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Modular Invariants, Graphs and $α$-Induction for Nets of Subfactors. III

J. Böckenhauer, D. E. Evans

TL;DR

The paper develops a two-sided α-induction framework by mixing the two braidings from chiral inductions, constructs a relative braiding on the ambichiral intersection, and expresses modular invariants via the intertwiner dimensions $Z_{\Lambda,\Lambda'}=\langle \alpha^+_{\Lambda},\alpha^-_{\Lambda'}\rangle_M$ for nets associated with SU$(n)_k$ conformal and orbifold embeddings. It proves equivalences between modular-invariant data, intersection properties of chiral induced sectors, and completeness of the full induced sector system, enabling the computation of principal and dual principal graphs from induced sectors. The methodology is applied to a broad array of SU$(n)$ embeddings, including all type I invariants for SU$(2)$ and SU$(3)$ and numerous orbifold cases, yielding new results such as the dual principal graph for SU$(3)_5\subset SU(6)_1$ and non-degenerate braiding on orbifold spectra. The results unify subfactor theory with conformal-field-theoretic invariants, providing tools to derive S-matrix-like structures and to analyze completeness of induced systems across both conformal and orbifold settings. The work advances the understanding of how α-induction encodes modular data and graph invariants, with potential extensions to other type I invariants and new subfactor examples.

Abstract

In this paper we further develop the theory of $α$-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two ``chiral'' induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices i.e. on the ``physical spectrum'' of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU(n) and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU(n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion $SU(3)_5 \subset SU(6)_1$ is computed for the first time.

Modular Invariants, Graphs and $α$-Induction for Nets of Subfactors. III

TL;DR

The paper develops a two-sided α-induction framework by mixing the two braidings from chiral inductions, constructs a relative braiding on the ambichiral intersection, and expresses modular invariants via the intertwiner dimensions for nets associated with SU conformal and orbifold embeddings. It proves equivalences between modular-invariant data, intersection properties of chiral induced sectors, and completeness of the full induced sector system, enabling the computation of principal and dual principal graphs from induced sectors. The methodology is applied to a broad array of SU embeddings, including all type I invariants for SU and SU and numerous orbifold cases, yielding new results such as the dual principal graph for SU and non-degenerate braiding on orbifold spectra. The results unify subfactor theory with conformal-field-theoretic invariants, providing tools to derive S-matrix-like structures and to analyze completeness of induced systems across both conformal and orbifold settings. The work advances the understanding of how α-induction encodes modular data and graph invariants, with potential extensions to other type I invariants and new subfactor examples.

Abstract

In this paper we further develop the theory of -induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two ``chiral'' induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices i.e. on the ``physical spectrum'' of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU(n) and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU(n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion is computed for the first time.

Paper Structure

This paper contains 23 sections, 49 theorems, 158 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries
  4. Sectors between different factors
  5. Braidings
  6. Nets of subfactors and $\alpha$-induction
  7. Mixing Two Inductions
  8. Subsectors of $[\alpha^+_\lambda]$ and $[\alpha^-_\lambda]$
  9. Comparing $[\alpha^+_\lambda]$ and $[\alpha^-_\mu]$
  10. A relative braiding
  11. Subsectors of $[\gamma]$ and $[\alpha^+_\lambda\circ\alpha^-_\mu]$
  12. Induction-Restriction Graphs and $\gamma$-Mul-ti-pli-ca-tion Graphs
  13. $\alpha$-induction and (dual) principal graphs
  14. Global indices
  15. Two Inductions and Modular Invariants
  16. ...and 8 more sections

Key Result

Lemma 3.1

Let $\lambda\in\Delta_\mathcal{N}(I_\circ)$ and $\beta\in\mathrm{End}(M)$ such that $[\beta]$ is a subsector of $[\alpha^\mp_\lambda]$. If there is a $\mu\in\Delta_\mathcal{N}(I_\circ)$ such that $[\beta]$ is a subsector of $[\alpha^\pm_\mu]$ as well, then $[\beta]$ is also subsector of $[\alpha^\pm

Figures (13)

  • Figure 1: A bi-unitary connection $W$
  • Figure 2: $\mathrm{E}_{6}$: Fusion graph of $[\alpha^+_{{1}}]$ and $[\alpha^-_{{1}}]$
  • Figure 3: $\mathrm{E}_{6}$: Principal graph for the conformal inclusion $\mathit{SU}(2)_{10}\subset\mathit{SO}(5)_1$
  • Figure 4: $\mathrm{E}_{6}$: Dual principal graph for the conformal inclusion $\mathit{SU}(2)_{10}\subset\mathit{SO}(5)_1$
  • Figure 5: $\mathrm{E}_{8}$: Fusion graph of $[\alpha^+_{{1}}]$ and $[\alpha^-_{{1}}]$
  • ...and 8 more figures

Theorems & Definitions (51)

  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Corollary 3.7
  • Lemma 3.8
  • Corollary 3.9
  • Lemma 3.10
  • ...and 41 more