Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes

In four-dimensional N = 2 compactifications of string theory or M-theory, m odifications of the Bekenstein-Hawking area law for black hole entropy in t he presence of higher-derivative interactions are crucial for finding agree ment between the macroscopic entropy obtained from supergravity and sublead ing corrections to the microscopic entropy obtained via state counting. Her e we compute the modifications to the area law for various classes of black holes, such as heterotic black holes, stemming from certain higher-derivat ive gravitational Wilsonian coupling functions. We consider the extension t o heterotic N = 4 supersymmetric black holes and their type-II duals and we discuss its implications for the corresponding micro-state counting. In th e effective field theory approach the Wilsonian coupling functions are know n to receive non-holomorphic corrections. We discuss how to incorporate suc h corrections into macroscopic entropy formulae so as to render them invari ant under duality transformations, and we give a concrete example thereof. (C) 2000 Elsevier Science B.V. All rights reserved.