On K(E_9)

TL;DR

This paper rigorously defines the maximal compact subgroup K(E_9) of the affine Lie group E_9(9), explores its on-shell realization as an R symmetry in N=16 supergravity, and discusses potential off-shell extensions and implications for M theory.

Contribution

It provides a rigorous definition of K(E_9), describes its action on supergravity fields, and suggests the existence of an off-shell spinor representation for K(E_9).

Findings

Fermions transform as spinors of K(E_9)

Fermionic equations fit into a covariant Dirac equation

Results imply possible off-shell realization of K(E_9)

Abstract

We study the maximal compact subgroup K(E_9) of the affine Lie group E_9(9) and its on-shell realization as an R symmetry of maximal N=16 supergravity in two dimensions. We first give a rigorous definition of the group K(E_9), which lives on the double cover of the spectral parameter plane, and show that the infinitesimal action of K(E_9) on the chiral components of the bosons and the fermions is determined in terms of an expansion of the Lie algebra of K(E_9) about the two branch points of this cover; this implies in particular that the fermions of N=16 supergravity transform in a spinor representation of K(E_9). The fermionic equations of motion can be fitted into the lowest components of a single K(E_9) covariant `Dirac equation', with the linear system of N=16 supergravity as the gauge connection. These results suggest the existence of an `off-shell' realization of K(E_9) in terms of an infinite component spinor representation. We conclude with some coments on `generalized holonomies' of M theory.