Supersymmetric theories with the same bosonic content but different fermions, aka emph{twins}, were thought to exist only for supergravity. Here we show that pairs of super conformal field theories, for example exotic $mathcal{N}=3$ and $mathcal{N}=1
$ theories in $D=4$ spacetime dimensions, can also be twin. We provide evidence from three different perspectives: (i) a twin S-fold construction, (ii) a double-copy argument and (iii) by identifying candidate twin holographically dual gauged supergravity theories. Furthermore, twin W-supergravity theories then follow by applying the double-copy prescription to exotic super conformal field theories.
We present a conformal isometry for static extremal black hole solutions in all four-dimensional Einstein-Maxwell-scalar theories with electromagnetic duality groups `of type $E_7$. This includes, but is not limited to, all supergravity theories with
$mathcal{N}>2$ supersymmetry and all $mathcal{N}=2$ supergravity theories with symmetric scalar manifolds. The conformal isometry is valid for arbitrary electromagnetic charge configurations and relies crucially on the notion of Freudenthal duality.
We establish duality between real forms of the quantum deformation of the 4-dimensional orthogonal group studied by Fioresi et al. and the classification work made by Borowiec et al.. Classically these real forms are the isometry groups of $mathbb{R}
^4$ equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named $q$-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.
We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals $mathbb{R}$, complexes $mathbb{C}$, ternions $mathbb{T}$, quaternions $mathbb{H}$, sextonions $mathbb{S}$ and octonions $mathbb{O}$. The ternionic and
sextonionic rows/columns of the magic square yield non-reductive Lie algebras, including $mathfrak{e}_{7scriptscriptstyle{frac{1}{2}}}$. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the $D=3$ maximal $mathcal{N}=16$, magic $mathcal{N}=4$ and magic non-supersymmetric theories, obtained by dimensionally reducing the $D=4$ parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebra $tilde{mathfrak{e}}_{7(7)scriptscriptstyle{frac{1}{2}}}$ (which is not a subalgebra of $mathfrak{e}_{8(8)}$) is the non-compact global symmetry algebra of $D=3$, $mathcal{N}=16$ supergravity as obtained by dimensionally reducing $D=4$, $mathcal{N}=8$ supergravity with $mathfrak{e}_{7(7)}$ symmetry on a circle. The ternionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the $D=4$ maximal $mathcal{N}=8$, magic $mathcal{N}=2$ and magic non-supersymmetric theories obtained by dimensionally reducing the parent $D=5$ theories on a circle. In particular, the Kantor-Koecher-Tits intermediate non-reductive Lie algebra $mathfrak{e}_{6(6)scriptscriptstyle{frac{1}{4}}}$ is the non-compact global symmetry algebra of $D=4$, $mathcal{N}=8$ supergravity as obtained by dimensionally reducing $D=5$, $mathcal{N}=8$ supergravity with $mathfrak{e}_{6(6)}$ symmetry on a circle.
We compute the modifications to the attractor mechanism due to fermionic corrections. In N=2, D=4 supergravity, at the fourth order, we find a new contribution to the horizon values of the scalar fields of the vector multiplets.
These lectures give an elementary introduction to the subject of four dimensional black holes (BHs) in supergravity and the Attractor Mechanism in the extremal case. Some thermodynamical properties are discussed and some relevant formulae for the cri
tical points of the BH effective potential are given. The case of Maxwell-Einstein-axion-dilaton (super)gravity is discussed in detail. Analogies among BH entropy and multipartite entanglement of qubits in quantum information theory, as well moduli spaces of extremal BH attractors, are also discussed.
We apply the entropy formalism to the study of the near-horizon geometry of extremal black p-brane intersections in D>5 dimensional supergravities. The scalar flow towards the horizon is described in terms an effective potential given by the superpos
ition of the kinetic energies of all the forms under which the brane is charged. At the horizon active scalars get fixed to the minima of the effective potential and the entropy function is given in terms of U-duality invariants built entirely out of the black p-brane charges. The resulting entropy function reproduces the central charges of the dual boundary CFT and gives rise to a Bekenstein-Hawking like area law. The results are illustrated in the case of black holes and black string intersections in D=6, 7, 8 supergravities where the effective potentials, attractor equations, moduli spaces and entropy/central charges are worked out in full detail.
We study N=2, d=4 attractor equations for the quantum corrected two-moduli prepotential $mathcal{F}=st^2+ilambda$, with $lambda$ real, which is the only correction which preserves the axion shift symmetry and modifies the geometry. In the classical c
ase the black hole effective potential is known to have a flat direction. We found that in the presence of D0-D6 branes the black hole potential exhibits a flat direction in the quantum case as well. It corresponds to non-BPS $Z eq 0$ solutions to the attractor equations. Unlike the classical case, the solutions acquire non-zero values of the axion field. For the cases of D0-D4 and D2-D6 branes the classical flat direction reduces to separate critical points which turn out to have a vanishing axion field.
The general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing central charge Z and for Z=0, are obtained for the so-called stu model, the minimal rank-3 N=2 symmetric supergravity in d=4 sp
ace-time dimensions. Comparisons with previous results, as well as the fake supergravity (first order) formalism and an analysis of the BPS bound all along the non-BPS attractor flows and of the marginal stability of corresponding D-brane configurations, are given.
We consider extremal black hole attractors (both BPS and non-BPS) for N=3 and N=5 supergravity in d=4 space-time dimensions. Attractors for matter-coupled N=3 theory are similar to attractors in N=2 supergravity minimally coupled to Abelian vector mu
ltiplets. On the other hand, N=5 attractors are similar to attractors in N=4 pure supergravity, and in such theories only 1N-BPS non-degenerate solutions exist. All the above mentioned theories have a simple interpretation in the first order (fake supergravity) formalism. Furthermore, such theories do not have a d=5 uplift. Finally we comment on the duality relations among the attractor solutions of Ngeq2 supergravities sharing the same full bosonic sector.
