Supersymmetric balance of forces and condensation of BPS states

Until now all known static multi-black-hole solutions described BPS states with charges of the same sign. Such solutions could not be related to flat directions in the space of BPS states: The total number of such states could not spontaneously increase because of charge conservation. We show that there exist static BPS configurations which remain in equilibrium even if they consist of states with opposite electric (or magnetic) charges from vector multiplets. This is possible because of the exact cancellation between the Coulomb and scalar forces. In particular, in the theories with N=4 or N=2 supersymmetry there exist stable massless multicenter configurations with vanishing total charge. Since such configurations have vanishing energy and charge independently of their number, they can be associated with flat directions in the space of all possible BPS states. For the N=2 case this provides a realization of the idea that BPS condensates could relate to each other different vacua of the string theory.