• attractor mechanism
• black holes
• duality
• supergravity

The low-energy limit of theories of gravity such as Superstring theory and $M$-theory is Supergravity, whose action contains the Einstein Hilbert term coupled to vector and scalar fields in the bosonic sector, and fermions of spin $1/2$ and spin-$3/2$ gravitinos in the fermionic sector, with a specific field content depending on the Supergravity theory under consideration.

Following Gaillard-Zumino construction, we find that the most general duality group, whose action leaves the equations of motion invariant, is the symplectic $Sp(2n, \bb R)$ group. When the effective duality group is a non compact subgroup of $Sp(2n, \bb R)$, it is necessary to include scalar fields in the theory as coordinates of a manifold characterizing the Supergravity theory, describing a non linear $\sigma$-model.

Black holes are states of the graviton spin 2 field in Supergravity spectrum. In the case of non thermal radiation these states are stable, as happens for electromagnetically charged black holes with zero temperature but finite entropy, a property called \textit{extremality}.
Among these solutions, we consider static spherically symmetric systems in $d=4$ space-time dimensions, for which the dynamic is one dimensional and allows the determination of an effective potential depending on electromagnetic charges and scalar (moduli) fields, written in terms of dressed central (in case, also matter) charges, in $\mc N=2$ quadratic, $\mc N=3,5$ extended Supergravity. All of these theories have a scalar manifold $\mc M_{scalar}$ which is a symmetric space and does not admit a $d=5$ uplift.

Even if the black hole has a scalar hair, its entropy does depend only on asymptotical degrees of freedom, namely electric and magnetic charges determined by vector field strengths fluxes at spatial infinity. This behaviour is explained by the \textit{Attractor Mechanism}, since the vector multiplets scalars radial trajectories approach a fixed point in the moduli space as they reach the black hole horizon, loosing all memories of initial conditions. The fixed point is an equilibrium point for the system, and a critical one for the black hole effective potential.

The supersymmetric (BPS) black hole in $\mc N=2$ quadratic Supergravity is the only solution in which all the scalars are stabilized and the moduli space is therefore empty.
All the other studied solutions present flat directions for the scalar fields, but the attractor equations cancel the moduli dependence in the dynamical configuration at the black hole horizon. Specifically, the black hole entropy is given in terms of the invariant of the relevant $U$-duality group, which turns out to be a quadratic expression (in the case of $\mc N=2$ quadratic and $\mc N=3$ Supergravity), or a perfect square of a quadratic expression (in the case of $\mc N=4$ and $\mc N=5$ Supergravity) in terms of the eigenvalues of the central charge matrix. Due to the peculiarity of these theories it is possible to write an alternative expression for the Bekenstein-Hawking Entropy in terms of the \textit{effective horizon radius} $R_{H}$, whose expression is a function of scalar charges and the geometrical radius of the horizon, $r_{H}$.

Non-BPS attractor flows of extremal black holes in $d=4$ can be described in the first order formalism, introducing a \textit{fake superpotential} $\mc W$ such that $\mc W(\phi_{\infty}, p,q)=~r_{H}(\phi_{\infty},p,q)$ that enters in the espression of the effective radius $R_{H}$; the latter turns out to be, for the above mentioned theories, a moduli independent quantity.

As a counterexample, $\mc N=4$, $d=4$ Supergravity coupled to $1$ vector multiplet admits an uplift to $\mc N=4$ pure Supergravity in $d=5$ dimensions, but has a quartic invariant which cannot be written as a quadratic expression of the skew-eigenvalues of the central charge matrix, and the effective radius description does not hold.

As a final discussion the dualities among bosonic sectors of extended Supergravities are presented to explicitely show that bosonic interacting theories do not have a unique supersymmetric extension.