Nonrational, Infinite-Index, and Logarithmic Net Problems

Outside complete rationality, the conformal-net axioms, modular theory, locality, covariance, and localized-endomorphism language can remain valid, but finite-sector conclusions do not. Infinite $\mu$ may coexist with good local algebras and positive energy; continuous or uncountable sector sets replace finite modular data; and logarithmic or nonunitary chiral algebras need not admit a positive Hilbert-space net realization at all.

Required background. Conformal Nets and Covariance Axioms supplies the unitary operator-algebraic baseline, and Split Property and Complete Rationality identifies the finite-index step that may fail. Helpful background. Full Two-Dimensional CFT from Chiral Nets gives the rational construction, Nonrational Modular Consistency and Spectral Densities replaces finite sums by kernels or densities, and Logarithmic CFT and Indecomposable Modules supplies the physical logarithmic structures.

What survives when the index is infinite

Let $\mathcal A$ be a Möbius or diffeomorphism-covariant vacuum net. Isotony, locality, positive energy, Reeh–Schlieder, and one-interval modular theory are meaningful without any rationality assumption. The split property can also hold without a finite sector set. Transportable localized endomorphisms still form a braided C*-tensor category when the representation and localization hypotheses are satisfied, but that category can have infinitely many simple objects and need not be a finite semisimple modular tensor category.

The two-interval inclusion

\mathcal A(E)\subset\mathcal A(E')'

still exists. Its Jones index may be infinite, in which case the finite canonical-endomorphism argument no longer bounds the sector set. Longo and Xu show, under split and finite-dimensional-sector assumptions, a sharp alternative: the net is completely rational or it has uncountably many irreducible sectors Longo and Xu 2004, Theorem 4.9, p. 340. They also extend the global-dimension relation so that an infinite sum of squared dimensions corresponds to infinite $\mu$ Longo and Xu 2004, Theorem 7.2, p. 350. These are conditional theorems, not a claim that every nonrational net has continuous charge labels.

Three regimes compared

Ising net. This net is completely rational. It has three irreducible sectors, dimensions $1,\sqrt2,1$ , a semisimple unitary modular tensor category, and $\mu=4$ . Finite Verlinde sums and commutative Q-systems are licensed.

U(1) current net. The vacuum current net has locally normal charged representations $\gamma_q$ indexed by $q\in\mathbb R$ . Carpi constructs them explicitly from current Weyl operators and proves inequivalence for distinct charges in Carpi 2004, §4, pp. 21–23. These sectors are automorphic and have dimension one, so even a countable subfamily already gives an unbounded partial sum

\sum_q d(\gamma_q)^2=\sum_q1.

Consequently the net cannot have finite $\mu$ or a finite modular tensor category. This failure does not erase its interval net, positive-energy vacuum, locality, or braided charged representations.

Logarithmic VOA candidate. A logarithmic vertex algebra can have indecomposable modules and non-diagonalizable $L_0$ . Standard conformal nets are represented on a positive Hilbert space with unitary covariance and self-adjoint nonnegative $L_0$ . A nonunitary module category or a Jordan block for the conformal Hamiltonian therefore does not automatically exponentiate to that structure. Some unitary strongly local VOAs do generate conformal nets, but the formal existence of logarithmic OPEs is not such a theorem. “No known net realization” and “proved impossible” are different statuses and must remain distinct.

This is the exact comparison returned to Logarithmic CFT and Indecomposable Modules: Ising has finite index and semisimple modularity; U(1) has a rigorous net but infinitely many charges; a logarithmic candidate may lack positivity, semisimplicity, or a proved bounded-operator realization.

Proof mechanisms that no longer transfer

The rational proof uses a finite-index canonical endomorphism, finite direct-sum decomposition, conjugates with finite dimensions, and nondegenerate finite $S$ . At infinite index the canonical endomorphism may require a direct integral rather than a finite sum. In a nonsemisimple setting, categorical dimension can vanish or cease to control Hilbert-space index, projective modules enter, and the usual Verlinde diagonalization is unavailable. In a continuous theory, modular transformations act by integral kernels rather than finite matrices. Each replacement requires its own convergence, positivity, and measure hypotheses.

An independent check for the U(1) example is elementary: choose distinct integer charges $q=n$ . Since each dimension is one, the first $N$ sectors contribute $N$ to the squared-dimension sum, so no finite global dimension can contain all of them.

Adversarial finite-category formula

Apply the finite Verlinde expression $N_{ij}^{\ k}=\sum_a S_{ia}S_{ja}S_{ka}^*/S_{0a}$ to the U(1) charge continuum as an ordinary finite sum. There is no finite index set $a$ , and replacing it by an integral demands a kernel, measure, distributional normalization, and convergence theorem. Apply the same formula to a logarithmic nonsemisimple category and $S$ need not diagonalize fusion. In both cases the computation fails before any numerical answer is licensed.

Exercises

Suppose a split conformal net has inequivalent irreducible automorphism sectors $\rho_n$ for every $n\in\mathbb Z$ . Show that it cannot be completely rational.

Solution

Automorphism sectors have $d(\rho_n)=1$ . If complete rationality held, the global-dimension formula would give finite $\mu=\sum_i d_i^2$ . The partial sum over $-N\le n\le N$ is $2N+1$ , which is unbounded. Hence finite $\mu$ and complete rationality are impossible.