Canonical structure of the E10 model and supersymmetry

Abstract

A coset model based on the hyperbolic Kac-Moody algebra E10 has been conjectured to underly eleven-dimensional supergravity and M theory. In this note we study the canonical structure of the bosonic model for finite- and infinite-dimensional groups. In the case of finite-dimensional groups like GL(n) we exhibit a convenient set of variables with Borel-type canonical brackets. The generalisation to the Kac-Moody case requires a proper treatment of the imaginary roots that remains elusive. As a second result, we show that the supersymmetry constraint of D=11 supergravity can be rewritten in a suggestive way using E10 algebra data. Combined with the canonical structure, this rewriting explains the previously observed association of the canonical constraints with null roots of E10. We also exhibit a basic incompatibility between local supersymmetry and the K(E10) `R symmetry', that can be traced back to the presence of imaginary roots and to the unfaithfulness of the spinor representations occurring in the present formulation of the E10 worldline model, and that may require a novel type of bosonisation/fermionisation for its resolution. This appears to be a key challenge for future progress with E10.