U-invariants, black-hole entropy and fixed scalars

The absolute (moduli-independent) U-invariants of all N > 2 extended supergravities at D = 4 are derived in terms of (moduli-dependent) central and matter charges. These invariants give a general definition of the ''topological'' Bekenstein-Hawking entropy formula for extremal black-holes and reduce to the square of the black-hole ADM mass for ''fixed scalars'' which extremize the black-hole ''potential'' energy. The Hessian matrix of the black-hole potential at ''fixed scalars'', in contrast to N = 2 theories, is shown to be degenerate, with rank (N - 2)(N - 3) + 2n (N being the number of supersymmetries and n the number of matter multiplets) and semipositive definite. (C) 1997 Elsevier Science B.V.