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Chiral structure of modular invariants for subfactors

J. Böckenhauer, D. E. Evans, Y. Kawahigashi

TL;DR

This work develops a unified framework for modular invariants in subfactor theory via braided endomorphisms and α-induction, linking the N–N fusion data to M–M representations and to nimreps of Verlinde fusion rules. By dissecting chiral and ambichiral sectors and exploiting the relative braiding within the double triangle algebra, the authors derive explicit decompositions, branching coefficients, and central projections that reproduce modular invariants, including detailed A–D–E classifications for SU(2) and concrete SU(3) conformal inclusion examples. The results illuminate when ambichiral braiding is non-degenerate, relate chiral branching to diagonal invariants (type I) and permutations (type II), and connect the spectral data of fusion graphs to the entries of Z. Collectively, the paper extends the α-induction program to a broad class of modular invariants, providing systematic constructions, illustrative examples, and a clear algebraic interpretation of modular data in terms of subfactor theory.

Abstract

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their ``ambichiral'' intersection, and we show that the ambichiral braiding is non-degenerate if the original braiding of the N-N morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of alpha-induced sectors. We show that modular invariants come along naturally with several non-negative integer valued matrix representations of the original N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU(2)_k modular invariants, some SU(3) conformal inclusions and the chiral conformal Ising model.

Chiral structure of modular invariants for subfactors

TL;DR

This work develops a unified framework for modular invariants in subfactor theory via braided endomorphisms and α-induction, linking the N–N fusion data to M–M representations and to nimreps of Verlinde fusion rules. By dissecting chiral and ambichiral sectors and exploiting the relative braiding within the double triangle algebra, the authors derive explicit decompositions, branching coefficients, and central projections that reproduce modular invariants, including detailed A–D–E classifications for SU(2) and concrete SU(3) conformal inclusion examples. The results illuminate when ambichiral braiding is non-degenerate, relate chiral branching to diagonal invariants (type I) and permutations (type II), and connect the spectral data of fusion graphs to the entries of Z. Collectively, the paper extends the α-induction program to a broad class of modular invariants, providing systematic constructions, illustrative examples, and a clear algebraic interpretation of modular data in terms of subfactor theory.

Abstract

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their ``ambichiral'' intersection, and we show that the ambichiral braiding is non-degenerate if the original braiding of the N-N morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of alpha-induced sectors. We show that modular invariants come along naturally with several non-negative integer valued matrix representations of the original N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU(2)_k modular invariants, some SU(3) conformal inclusions and the chiral conformal Ising model.

Paper Structure

This paper contains 21 sections, 34 theorems, 112 equations, 53 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chiral analysis
  4. Chiral horizontal projectors and chiral global indices
  5. Chiral branching coefficients
  6. Chiral vertical algebras
  7. Chiral representations
  8. Chiral structure of the center ${\cal{Z}}_h$
  9. Non-degeneracy of the ambichiral braiding
  10. Chiral matrix units
  11. Representations of fusion rules and exponents
  12. The A-D-E classification of ${\it{SU}}(2)$ modular invariants
  13. The local inclusions: ${\rm{A}}_\ell$, ${\rm{D}}_{2\ell}$, ${\rm{E}}_6$ and ${\rm{E}}_8$
  14. The non-local simple current extensions: ${\rm{D}}_{2\ell+1}$
  15. $\rm E_7$
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $\lambda,\mu,\nu\in{\Sigma({\Delta})}$, $\beta_\pm\in{\rm{Mor}}(M,M)$$a,b\in{\rm{Mor}}(M,N)$ such that $[\beta_\pm]$, $[a]$, $[b]$ are subsectors of $[\alpha^\pm_\lambda]$, $[\mu\bar{{\iota}}]$ and $[\nu\bar{{\iota}}]$ respectively. Let also $\bar{{a}},\bar{{b}}\in{\rm{Mor}}(N,M)$ be conjugates and for all $\rho\in{\Sigma({\Delta})}$.

Figures (53)

  • Figure 1: Wire diagrams for ${\cal E}_{{\rm{r}}}({\beta_+},{\beta_-})$ and ${\cal E}_{{\rm{r}}}({\beta_+},{\beta_-}) ^*$
  • Figure 2: Unitarity of relative braiding operators
  • Figure 3: The first braiding fusion relation for the relative braiding
  • Figure 4: The first intertwining braiding fusion relation of Eq. (\ref{['newibfe1']})
  • Figure 5: The second intertwining braiding fusion relation of Eq. (\ref{['newibfe2']})
  • ...and 48 more figures

Theorems & Definitions (58)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 48 more