The Complete Form of N=2 Supergravity and its Place in the General Framework of D=4 N--Extended Supergravities

Relying on the geometrical set up of Special K\"ahler Geometry and Quaternionic Geometry, which I discussed at length in my Lectures at the 1995 edition of this Spring School, I present here the recently obtained fully general form of N=2 supergravity with completely arbitrary couplings. This lagrangian has already been used in the literature to obtain various results: notably the partial breaking of supersymmetry and various extremal black--hole solutions. My emphasis, however, is only on providing the reader with a completely explicit and ready to use component expression of the supergravity action. All the details of the derivation are omitted but all the definitions of the items entering the lagrangian and the supersymmetry transformation rules are given.Comment: 11 pages, LaTeX espcrc2, Seminar at Trieste Spring School 199

The full integration of black hole solutions to symmetric supergravity theories

We prove that all stationary and spherical symmetric black hole solutions to theories with symmetric target spaces are integrable and we provide an explicit integration method. This exact integration is based on the description of black hole solutions as geodesic curves on the moduli space of the theory when reduced over the time-like direction. These geodesic equations of motion can be rewritten as a specific Lax pair equation for which mathematicians have provided the integration algorithms when the initial conditions are described by a diagonalizable Lax matrix. On the other hand, solutions described by nilpotent Lax matrices, which originate from extremal regular (small) D = 4 black holes can be obtained as suitable limits of solutions obtained in the diagonalizable case, as we show on the generating geodesic (i.e. most general geodesic modulo global symmetries of the D = 3 model) corresponding to regular (and small) D = 4 black holes. As a byproduct of our analysis we give the explicit form of the Wick rotation connecting the orbits of BPS and non-BPS solutions in maximally supersymmetric supergravity and its STU truncation.Comment: 27 pages, typos corrected, references added, 1 figure added, Discussion on black holes and the generating geodesic significantly extended. Statement about the relation between the D=3 geodesics from BPS and non-BPS extreme black holes made explicit by defining the Wick rotation mapping the corresponding orbit

On the Gauged Kahler Isometry in Minimal Supergravity Models of Inflation

In this paper we address the question how to discriminate whether the gauged isometry group G_Sigma of the Kahler manifold Sigma that produces a D-type inflaton potential in a Minimal Supergravity Model is elliptic, hyperbolic or parabolic. We show that the classification of isometries of symmetric cosets can be extended to non symmetric Sigma.s if these manifolds satisfy additional mathematical restrictions. The classification criteria established in the mathematical literature are coherent with simple criteria formulated in terms of the asymptotic behavior of the Kahler potential K(C) = 2 J(C) where the real scalar field C encodes the inflaton field. As a by product of our analysis we show that phenomenologically admissible potentials for the description of inflation and in particular alpha-attractors are mostly obtained from the gauging of a parabolic isometry, this being, in particular the case of the Starobinsky model. Yet at least one exception exists of an elliptic alpha-attractor, so that neither type of isometry can be a priori excluded. The requirement of regularity of the manifold Sigma poses instead strong constraints on the alpha-attractors and reduces their space considerably. Curiously there is a unique integrable alpha-attractor corresponding to a particular value of this parameter.Comment: 85 pages, LaTex, 32 jpg figures, 4 tables; v2: title and abstract slightly modified, some assessments improved and made more precise, two figures and one reference added, several misprints correcte

Global U(1) R-Symmetry And Conformal Invariance Of (0,2) Models

We derive a condition under which (0,2) linear sigma models possess a ``left-moving'' conformal stress tensor in \bq cohomology (i.e. which leaves invariant the ``right-moving'' ground states) even away from their critical points. At the classical level this enforces quasihomogeneity of the superpotential terms. The persistence of this structure at the quantum level on the worldsheet is obstructed by an anomaly unless the charges and superpotential degrees satisfy a condition which is equivalent to the condition for the cancellation of the anomaly in a particular ``right-moving'' U(1) R-symmetry.Comment: 8 page

Integrability of Supergravity Black Holes and New Tensor Classifiers of Regular and Nilpotent Orbits

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of N=2 supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H*-orbits in which the space of possible Lax operators decomposes, H* being the isotropy group of scalar manifold originating from time-like dimensional reduction of supergravity from D=4 to D=3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H* orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S^3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple and practical discriminating method. We present a detailed analysis of the S^3 model and its black hole solutions, discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal black-holes and are nilpotent. We analyze these orbits in some detail working out different representatives thereof and showing that the relation between H* orbits and critical points of the geodesic potential is not one-to-one. Finally we present the conjecture that our newly identified tensor classifiers are universal and able to label all regular and nilpotent orbits in all homogeneous symmetric Special Geometries.Comment: Analysis of nilpotent orbits in terms of tensor classifiers in section 8.1 corrected. Table 1 corrected. Discussion in section 11 extende