E10 and SO(9,9) invariant supergravity

TL;DR

This paper reveals hidden symmetries in massive D=10 type IIA supergravity, connecting it to E10/K(E10) sigma-models and hyperbolic Kac--Moody algebras, and explores their algebraic structures and truncations.

Contribution

It explicitly constructs a one-dimensional supergravity Lagrangian with hidden SO(9,9) symmetry and relates it to E10/K(E10) models and hyperbolic Kac--Moody algebras, extending previous symmetry analyses.

Findings

Identifies SO(9,9) and SO(9) x SO(9) symmetries in supergravity

Constructs a supersymmetric Lagrangian in one dimension

Shows the relation to E10/K(E10) sigma-model truncations

Abstract

We show that (massive) D=10 type IIA supergravity possesses a hidden rigid SO(9,9) symmetry and a hidden local SO(9) x SO(9) symmetry upon dimensional reduction to one (time-like) dimension. We explicitly construct the associated locally supersymmetric Lagrangian in one dimension, and show that its bosonic sector, including the mass term, can be equivalently described by a truncation of an E10/K(E10) non-linear sigma-model to the level \ell<=2 sector in a decomposition of E10 under its so(9,9) subalgebra. This decomposition is presented up to level 10, and the even and odd level sectors are identified tentatively with the Neveu--Schwarz and Ramond sectors, respectively. Further truncation to the level \ell=0 sector yields a model related to the reduction of D=10 type I supergravity. The hyperbolic Kac--Moody algebra DE10, associated to the latter, is shown to be a proper subalgebra of E10, in accord with the embedding of type I into type IIA supergravity. The corresponding decomposition of DE10 under so(9,9) is presented up to level 5.