[Paper Review] Classification of Local Conformal Nets. Case c < 1

Yasuyuki Kawahigashi, Roberto Longo|arXiv (Cornell University)|Jan 8, 2002

Advanced Operator Algebra ResearchMathematics36 references18 citations

TL;DR

This paper classifies diffeomorphism covariant local conformal nets on the circle with central charge c < 1, proving that irreducible nets are in bijective correspondence with pairs of A-D2n-E6,8 Dynkin diagrams whose Coxeter numbers differ by 1. The classification relies on identifying Virasoro nets with coset constructions and applying α-induction and modular invariance from Cappelli-Itzykson-Zuber to classify finite-index extensions.

ABSTRACT

We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of A-D_{2n}-E_{6,8} Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by Cappelli-Itzykson-Zuber and the method of alpha-induction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.

Motivation & Objective

• To provide a complete classification of diffeomorphism covariant local conformal nets on the circle with central charge c < 1.
• To establish that irreducible nets in this class are in one-to-one correspondence with pairs of A-D2n-E6,8 Dynkin diagrams whose Coxeter numbers differ by 1.
• To demonstrate that Virasoro nets with c < 1 are completely rational and serve as minimal subnets within all such conformal nets.
• To use α-induction and modular invariants from Cappelli-Itzykson-Zuber to classify finite-index extensions of Virasoro nets.
• To identify concrete coset nets in the classification list, linking abstract classification to known models in the literature.

Proposed method

• Identify Virasoro nets with central charge c < 1 as coset nets arising from the diagonal embedding SU(2)m−1 ⊂ SU(2)m−2 × SU(2)1.
• Apply the classification of modular invariants for minimal models by Cappelli-Itzykson-Zuber to enumerate possible extension structures.
• Use α-induction from subfactor theory to classify irreducible finite-index extensions of Virasoro nets.
• Leverage the complete rationality of Virasoro nets (c < 1) to ensure finitely many irreducible representations and finite statistical dimensions.
• Utilize the 2-interval inclusion and µ-index finiteness to ensure modular tensor category structure via DHR endomorphisms.
• Establish correspondence between Dynkin diagram pairs and net extensions using Coxeter number differences.

Experimental results

Research questions

• RQ1 What is the complete classification of irreducible local conformal nets on the circle with central charge c < 1?
• RQ2 How do finite-index extensions of Virasoro nets with c < 1 relate to modular invariants and Dynkin diagrams?
• RQ3 What is the role of α-induction in classifying extensions of Virasoro nets in the context of subfactor theory?
• RQ4 How does the difference in Coxeter numbers of A-D2n-E6,8 Dynkin diagrams characterize the structure of conformal nets with c < 1?
• RQ5 Which known coset models in the literature are realized within the classification framework of this paper?

Key findings

• Irreducible local conformal nets with c < 1 are in bijective correspondence with pairs of A-D2n-E6,8 Dynkin diagrams whose Coxeter numbers differ by 1.
• The number of finite-index conformal subnets (up to conjugacy) for a given net with c = 1 − 6/m(m+1) is finite and takes values in {1, 2, 3}.
• For m ≡ 1, 2 mod 4 or m = 11, 12, there are exactly two finite-index conformal subnets; for m = 29, 30, there are exactly three.
• The Virasoro net with c < 1 is completely rational, implying finite index for the 2-interval inclusion and finite statistical dimensions for all irreducible representations.
• The classification is achieved by identifying Virasoro nets as specific coset nets and applying α-induction to classify extensions via modular invariants.
• Concrete coset nets from the literature, such as those from SU(2)m−1 ⊂ SU(2)m−2 × SU(2)1, are explicitly identified within the classification list.