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10.4 Cosmological solutions with magnetic flux

We will now briefly sketch how one can also obtain the SM 5-brane solutions from geometric configurations and regular subalgebras of E10. In order to do this we consider “magnetic” subalgebras of E 10, constructed only from simple root generators at level two in the level decomposition of E 10. To the best of our knowldege, there is no theory of geometric configurations developed for the case of having 6 points on each line, which would be needed here. However, we may nevertheless continue to investigate the simplest example of such a configuration, namely (61,16), displayed in Figure 53View Image.
View Image

Figure 53: The simplest “magnetic configuration” (61,16), dual to the algebra A1.The associated supergravity solution describes an SM 5-brane, whose world volume is extended in the directions x1,⋅⋅⋅ ,x6.

The algebra dual to this configuration is an A1-subalgebra of E10 with the following generators:

e = E123456 = F123456, 1 ∑ 1 h ≡ [E123456,F123456] = − -- Kaa + --(K11 + ⋅⋅⋅ + K66). (10.65 ) 6 a⁄=1,⋅⋅⋅,6 3
Although the embedding of this algebra is different from the electric cases considered previously, the sigma model solution is still associated to an SL (2,ℝ )∕SO (2) coset space and therefore the solutions for φ(t) and &tidle;φ(t) are the same as before. Because of the embedding, however, the sigma model translates to a different type of supergravity solution, namely a spacelike five-brane whose world volume is extended in the directions 1 6 x ,⋅⋅⋅ ,x. The metric is given by
6 10 2 −4∕3 23√E-−t 2 −1∕3 ∑ a′ 2 1∕6 16√E-−t∑ ¯a2 ds = − H (t) e dt + H (t) (dx ) + H (t)e (dx ). (10.66 ) a′=1 ¯a=7
This solution coincides with the SM 5-brane found by Strominger and Gutperle in [90Jump To The Next Citation Point]40. Note that the correct power of H (t) for the five-brane arises here entirely due to the embedding of h into E10 through Equation (10.65View Equation).

Because of the existence of electric-magnetic duality on the supergravity side, it is suggestive to expect the existence of a duality between the two types of configurations (n ,g ) m 3 and (nm, g6), of which we have here seen the simplest realisation for the configurations (31,13) and (61,16).

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