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Unitarity of minimal $W$-algebras

Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi

Abstract

We obtain a complete classification of minimal simple unitary $W$-algebras.

Unitarity of minimal $W$-algebras

Abstract

We obtain a complete classification of minimal simple unitary -algebras.

Paper Structure

This paper contains 10 sections, 12 theorems, 93 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Basic simple Lie superalgebras
  4. Conjugate linear involutions and real forms
  5. Invariant Hermitian forms on vertex algebras
  6. The almost compact conjugate linear involution of $\mathfrak g$
  7. Explicit expressions for almost compact forms
  8. Uniqueness of the almost compact involution
  9. The bilinear form $\langle \cdot ,\cdot \rangle$ on $\mathfrak g_{-1/2}$
  10. Unitarity

Key Result

Theorem 1.2

The complete list of unitary $N=0,1,$ and $2$ vertex algebras is as follows: either $c(k)$ is given by 1, 22, or 3 respectively for $p\in \mathbb Z_{\ge 2}$ or $c(k)\ge 1, \frac{3}{2}, 3$ respectively.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Proposition 4.2
  • ...and 19 more