Superpotential from black holes.

BPS monopoles in N=2 SUSY theories may lead to monopole condensation and confinement. We have found that supersymmetric black holes with non-vanishing area of the horizon may stabilize the moduli in theories where the potential is proportional to the square of the graviphoton central charge. In particular, in known models of spontaneous breaking of N=2 to N=1 SUSY theories, the parameters of the electric and magnetic Fayet–Iliopoulos terms can be considered proportional to electric and magnetic charges of the dyonic black holes. Upon such identification the potential is found to be proportional to the square of the black hole mass. The fixed values of the moduli near the black hole horizon correspond exactly to the minimum of this potential. The value of the potential at the minimum is proportional to the black hole entropy. 1 E-mail: [email protected] 1 N=2 supersymmetric theories may be viewed as low-energy effective actions describing the non-perturbative dynamics of more fundamental theories. In particular the vacuum structure and dyon spectrum of these theories has been studied by Seiberg and Witten [1] and the effect of BPS monopoles in N=2 supersymmetric Yang-Mills theories was understood. The relevance of generic supersymmetric black holes to low-energy effective theories was not studied yet, however, one may expect that such relation exists. In particular, one could have guessed that the black holes of N=2 theory with one half of supersymmetry unbroken may be somehow relevant to models with spontaneous breaking of N=2 supersymmetry to N=1. We will find indeed that this is the case and that the effect of supersymmetric dyonic black holes is to stabilize the moduli. The choice of the superpotential in such models will be related to the central charge of the graviphoton, i.e. to the black hole mass as the function of moduli and conserved charges of the black hole. The main difference between our study of the black holes and the study of non-perturbative phenomena in N=2 theories of rigid supersymmetries in [1] is in the properties of the relevant central charge. In ref. [1] the central charge defining the mass of the dyon in rigid supersymmetric theory is defined by charge of the vector multiplet. It is defined by the symplectic section of a given N=2 theory (X, FΛ) and by conserved charges (qΛ, p ) of the dyon and is given by (M YM ) 2 = |Z(t, t̄, q, p)| = ∣ ∣ ∣X(t)qΛ − FΛ(t) p ∣