The Seven-sphere and its Kac-Moody Algebra

TL;DR

This paper explores the mathematical structure of the seven-sphere, extending it to a Kac-Moody algebra, and examines its implications for tensorial spinor properties, Malcev algebras, and current algebra anomalies.

Contribution

It introduces a novel extension of the seven-sphere to a Kac-Moody-like algebra and analyzes its covariance, tensorial properties, and anomalies in a unified framework.

Findings

Established the relation to Malcev algebras.

Formulated current algebras with unique anomalies.

Provided a Sugawara construction for the algebra.

Abstract

We investigate the seven-sphere as a group-like manifold and its extension to a Kac-Moody-like algebra. Covariance properties and tensorial composition of spinors under $S^7$ are defined. The relation to Malcev algebras is established. The consequences for octonionic projective spaces are examined. Current algebras are formulated and their anomalies are derived, and shown to be unique (even regarding numerical coefficients) up to redefinitions of the currents. Nilpotency of the BRST operator is consistent with one particular expression in the class of (field-dependent) anomalies. A Sugawara construction is given.