Gradient Representations and Affine Structures in AE(n)

Details Gradient Representations and Affine Structures in AE(n) Gradient Representations and Affine Structures in AE(n)

2005-06-28

arxiv.org/abs/hep-th/0506238v2

Class.Quant.Grav. 22 (2005) 4457-4488

Physics

High Energy Physics

High Energy Physics - Theory

43 pages

Scientific paper

10.1088/0264-9381/22/21/004

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.

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