By V. S. Varadarajan (auth.), Sergio Ferrara, Rita Fioresi, V.S. Varadarajan (eds.)

Supersymmetry used to be created by way of the physicists within the 1970's to offer a unified remedy of fermions and bosons, the fundamental components of subject. because then its mathematical constitution has been well-known as that of a brand new improvement in geometry, and mathematicians have busied themselves with exploring this element. This quantity collects contemporary advances during this box, either from a actual and a mathematical standpoint, with an accessory on a rigorous therapy of some of the questions raised.

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**Example text**

G0 ? b Œh0 ? Z O . g0 ? ; P/ D W r ; P/ (44) The above equality holds for any g0 2 G0 and thus for any h0 2 H0 . We conclude from this that W can be characterized, for a given orbit of solutions, as an H0 -invariant function of the central and matter charges. This is consistent with what was found in [4, 14]. Let us stress once more that we have started from a generic charge vector P, so that the definition of G0 , and thus of H0 , is charge dependent. We could have started from a given G0 inside G and worked out the representative P0 of the G-orbit having G0 as manifest little group.

H0 /. As far as the choice of the parametrization is concerned, for the BPS and non-BPS (I4 > 0) solutions, we choose the coset representative as follows: L. r / D L0 . ˛ / L1 . k / 2 e K0 e K1 ; (46) that is L0 . ˛ / is an element of e K0 Á G0 =H0 and L1 . k / is an element of e K1 . This in particular implies that ˛ and k transform in the representations R0 , R1 of H0 , respectively (see Table 1 for a list of these representations). For the nonBPS (I4 < 0) solutions, it is more convenient to adopt a parametrization of the coset which is different from (46), in which k can be defined to transform linearly with respect to the whole G0 .

D 0/ D r 0: We shall therefore simply denote them by: U D U. I 0 / and ADM mass and the scalar charges at infinity are given by: (10) r D r . I 0 /. The MADM . 0 ; P/ D UP . D 0/ D W . 0 ; P/; @W †r . 0 ; P/ D P r . D 0/ D 2 G rs . 0 / r . 0 ; P/: @ (11) non-trivial solutions on which the Hamiltonian vanishes. These correspond to the extremal black holes. W 2 5 In the case of non-extremal solutions the Hamilton-Jacobi equation reads @@U C 2 G rs . / @W @W 2U 2 D 4 e V C 4 c , and the corresponding first order equations have the form UP D @ r @ s 1 @W P r ; D G rs .