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Conformal Nets and Operator-Algebraic CFT

Conformal nets describe chiral CFT through interval algebras on a positive-energy Hilbert space. The chapter traces how locality, index, and rationality control charge categories and extensions without assuming that finite modular data determine the whole theory.

Enter this chapter

Section titled “Enter this chapter”

Begin with interval covariance and distinguish the vacuum net from its charged representations before studying complete rationality. 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.

Topics

Section titled “Topics”
  1. Conformal Nets and Covariance Axioms — Defines interval localization, vacuum, energy, and covariance.
  2. Circle and Interval Nets: Möbius and Diffeomorphism Covariance — Separates Möbius covariance from its diffeomorphism extension.
  3. Strong Additivity, Haag Duality, and the μ-Index — Distinguishes puncture removal, duality, and disconnected-region index.
  4. Split Property and Complete Rationality — States the finiteness package behind completely rational theories.
  5. DHR Sectors and Modular Tensor Categories of Nets — Derives the finite modular sector category under rationality hypotheses.
  6. Subfactors, Jones Index, and Q-Systems — Encodes finite-index extensions in positive algebra objects.
  7. Extensions, Orbifolds, Cosets, and Alpha-Induction — Constructs and tests new nets from specified extension data.
  8. Full Two-Dimensional CFT from Chiral Nets — Combines chiral theories with a local left–right extension.
  9. Nonrational, Infinite-Index, and Logarithmic Net Problems — Identifies what changes beyond finite semisimple sectors.
  10. Conformal-Net Classification: Invariants and Limits — States the scope of known classifications and their invariants.

How the ideas fit together

Section titled “How the ideas fit together”

Strong additivity and split control localization; finite two-interval index completes the rationality hypotheses. They imply finitely many sectors and nondegenerate braiding Kawahigashi, Longo, and Müger 2001, Theorem 33 and Corollaries 37–39. Constructing a local extension then requires a suitable Q-system, including multiplication and locality data Longo and Rehren 1995, Theorem 4.9. A modular-invariant matrix or list of fusion coefficients does not supply those maps.

From interval localization to finite sectors

Section titled “From interval localization to finite sectors”
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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Review the chapter

Section titled “Review the chapter”

Why is a nonnegative integer matrix commuting with modular S and T insufficient to establish a local full CFT?

Solution

It constrains sector multiplicities and modular consistency. It does not specify an associative positive multiplication, its unit, or the locality and sewing conditions of an extension. Existence requires the corresponding algebraic construction, not only the matrix test.

Where to go next

Section titled “Where to go next”

Continue to vertex operator algebras and net bridges to compare interval algebras with formal chiral fields.

References

Section titled “References”