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Supersymmetry in Mathematics and Physics: UCLA Los Angeles, by V. S. Varadarajan (auth.), Sergio Ferrara, Rita Fioresi,

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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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 .

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