Transformation of black-hole hair under duality and supersymmetry

Abstract

We study the transformation under the full string theory duality group of the observable charges (including mass, angular momentum, NUT charge, electric, magnetic and different scalar charges) of four-dimensional point-like objects whose asymptotic behavior constitutes a subclass closed under duality. We find that those charges fall into two complex four-dimensional representations of the duality group. T- duality (including Buscher's transformations) has an O(1, 2) action on them and S- duality a U(1) action. The generalized Bogomol'nyi bound is an U(2, 2)- invariant built out of one representation while the other representation (which includes the angular momentum) never appears in it. The bound is manifestly duality-invariant. Consistency between T- duality and supersymmetry seems to require that primary scalar hair is included in the generalized Bogomol'nyi bound. We also find that all known four-dimensional supersymmetric massless black holes are the T- duals in the time direction of the usual massive supersymmetric black holes. Non-extreme massless “black holes” can be found as the T- duals of the non-extreme black holes. All of them have primary scalar hair and naked singularities.