Black Holes in Six-dimensional Conformal Gravity

We study conformally-invariant theories of gravity in six dimensions. In four dimensions, there is a unique such theory that is polynomial in the curvature and its derivatives, namely Weyl-squared, and furthermore all solutions of Einstein gravity are also solutions of the conformal theory. By contrast, in six dimensions there are three independent conformally-invariant polynomial terms one could consider. There is a unique linear combination (up to overall scale) for which Einstein metrics are also solutions, and this specific theory forms the focus of our attention in this paper. We reduce the equations of motion for the most general spherically-symmetric black hole to a single 5th-order differential equation. We obtain the general solution in the form of an infinite series, characterised by 5 independent parameters, and we show how a finite 3-parameter truncation reduces to the already known Schwarzschild-AdS metric and its conformal scaling. We derive general results for the thermodynamics and the first law for the full 5-parameter solutions. We also investigate solutions in extended theories coupled to conformally-invariant matter, and in addition we derive some general results for conserved charges in cubic-curvature theories in arbitrary dimensions.Comment: 28 pages. References adde

Holographic RG Flow in a New SO(3)×SO(3)SO(3)\times SO(3) Sector of ω\omega-Deformed SO(8)SO(8) Gauged N=8{\cal N}=8 Supergravity

We consider a certain ${\cal N}=1$ supersymmetric, $SO(3)\times SO(3)$ invariant, subsector of the $\omega$-deformed family of $SO(8)$-gauged ${\cal N}=8$ four-dimensional supergravities. The theory contains two scalar fields and two pseudoscalar fields. We look for stationary points of the scalar potential, corresponding to AdS vacua in the theory. One of these, which breaks all supersymmetries but is nonetheless stable, is new. It exists only when $\omega\ne 0$. We construct supersymmetric domain wall solutions in the truncated theory, and we give a detailed analysis of their holographic dual interpretations using the AdS/CFT correspondence. Domain walls where the pseudoscalars vanish were studied previously, but those with non-vanishing pseudoscalars, which we analyse numerically, are new. The pseudoscalars are associated with supersymmetric mass deformations in the CFT duals. When $\omega$ is zero, the solutions can be lifted to M-theory, where they approach the Coulomb-branch flows of dielectric M5-branes wrapped on $S^3$ in the deep IR.Comment: 40 pages, 10 figure

An ω\omega Deformation of Gauged STU Supergravity

Four-dimensional ${\cal N}=2$ gauged STU supergravity is a consistent truncation of the standard ${\cal N}=8$ gauged $SO(8)$ supergravity in which just the four $U(1)$ gauge fields in the Cartan subgroup of $SO(8)$ are retained. One of these is the graviphoton in the ${\cal N}=2$ supergravity multiplet and the other three lie in three vector multiplets. In this paper we carry out the analogous consistent truncation of the newly-discovered family of $\omega$-deformed ${\cal N}=8$ gauged $SO(8)$ supergravities, thereby obtaining a family of $\omega$-deformed STU gauged supergravities. Unlike in some other truncations of the deformed ${\cal N}=8$ supergravity that have been considered, here the scalar potential of the deformed STU theory is independent of the $\omega$ parameter. However, it enters in the scalar couplings in the gauge-field kinetic terms, and it is non-trivial because of the minimal couplings of the fermion fields to the gauge potentials. We discuss the supersymmetry transformation rules in the $\omega$-deformed supergravities, and present some examples of black hole solutions.Comment: 31 pages. Derivation of the range of \omega corrected; discussion of supersymmetry of solutions extended, and a reference adde

AdS Dyonic Black Hole and its Thermodynamics

We obtain spherically-symmetric and $\R^2$-symmetric dyonic black holes that are asymptotic to anti-de Sitter space-time (AdS), which are solutions in maximal gauged four-dimensional supergravity, with just one of the U(1) fields carrying both the electric and magnetic charges $(Q,P)$. We study the thermodynamics, and find that the usually-expected first law does not hold unless P=0, Q=0 or P=Q. For general values of the charges, we find that the first law requires a modification with a new pair of thermodynamic conjugate variables. We show that they describe the scalar hair that breaks some of the asymptotic AdS symmetries.Comment: 21 pages, typos corrected, discussion of Euclidean action adde

