Gradient Representations and Affine Structures in AE(n)

Abstract

We study the indefinite Kac-Moody algebras AE(n), arising in the reduction of Einstein's theory from (n+1) space-time dimensions to one (time) dimension, and their distinguished maximal regular subalgebras sl(n) and affine A_{n-2}^{(1)}. The interplay between these two subalgebras is used, for n=3, to determine the commutation relations of the `gradient generators' within AE(3). The low level truncation of the geodesic sigma-model over the coset space AE(n)/K(AE(n)) is shown to map to a suitably truncated version of the SL(n)/SO(n) non-linear sigma-model resulting from the reduction Einstein's equations in (n+1) dimensions to (1+1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space H/K(H), where H is the (extended) Heisenberg group, and K(H) its maximal compact subgroup. We clarify the relation between H and the corresponding subgroup of the Geroch group.

43 pages

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Cited by in corpus (20)

• K(E10), Supergravity and Fermions
• Counting supersymmetric branes
• Integrability of Five Dimensional Minimal Supergravity and Charged Rotating Black Holes
• Beyond E11
• Inverse Scattering and the Geroch Group
• IIA and IIB spinors from K(E10)
• E10 Cosmology
• On the E10/Massive Type IIA Supergravity Correspondence
• K(E9) from K(E10)
• Eisenstein series for infinite-dimensional U-duality groups
• Fermionic Kac-Moody Billiards and Supergravity
• E10 and Gauged Maximal Supergravity
• On higher spin realizations of K(E10)
• Tensor hierarchy algebra extensions of over-extended Kac--Moody algebras
• Representations of involutory subalgebras of affine Kac-Moody algebras
• Real forms of extended Kac-Moody symmetries and higher spin gauge theories
• An M-theory solution from null roots in E11
• Gravitational Coset Models
• A note on gauge fixing in supergravity/Kac-Moody correspondences
• Remarks on E11 approach