U-Invariants, Black-Hole Entropy and Fixed Scalars

TL;DR

The paper derives moduli-independent U-duality invariants for all $N>2$ in $D=4$ supergravity, expressing them in terms of moduli-dependent central charges $Z_{AB}$ and matter charges $Z_I$ to define a topological entropy for extremal black holes. It provides explicit invariant formulas for $N=3,4,5,6,8$ and shows these reduce to the squared ADM mass at fixed scalars (the attractor point), with the Hessian of the black-hole potential degenerating yet semipositive and possessing rank $(N-2)(N-3) + 2n$. The invariants are constructed via Cartan elements of the coset $G/H$ and align with quartic or higher-order invariants of the respective U-duality groups, enabling a unified, duality-invariant entropy framework across $N>2$ theories. The work also connects these macroscopic results to string/M-theory compactifications, illustrating how the invariants reflect the underlying geometry of the scalar manifolds and offering a bridge between attractor physics and microscopic state counting.

Abstract

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) + 2 n (N being the number of supersymmetries and n the number of matter multiplets) and semipositive definite.