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Extremal unitary representations of big $N=4$ superconformal algebra

Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi

Abstract

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big $N=4$ superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal $W$-algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big $N=4$ superconformal algebra.

Extremal unitary representations of big $N=4$ superconformal algebra

Abstract

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal -algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big superconformal algebra.

Paper Structure

This paper contains 10 sections, 16 theorems, 380 equations.

Table of Contents

  1. Introduction
  2. Unitarity of highest weight representations of minimal $W$-algebras
  3. Compact complex homogeneous spaces and cubic Dirac operators
  4. The $N=2$ vertex algebra
  5. Coset realisation of $N=2$ vertex algebra
  6. The Joyce construction
  7. The coset realisation of the $big\ N=4$ vertex algebra
  8. The vertex algebra $W^{k}_{\min}(D(2,1;a))$
  9. Application to unitarity of massless non-twisted representations
  10. Application to unitarity of massless representations in the Ramond sector

Key Result

Theorem 2.1

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 $c(k)\ge 1, \frac{3}{2}$ or $3$, respectively.

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Conjecture 2.3
  • Theorem 2.4
  • Conjecture 2.5
  • Remark 2.6
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • ...and 29 more