Higher spin representations of K(E10)

TL;DR

This paper reviews fermionic representations of the infinite-dimensional algebra K(E10), exploring their mathematical structure, decompositions, and potential physical significance in supergravity and M-theory.

Contribution

It introduces new finite-dimensional fermionic representations of K(E10), including their decompositions and projectors, with implications for supergravity and M-theory.

Findings

Presented decompositions under finite-dimensional subgroups.

Constructed projectors for 'spin-7/2' representations.

Highlighted potential roles of K(E10) in M-theory.

Abstract

We review the recently constructed non-trivial fermionic representations of the infinite-dimensional subalgebra K(E10) of the hyperbolic Kac--Moody algebra E10. These representations are all unfaithful (and more specifically, of finite dimension). In addition we present their decompositions under the various finite-dimensional subgroups associated with some maximal supergravities in dimensions D<=11, and the projectors for `spin-7/2' which have not been given before. Those representations that have not been derived from supergravity still have to find a role and a proper physical interpretation in the conjectured correspondence between E10 and M-theory. Nevertheless, they provide novel mathematical structures that could shed some light on fundamental questions in supergravity and on the possible role of K(E10) as an `R-symmetry' of M-theory, and perhaps also on the algebra E10 itself.