On spinorial representations of involutory subalgebras of Kac-Moody algebras

TL;DR

This paper reviews recent efforts to understand spinorial representations of involutory subalgebras of Kac-Moody algebras, especially $K(E_{10})$, which may relate to fermionic sectors in M theory and Standard Model fermions.

Contribution

It systematically studies spinorial representations of involutory subalgebras of Kac-Moody algebras, highlighting their potential role in M theory and fermion structure.

Findings

Progress in understanding spinorial representations of $K(E_{10})$

Potential links to fermionic sectors of supergravity and M theory

Initial insights into fermion structure in the Standard Model

Abstract

The representation theory of involutory (or 'maximal compact') subalgebras of infinite-dimensional Kac-Moody algebras is largely terra incognita, especially with regard to fermionic (double-valued) representations. Nevertheless, certain distinguished such representations feature prominently in proposals of possible symmetries underlying M theory, both at the classical and the quantum level. Here we summarise recent efforts to study spinorial representations systematically, most notably for the case of the hyperbolic Kac-Moody algebra $E_{10}$ where spinors of the involutory subalgebra $K(E_{10})$ are expected to play a role in describing algebraically the fermionic sector of $D=11$ supergravity and M theory. Although these results remain very incomplete, they also point towards the beginning of a possible explanation of the fermion structure observed in the Standard Model of Particle Physics.