Geometry of the Frenkel-Kac-Segal cocycle

TL;DR

This paper analyzes the cocycle in vertex operator representations of affine Kac-Moody algebras, framing it within R-commutative geometry and exploring its connections to homology theories and string theory.

Contribution

It demonstrates that the cocycle can be described as a strong R-commutative algebra within R-commutative geometry, linking algebraic structures to non-commutative geometry and string theory.

Findings

Cocycle described as a strong R-commutative algebra

Connections made between homology theories and string theory

Framework established for R-commutative geometry in affine algebras

Abstract

We present an analysis of the cocycle appearing in the vertex operator representation of simply-laced, affine, Kac-Moody algebras. We prove that it can be described in the context of $R$-commutative geometry, where $R$ is a Yang-Baxter operator, as a strong $R$-commutative algebra. We comment on the Hochschild, cyclic and dihedral homology theories that appear in non-commutative geometry and their potential relation to string theory.