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The $α$-induction of Graded Local Conformal Nets

Ziyun Xu

TL;DR

This work extends $α$-induction to graded local conformal nets by establishing a braided tensor framework for graded endomorphisms and showing that graded locality survives simple extensions. It develops the $Q$-system formalism for inclusions of graded nets and uses the ambichiral/relative braiding structure to relate inclusions to modular invariants. Applying these tools to $N=2$ super-Virasoro nets in the discrete series, the authors provide a concise derivation of their modular invariant data and a shorter, conceptually transparent classification, leveraging coset constructions and simple current extensions. The results elucidate how graded locality interacts with extension and classification problems in AQFT, with explicit outcomes for the discrete-series $N=2$ nets and their modular-invariant correspondences.

Abstract

The $α$-induction of graded local conformal nets is studied. We show that inclusions of graded local conformal nets give rise to braided subfactors so that the $α$-induction is still effective for graded local conformal nets. As an application, we give a shorter proof of classification of $N=2$ superconformal nets in the discrete series.

The $α$-induction of Graded Local Conformal Nets

TL;DR

This work extends -induction to graded local conformal nets by establishing a braided tensor framework for graded endomorphisms and showing that graded locality survives simple extensions. It develops the -system formalism for inclusions of graded nets and uses the ambichiral/relative braiding structure to relate inclusions to modular invariants. Applying these tools to super-Virasoro nets in the discrete series, the authors provide a concise derivation of their modular invariant data and a shorter, conceptually transparent classification, leveraging coset constructions and simple current extensions. The results elucidate how graded locality interacts with extension and classification problems in AQFT, with explicit outcomes for the discrete-series nets and their modular-invariant correspondences.

Abstract

The -induction of graded local conformal nets is studied. We show that inclusions of graded local conformal nets give rise to braided subfactors so that the -induction is still effective for graded local conformal nets. As an application, we give a shorter proof of classification of superconformal nets in the discrete series.
Paper Structure (10 sections, 15 theorems, 31 equations)

This paper contains 10 sections, 15 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braided subfactors and $\alpha$-induction
  4. $Q$-systems and classification
  5. Graded Local Conformal Nets
  6. Representations of Graded Local Conformal Nets
  7. Simple Current Extensions of Graded Local Conformal Nets
  8. From Inclusion of Nets to Modular Invariants
  9. $N=2$ Super-Virasoro Nets and Their Representations
  10. Classification of $N=2$ Superconformal Nets

Key Result

Proposition 1

Let $\lambda$ be an irreducible DHR representation of $\mathcal{A}$. Then $\lambda$ is graded and diffeomorphism covariant.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1: ckl, Proposition 12
  • Definition 4
  • Proposition 2
  • proof
  • Definition 5
  • Theorem 1
  • proof
  • ...and 24 more