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26) Proceeding in this fashion we find the minimum mass for any D-brane M pD = R10−p · · · R9 . 24) we get the mass of the solitonic fivebrane M s5 = R5 · · · R9 . 11). 22). 22) by setting k (d) = 0 we can read off the mass of these objects in terms of the coefficients in the harmonic function f (d) . 29) 2π . This determines the where ωn is the volume of the unit sphere Sn , ωn = Γ(n/2) coefficients for all excitations. 7). In order to do that, we use Dirac duality of the fundamental string and the solitonic fivebrane.

One can think that what we did was to determine the dimension of the classical moduli space of vacua of this theory and then considered oscillations around a given vacuum. Let us remark for later use that in this picture the momentum is carried by the hypermultiplets. The bosonic components of the hypermultiplets do not carry angular momentum under SO(4)E of the external rotations, while the fermions do indeed carry it. In fact the SO(4)E ∼SU(2)L ×SU(2)R symmetry appears as 55 an R-symmetry of the theory and left or right moving fermions carry spin under SU(2)L,R respectively [15].

64) The solution with angular momentum can be found in [15], [16], [64], we will be just interested in the entropy of that solution in the extremal limit, for which the mass is the minimum consistent with a given angular momentum and charges. The entropy is then Sext = 2π N Q1 Q5 − J1 J2 . For J1 = J2 the solution is also BPS [15]. 6. Black hole solutions in four dimensions Now we turn to the more realistic case of four dimensional black holes. It is still not totally realistic since the compactification we will consider is on T 6 which is not the one that describes our four dimensional world.

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