Dual Gauged Supergravities

We shall review a novel formulation of four dimensional gauged supergravity which is manifestly covariant with respect to the non-perturbative electric-magnetic duality symmetry transformations of the ungauged theory, at the level of the equations of motion and Bianchi identities. We shall also discuss the application of this formalism to the description of M-theory compactified on a twisted torus in the presence of fluxes and to the interpretation from a M/Type IIA theory perspective of the D=5 --> D=4 generalized Scherk-Schwarz reduction. This latter analysis will bring up the issue of non-geometric fluxes.Comment: 31 pages, LaTeX file, typo in formula (70) of the Appendix corrected, references update

Branes in Anti de Sitter Space-Time

An intense study of the relationship between certain quantum theories of gravity realized on curved backgrounds and suitable gauge theories, has been originated by a remarkable conjecture put forward by Maldacena almost one year ago. Among the possible curved vacua of superstring or M-theory, spaces having the form of an Anti-de Sitter space-time times a compact Einstein manifold, have been playing a special role in this correspondence, since the quantum theory realized on them, in the original formulation of the conjecture, was identified with the effective superconformal theory on the world volume of parallel p-branes set on the boundary of such a space (holography). An important step in order to verify such a conjecture and eventually generalize it, consists in a precise definition of the objects entering both sides of the holographic correspondence. In the most general case indeed it turns out that important features of the field theory on the boundary of the curved background, identified with the quantum theory of gravity in the bulk, are encoded in the dynamics of the coinciding parallel p-branes set on the boundary of the same space. The study of p-brane dynamics in curved space-times which are vacua of superstring of M-theory, turns out therefore to be a relevant issue in order to verify the existence of the holographic correspondence. In the present paper, besides providing a hopefully elementary introduction to Maldacena's duality, I shall deal in a tentatively self contained way with a particular aspect of the problem of p-brane dynamics in Anti-de Sitter space-time, discussing some recent results.Comment: Talk given at the XII Congress of General Relativity (SIGRAV), Bari 21-25 Sept. 1998, 1 LaTeX file, 16 page

Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed

Numerical comparisons among some methods for Hamiltonian problems

We report a few sumerical tests comparing some newly defined energy-preserving integrators and symplectic methods, using either constant and variable stepsize.Comment: 5 pages, 8 figure

N=8 BPS black holes preserving 1/8 supersymmetry

In the context of N=8 supergravity we consider BPS black-holes that preserve 1/8 supersymmetry. It was shown in a previous paper that, modulo U-duality transformations of E_{7(7)} the most general solution of this type can be reduced to a black-hole of the STU model. In this paper we analize this solution in detail, considering in particular its embedding in one of the possible Special K\"ahler manifold compatible with the consistent truncations to N=2 supergravity, this manifold being the moduli space of the T^6/Z^3 orbifold, that is: SU(3,3)/SU(3)*U(3). This construction requires a crucial use of the Solvable Lie Algebra formalism. Once the group-theoretical analisys is done, starting from a static, spherically symmetric ans\"atz, we find an exact solution for all the scalars (both dilaton and axion-like) and for gauge fields, together with their already known charge-dependent fixed values, which yield a U-duality invariant entropy. We give also a complete translation dictionary between the Solvable Lie Algebra and the Special K\"ahler formalisms in order to let comparison with other papers on similar issues being more immediate. Although the explicit solution is given in a simplified case where the equations turn out to be more manageable, it encodes all the features of the more general one, namely it has non-vanishing entropy and the scalar fields have a non-trivial radial dependence.Comment: 29+1 pages, 1 Latex file; a misprint in the entropy formula, eq.(5.14), correcte

Fifty Years of Stiffness

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.Comment: 24 pages, 11 figure

On Extremal Limits and Duality Orbits of Stationary Black Holes

With reference to the effective three-dimensional description of stationary, single center solutions to (ungauged) symmetric supergravities, we complete a previous analysis on the definition of a general geometrical mechanism for connecting global symmetry orbits (duality orbits) of non-extremal solutions to those of extremal black holes. We focus our attention on a generic representative of these orbits, providing its explicit description in terms of D=4 fields. As a byproduct, using a new characterization of the angular momentum in terms of quantities intrinsic to the geometry of the D=3 effective model, we are able to prove on general grounds its invariance, as a function of the boundary data, under the D=4 global symmetry. In the extremal under-rotating limit it becomes moduli-independent. We also discuss the issue of the fifth parameter characterizing the four-dimensional seed solution, showing that it can be generated by a transformation in the global symmetry group which is manifest in the D=3 effective description.Comment: 24 pages, LaTeX source, typos corrected, expressions for the entropy of the non-extremal solutions adde