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The corresponding unbroken, maximal N = 2, d = 4 SUSY algebra is the N = 2, d = 4 superPoincar´e one (asymptotical rigid N = 2 SUSY ). 6) i i i z = 0 ⇔ z = z ∈ C, ∂ µ ∞ u ∈ H. ∂µ q u = 0 ⇔ q u = q∞ i z∞ is an unconstrained scalar ﬁeld conﬁguration in the nV -d K¨ahler– Hodge complex moduli space MnV of the N = 2, d = 4, nV -fold MESGT. , global. Thus, the unbroken SUSY manifests itself in the fact that each nonvanishing scalar ﬁeld is the ﬁrst component of a covariantly constant N = 2 superﬁeld for the vector and/or the hypermultiplet, but the supergravity superﬁeld vanishes.
G. to [206–208]. 11) where, as usual, the contravariant and covariant metric tensors are related by the orthonormality condition Gij Glj = Gij ∂ j ∂l K = δli . 12) Third, since MnV is a K¨ ahler–Hodge manifold, it admits a U (1) line (Hodge) bundle , whose ﬁrst Chern class coincides with the K¨ ahler class of MnV c1 [ ] = K [MnV ] . 13) Such a property allows one to locally write the U (1) connection Q as 3 It should also be recalled that in a K¨ ahlerian manifold the (completely covariant) Riemann-Christoﬀel tensor, as well as its contractions Rij and R and the (completely covariant) Weyl conformal curvature tensor Cijkl , are all real  Rijkl = Rijkl ; R = R; Rij = Rij ; Cijkl = Cijkl .
The action of such a group. Otherwise speaking, the complex coniugation and/or the Hodge Abelian dualization do not have any eﬀect on the symplectic covariance. 31) implying, for instance GΛ ≡ Re GΛ− = Re (NΛΣ ) F Σ − Im (NΛΣ ) ∗ F Σ . 33) d Re Z − = 0 . 2 Electric–Magnetic Duality, Central Charge, and Attractor Mechanism 53 2 where the integration is performed in the physical space, and S∞ is the 2sphere at the inﬁnity. 34) may respectively be considered as the deﬁnition, in a suitable system of units, of the asymptotical values of the magnetic and electric charges characterizing the charge conﬁguration of the nV + 1 Maxwell vector ﬁelds of the theory (indeed we get a vector potential from the gravity multiplet plus another one for each considered vector multiplet).