All WZW Mooels in $D\leq5$

TL;DR

This paper classifies all low-dimensional real algebras suitable for WZW models, identifying unique five- and six-dimensional cases, and explores their associated string backgrounds, including plane-wave and flat geometries.

Contribution

It provides a complete classification of real algebras up to dimension five and specific six-dimensional nilpotent algebras with invariant metrics for WZW models, and analyzes their string theory backgrounds.

Findings

Classified all real algebras with dim ≤ 5 for WZW models.

Identified unique five- and six-dimensional nilpotent algebras with invariant metrics.

Derived string backgrounds, including plane-wave and flat geometries, from these algebras.

Abstract

We present here all the real algebras $\cal{A}$ with dim$\cal{A}\leq $5 and all 6-dimensional nilpotent ones with symmetric, invariant and non-degenerate metrics for which a WZW model can be constructed. In three and four dimensions there are no other algebras than the well known $SU(2)$, $SU(1,1)$, $E_2^c$ and $H_4$. There exist only one five-dimensional and one six-dimensional nilpotent algebra with invariant non-degenerate metric and central charge $c=5, \,6$, respectively. We examine in details the five-dimensional case and, by gauging an appropriate subgroup, four-dimensional plane-wave string backgrounds are obtained. The corresponding background for the six-dimensional case is flat.