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) spacetime dimensions to one (time) dimension, and their distinguished maximal regular subalgebras A_{n1}\equiv\mathfrak{sl}_n and A^{(1)}_{n2}. The interplay between these two subalgebras is used, for n = 3, to determine the commutation relations of the 'gradient generators' within AE_{3}. The lowlevel truncation of the geodesic σmodel over the coset space AE_{n}/K(AE_{n}) is shown to map to a suitably truncated version of the SL(n)/SO(n) nonlinear σmodel 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 onetoone correspondence between such solutions, and null geodesic trajectories on the infinitedimensional coset space {\mathfrak{H}}/K({\mathfrak{H}}) , where {\mathfrak{H}} is the (extended) Heisenberg group, and K({\mathfrak{H}}) its maximal compact subgroup. We clarify the relation between {\mathfrak{H}} and the corresponding subgroup of the Geroch group.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 November 2005
 DOI:
 10.1088/02649381/22/21/004
 arXiv:
 arXiv:hepth/0506238
 Bibcode:
 2005CQGra..22.4457K
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology
 EPrint:
 43 pages