Unitary minimal models of SW(3/2,3/2,2) superconformal algebra and manifolds of G_2 holonomy

TL;DR

This paper explores the structure and representations of the SW(3/2,3/2,2) superconformal algebra, linking it to G_2 holonomy manifolds, and identifies its minimal models and fusion rules.

Contribution

It introduces the unitary minimal models of the SW(3/2,3/2,2) algebra and analyzes their fusion structure and spectrum, connecting algebraic and geometric aspects.

Findings

Unitary minimal models of the algebra are classified.

Fusion rules for the models are determined.

Spectrum of G_2 holonomy algebra representations is obtained.

Abstract

The SW(3/2,3/2,2) superconformal algebra is a W algebra with two free parameters. It consists of 3 superconformal currents of spins 3/2, 3/2 and 2. The algebra is proved to be the symmetry algebra of the coset (su(2)+su(2)+su(2))/su(2). At the central charge c=21/2 the algebra coincides with the superconformal algebra associated to manifolds of G_2 holonomy. The unitary minimal models of the SW(3/2,3/2,2) algebra and their fusion structure are found. The spectrum of unitary representations of the G_2 holonomy algebra is obtained.