Thermodynamics of Magnetised Kerr-Newman Black Holes

The thermodynamics of a magnetised Kerr-Newman black hole is studied to all orders in the appended magnetic field $B$. The asymptotic properties of the metric and other fields are dominated by the magnetic flux that extends to infinity along the axis, leading to subtleties in the calculation of conserved quantities such as the angular momentum and the mass. We present a detailed discussion of the implementation of a Wald-type procedure to calculate the angular momentum, showing how ambiguities that are absent in the usual asymptotically-flat case may be resolved by the requirement of gauge invariance. We also present a formalism from which we are able to obtain an expression for the mass of the magnetised black holes. The expressions for the mass and the angular momentum are shown to be compatible with the first law of thermodynamics and a Smarr type relation. Allowing the appended magnetic field $B$ to vary results in an extra term in the first law of the form $-\mu dB$ where $\mu$ is interpreted as an induced magnetic moment. Minimising the total energy with respect to the total charge $Q$ at fixed values of the angular momentum and energy of the seed metric allows an investigation of Wald's process. The Meissner effect is shown to hold for electrically neutral extreme black holes. We also present a derivation of the angular momentum for black holes in the four-dimensional STU model, which is ${\cal N}=2$ supergravity coupled to three vector multiplets.Comment: 27 page

Correlation Functions in ω\omega-Deformed N=6 Supergravity

Gauged N=8 supergravity in four dimensions is now known to admit a deformation characterized by a real parameter $\omega$ lying in the interval $0\le\omega\le \pi/8$. We analyse the fluctuations about its anti-de Sitter vacuum, and show that the full N=8 supersymmetry can be maintained by the boundary conditions only for $\omega=0$. For non-vanishing $\omega$, and requiring that there be no propagating spin s>1 fields on the boundary, we show that N=3 is the maximum degree of supersymmetry that can be preserved by the boundary conditions. We then construct in detail the consistent truncation of the N=8 theory to give $\omega$-deformed SO(6) gauged N=6 supergravity, again with $\omega$ in the range $0\le\omega\le \pi/8$. We show that this theory admits fully N=6 supersymmetry-preserving boundary conditions not only for $\omega=0$, but also for $\omega=\pi/8$. These two theories are related by a U(1) electric-magnetic duality. We observe that the only three-point functions that depend on $\omega$ involve the coupling of an SO(6) gauge field with the U(1) gauge field and a scalar or pseudo-scalar field. We compute these correlation functions and compare them with those of the undeformed N=6 theory. We find that the correlation functions in the $\omega=\pi/8$ theory holographically correspond to amplitudes in the U(N)_k x U(N)_{-k} ABJM model in which the U(1) Noether current is replaced by a dynamical U(1) gauge field. We also show that the $\omega$-deformed N=6 gauged supergravities can be obtained via consistent reductions from the eleven-dimensional or ten-dimensional type IIA supergravities.Comment: 38 pages, one figur

Supersymmetric Solutions in Four-Dimensional Off-Shell Curvature-Squared Supergravity

Off-shell formulations of supergravities allow one to add closed-form higher-derivative super-invariants that are separately supersymmetric to the usual lower-derivative actions. In this paper we study four-dimensional off-shell N=1 supergravity where additional super-invariants associated with the square of the Weyl tensor and the square of the Ricci scalar are included. We obtain a variety of solutions where the metric describes domain walls, Lifshitz geometries, and also solutions of a kind known as gyratons. We find that in some cases the solutions can be supersymmetric for appropriate choices of the parameters. In some solutions the auxiliary fields may be imaginary. One may reinterpret these as real solutions in an analytically-continued theory. Since the supersymmetry transformation rules now require the gravitino to be complex, the analytically-continued theory has a "fake supersymmetry" rather than a genuine supersymmetry. Nevertheless, the concept of pseudo-supersymmetric solutions is a useful one, since the Killing spinor equations provide first-order equations for the bosonic fields.Comment: 28 page

Inductive vs transductive inference, global vs local models: SVM, TSVM, and SVMT for gene expression classification problems

This paper compares inductive-, versus transductive modeling, and also global-, versus local models with the use of SVM for gene expression classification problems. SVM are used in their three variants - inductive SVM, transductive SVM (TSVM), and SVM tree (SVMT) -the last two techniques being recently introduced by the authors. The problem of gene expression classification is used for illustration and four benchmark data sets are used to compare the different SVM methods. The TSVM outperforms the inductive SVM models applied on a small to medium variable (gene) set and a small to medium sample set, while SVMT is superior when the problem is defined with a large data set, or - a large set of variables (e.g. 7,000 genes, with little or no variable pre-selection)