical endomorphism controls localized sectors and bounds their total squared dimensions. Strong additivity moves localization past a deleted point, while split turns separated components into a tensor-product inclusion. Ocneanu compactness and canonical-endomorphism arguments then show that no additional irreducible finite-index sectors are missing; the two-interval inclusion becomes the Longo–Rehren inclusion of the sector category. Nondegeneracy follows because a sector transparent to all others would create an additional relative-commutant contribution incompatible with that identification. The detailed chain is completed in Kawahigashi, Longo, and Müger 2001, Theorem 33 and Corollaries 37–39, pp. 654–658.

There are useful variants. Longo and Xu prove that finite $\mu$ together with the relevant conformal-net hypotheses forces strong additivity, so in that setting it need not be checked separately Longo and Xu 2004, Theorem 5.3, p. 343. This does not make split automatic, and the theorem should not be quoted after discarding its covariance and factor assumptions.

WZW nets as the first application

Let $\mathcal A_{G,k}$ be the conformal net associated with a compact, simply connected simple Lie group $G$ at positive integer level $k$ . The local algebras are generated in the vacuum positive-energy loop-group representation. The exact application to Affine Current Algebras and WZW Models is to verify split, strong additivity, and finite $\mu$ , then conclude that the DHR category is finite.

For all simple Lie types and positive integral levels, rationality of the WZW conformal nets—and in particular finiteness of the Jones–Wassermann subfactors—was proved by Tener 2024, Theorem A, pp. 1227–1229. The new input is a positivity-compatible comparison between conformal-net Connes fusion and VOA fusion using bounded localized vertex operators. Combined with the established split and additivity properties, it supplies the missing finite-index step. This current theorem avoids extrapolating older type- $A$ results to all $G$ .

For $SU(2)_k$ , an independent numerical check is

d_j=\frac{\sin\!\bigl((j+1)\pi/(k+2)\bigr)}{\sin\!\bigl(\pi/(k+2)\bigr)}, \qquad j=0,\ldots,k.

At $k=1$ , both dimensions equal $1$ and $\mu=2$ ; at $k=2$ , they are $1,\sqrt2,1$ and $\mu=4$ . These agree with the positive-index constraint and the known fusion rules.

The theorem’s conclusion is categorical finiteness in the vacuum net, not a finite-dimensional local Hilbert space. Every interval algebra remains type III, and the split type-I factor depends on a collar. Nor does complete rationality identify a unique stress tensor or a unique net from the modular category alone. Those require separate extension and classification theorems.

Counterexample boundary

Split and strong additivity do not imply finite $\mu$ . The U(1) current net has continuously labeled charge representations; finite-sector modular-category formulas therefore cannot be imported merely because its local algebras and energy behavior are well controlled. Longo and Xu’s dichotomy makes this sharp: under split and finite-dimensional-sector assumptions, one obtains complete rationality or uncountably many sectors, not an intermediate automatic finiteness theorem Longo and Xu 2004, Theorem 4.9, p. 340. Conversely, a finite-looking table of characters does not establish split, strong additivity, or finite Jones index.

Exercises

Assume complete rationality and $\mu<5$ . Show that there can be at most four nontrivial irreducible sectors in addition to the vacuum, and sharpen the statement if every nonvacuum sector has dimension at least $\sqrt2$ .

Solution

Every irreducible statistical dimension satisfies $d_i\ge1$ , and the vacuum contributes $1$ , so fewer than five total squared dimension allows at most four further sectors. If each nonvacuum sector has $d_i^2\ge2$ , then $1+2n<5$ gives $n\le1$ .