Duality-invariant non-extreme black holes in toroidally compactified string theory

We present a duality-invariant structure of the thermodynamic quantities of non-extreme black hole solutions of toroidally compactified Type II (M-theory) and heterotic string in five and four dimensions. These quantities are parametrized by duality-invariant combination of charges and the non-extremality parameter, which measures a deviation from the BPS-saturated limit. In particular, in D = 5 we find explicit S- and T-duality- [U-duality]-invariant expressions for solutions of toroidally compactified heterotic string [Type II string]. In D = 4, we consider general S-duality-invariant expressions for non-extreme solutions of pure N = 4 supergravity and find to the leading order in non-extremality parameter the T- and S-duality-invariant expressions of toroidally compactified heterotic string. General non-extreme solutions of toroidally compactified string in D = 4 are awaiting further investigation. (C) 1997 Elsevier Science B.V.