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Conformal Nets and Operator-Algebraic CFT

Operator-algebraic chiral CFT begins with interval-indexed von Neumann algebras and asks exactly which covariance, energy, locality, duality, split, and index hypotheses justify each structural conclusion. The chapter builds from the vacuum net to completely rational sector categories and local extensions, then marks the point where infinite-index and nonunitary theories require different mathematics. Physical correlators, OPE coefficients, and bootstrap constraints remain with their two-dimensional CFT treatments.

Helpful background. Minimal Models and Fusion Rules supplies the rational examples, Chiral Blocks, Sewing, and Modular Invariance supplies left–right CFT data, and Modular Crossing and Spectral Bounds separates modular consistency from existence of a local net.

From interval localization to finite sectors

Section titled “From interval localization to finite sectors”

The first implication chain is one-way. A vacuum conformal net fixes local algebras, projective covariance, positive energy, and the vacuum. Strong additivity and split add independent control of punctures and separated regions. Finite two-interval index then completes the rationality package. Only within that package does the DHR category become a finite unitary modular tensor category, and only a positive commutative Q-system constructs a local extension.

The dependency map makes those domains visible. Follow the main row for the rational theorem chain and the branch for full two-dimensional extensions; neither path claims that modular data reconstruct the initial net.

A one-way chain from interval-net axioms through split, strong additivity, finite mu-index, modular DHR sectors, and commutative Q-systems, with a separate branch to full two-dimensional nets

The theorem dependencies for rational conformal nets are schematic and not to scale. Split, strong additivity, and finite two-interval index are separate inputs; modular sector data and local full-CFT extensions follow only after the displayed positive and commutative Q-system conditions.

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The central comparison is summarized below. “Excluded converse” means that the conclusion alone does not recover the hypotheses in that row.

Hypotheses, licensed conclusions, and failure checks for conformal nets
Object or domain Spectral, locality, or index hypotheses Licensed conclusion Excluded converse Adversarial check
Interval assignment on the vacuum Hilbert space Isotony, locality, Möbius covariance, nonnegative rotation generator, unique cyclic vacuum A positive-energy Möbius conformal net One global covariant algebra does not recover localization Delete the interval labels and verify that locality becomes unstated
Punctured interval and complementary interval Strong additivity and correctly formulated circle Haag duality Point deletion preserves the interval algebra; complements have the stated commutants Ordinary additivity does not imply strong additivity Cover only by overlapping intervals and observe that the deleted point is never tested
Two disjoint intervals Split, strong additivity, finite Jones index of the dual inclusion Complete rationality and finite total squared sector dimension Split and additivity alone do not force finite index Use the U(1) current net with uncountably many charge sectors
DHR endomorphism category Complete rationality, finite dimensions, conjugates, nondegenerate braiding A unitary modular tensor category An abstract modular category need not determine or realize a net Add a transparent nonvacuum sector and watch the S matrix become degenerate
Finite-index inclusion Positive special Q-system; braided commutativity for locality A finite-index extension; a local extension in the commutative case A sector sum or Jones index does not supply multiplication and positivity Use the chiral Ising fermion sum and detect its nontrivial exchange sign
Alpha-induced coupling matrix Finite braided system, nondegenerate braiding, chiral locality Nonnegative integral matrix commuting with S and T Modular commutation alone does not produce a local Q-system Supply a matrix but no associative commutative multiplication
Left–right chiral product Completely rational chiral nets and a positive commutative full-center Q-system A local finite-index full two-dimensional net A multiplicity matrix commuting with one modular generator is insufficient For Ising, test diag(1,0,1): it commutes with T but not S

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Where the implications stop

Section titled “Where the implications stop”

The failure map organizes the most common overextensions. Some failures remove a base axiom, others preserve a net but destroy rationality, and still others keep numerical modular data while losing a local construction. These are different diagnoses and leave different conclusions intact.

Paired checkpoints and failures showing loss of localization, finite index, nondegenerate braiding, Q-system locality, and positive net realization

Dropped hypotheses fail at distinct checkpoints, schematically and not to scale. A global algebra loses localization; the U(1) current preserves a rigorous net but has infinite sector content; noncommutative algebra objects give only nonlocal or relatively local extensions; logarithmic chiral data need a separate positivity and realization theorem.

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Reading sequence

Section titled “Reading sequence”
  1. Conformal Nets and Covariance Axioms defines the interval assignment, vacuum, locality, and positive energy.
  2. Circle and Interval Nets: Möbius and Diffeomorphism Covariance distinguishes geometric relabeling from projective implementation.
  3. Strong Additivity, Haag Duality, and the μ-Index builds the two-interval inclusion.
  4. Split Property and Complete Rationality states the finiteness package and its theorem.
  5. DHR Sectors and Modular Tensor Categories of Nets derives fusion, braiding, dimensions, and modularity.
  6. Subfactors, Jones Index, and Q-Systems reconstructs extensions and isolates braided commutativity.
  7. Extensions, Orbifolds, Cosets, and Alpha-Induction computes induced sectors and modular couplings.
  8. Full Two-Dimensional CFT from Chiral Nets constructs local left–right extensions through full centers.
  9. Nonrational, Infinite-Index, and Logarithmic Net Problems records what survives outside the finite regime.
  10. Conformal-Net Classification: Invariants and Limits states the exact c<1c<1 theorem and its nonconverses.

Review the chapter

Section titled “Review the chapter”

A proposed operator-algebraic CFT statement should answer five questions. What intervals and Hilbert-space representation define its algebras? Which group is implemented, and is the rotation generator nonnegative? Are strong additivity, split, and finite μ\mu proved or merely expected? Does a sector statement use finite-index conjugates and nondegenerate braiding? Does an extension come with a positive commutative Q-system, rather than only fusion rules or a modular matrix?

For rational examples, verify the independent identity μ=∑idi2\mu=\sum_i d_i^2 and compare the extension index with the Q-system dimension. For nonrational candidates, replace finite sums only after specifying the relevant direct integral, kernel, measure, and positivity structure. For logarithmic or nonunitary input, state whether a positive-energy conformal-net realization is proved, open, or obstructed; formal similarity to rational formulas is not evidence of realization.

References

Section titled “References”