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In these lectures we discuss N-extended conformal supergravity and its spectrum in four dimensions. These theories can be considered as the massless limit of Einstein-Weyl supergravity and by taking into account their enhanced gauge symmetries, we derive their massless spectrum, which in general contains a dipole-ghost graviton multiplet and an N-fold tripole-ghost gravitino multiplet.

We discuss the Einstein tensor, the supercurrent and their conservation laws of old and new minimal formulations of supergravity in the superconformal approach. The variation of the action with respect to the gauge field of the $R$-symmetry in the conformal approach (the auxiliary field in the super-Poincaré action) allows to find the Einstein tensor and supercurrent in any curved background. Hence generalized expressions for their Ward identities follow. This proceeding is based on arXiv:1805.09228, arXiv:1705.02272.

Supergravity theories at D=4 allow to formulate the Swampland de Sitter conjectures in the complex field space of scalar components of chiral multiplets. We formulate the de Sitter and refined de Sitter conjecture by using the Kähler invariant ${\cal G}$-function and explore a class of models in the Landscape/Swampland scenario which obey and/or violate such conjectures. Furthermore we give a new construction of single exponential potentials in supergravity. These depend on a chiral superfield with a Kähler potential parametrizing an SU(1,1)/U(1) geometry. We show that the construction allows for modifications to supergravity theories causing them to obey the de Sitter conjectures.

Some general aspects of supersymmetry and supergravity are briefly reviewed with emphasis on Noether supercurrents and their role in the discussion of supersymmetry breaking.

We discuss the connection between Weyl$^2$ supergravity and superstrings and further discuss holography between 4-dimensional, ${\cal N}=4$ superconformal Weyl$^2$ supergravity and ${\cal N}=8$, higher spin-four theory on $AdS_5$. The Weyl$^2$ plus Einstein supergravity theory is a special kind of a bimetric gravity theory and consists of a massless graviton multiplet plus an additional massive spin-two supermultiplet. Here, we argue that the additional spin-two field and its superpartners originate from massive excitations in the open string sector; just like the ${\cal N}=4$ super Yang-Mills gauge fields, they are localized on the world volume of D3-branes. The ghost structure of the Weyl action should be considered as an artifact of the truncation of the infinitely many higher derivative terms underlying the massive spin 2 action. In field theory, ${\cal N}=4$ Weyl$^2$ supergravity exhibits superconformal invariance in the limit of vanishing Planck mass. In string theory the additional spin-two fields become massless in the tensionless limit. Therefore low string scale scenarios with large extra dimensions provide (almost) superconformal field theories with almost massless open string spin-two fields. The full ${\cal N}=4$ scalar potential including the Yang-Mills matter multiplets is presented and the supersymmetric vacua of Einstein Supergravity are shown, as expected, to be vacua of massive Weyl supergravity. Other vacua are expected to exist which are not vacua of Einstein supergravity. Finally, we identify certain spin-four operators on the 4-dimensional boundary theory that could be the holographic duals of spin-four fields in the bulk.

In this paper we study the spectrum of all conformal, ${\cal N}$-extended supergravities (${\cal N}=1,2,3,4$) in four space-time dimensions. When these theories are obtained as massless limit of Einstein plus Weyl$^2$supergravity, the appropriate counting of the enhanced gauge symmetries allow us to derive the massless spectrum which consist of a dipole ghost graviton multiplet, a ${\cal N}$-fold tripole ghost gravitino, the third state belonging to a spin 3/2 multiplet and a residual vector multiplet present for non-maximal ${\cal N}<4$ theories. These theories are not expected to have a standard gravity holographic dual in five dimensions.

In the present investigation we consider the possibility of having new massive, higher spin W-supergravity theories, which do not exist as four-dimensional perturbative models. These theories are based on a double copy construction of two supersymmetric field theories, where at least one factor is given by a N=3 field theory, which is a non-perturbative S-fold of N=4 super Yang-Mills theory. In this way, we can obtain as S-folds a new N=7 (corresponding to 28 supercharges) W-supergravity and its N=7 W-superstring counterpart, which both do not exist as four-dimensional perturbative models with an (effective) Langrangian description. The resulting field resp. string theory does not contain any massless states, but instead a massive higher spin-four supermultiplet of the N=7 supersymmetry algebra. Furthermore we also construct a four-dimensional heterotic S-fold with N=3 supersymmetry. It again does not exist as perturbative heterotic string model and can be considered as the heterotic counterpart of the N=3 superconformal field theories, which were previously constructed in the context of type I orientfold models.

We discuss local supercurrents as sources of the super-Einstein equations in the superconformal approach in the old and new minimal (auxiliary fields) formulation. Modifications of the Ward identity giving the covariant divergence of the Einstein multiplet are considered in presence of a Fayet-Iliopoulos term. Curvature multiplets can be used as alternative to the gravitino variation in the search for rigid supersymmetric curved backgrounds.

We consider an expression for the supercurrent in the superconformal formulation of N=1 supergravity. A chiral compensator provides the supersymmetric formulation of the Callan-Coleman-Jackiw (CCJ) improved stress energy tensor, when the conformal gauge is used. Superconformal and non-superconformal matter give different conservation laws of the supercurrent, when coupled to the curvature supermultiplets which underlie the local superspace geometry. This approach can be applied to any set of auxiliary fields and it is useful to classify rigid curved superspace geometries. Examples with four supersymmetries are briefly described.

N=1, D=4 non linear sigma models, parametrized by chiral superfields, usually describe Kaehlerian geometries, provided that Einstein frame supergravity is used. The sigma model metric is no longer Kaehler when local supersymmetry becomes nonlinearly realized through the nilpotency of the supergravity auxiliary fields. In some cases the nonlinear realization eliminates one scalar propagating degree of freedom. This happens when the sigma model conformal-frame metric has co-rank 2. In the geometry of the inflaton, this effect eliminates its scalar superpartner. We show that the sigma model metric remains semidefinite positive in all cases, due the to positivity properties of the conformal-frame sigma model metric.

We focus on non-linear realizations of local supersymmetry as obtained by using constrained superfields in supergravity. New constraints, beyond those of rigid supersymmetry, are obtained whenever curvature multiplets are affected as well as higher derivative interactions are introduced. In particular, a new constraint, which removes a very massive gravitino is introduced, and in the rigid limit it merely reduces to an explicit supersymmetry breaking. Higher curvature supergravities free of ghosts and instabilities are also obtained in this way. Finally, we consider direct coupling of the goldstino multiplet to the super Gauss--Bonnet multiplet and discuss the emergence of a new scalar degree of freedom.

We identify a cubic holomorphic constraint that subtends the total breaking of N=2 supersymmetry in a vector multiplet and exhibit its microscopic origin. The new constraint leaves behind, at low energies, a vector and the two goldstini, in a non-linear Lagrangian that generalizes the N=2 Volkov-Akulov model.

We give a new expression for the supercurrent and its conservation in curved ${\cal N}=1$, $D=4$ superspace using the superconformal approach. The first component of the superfield, whose lowest component is the vector auxiliary field gives the (super)Einstein equations. Its trace and couplings to conformal and non-conformal matter is presented. In a suitable dilatational gauge, the conformal gauge, we obtain an update of the Callan-Coleman-Jackiw improved currents for conformal matter, containing $R$-symmetry corrections for a new traceless covariantly conserved energy--momentum tensor. We observe that in the Poincaré gauge, where standard Poincaré supergravity is usually formulated, the currents are not improved and then the higher conformal symmetry of the matter sector is obscured. The curvature multiplets are used to find supersymmetric curved backgrounds and some examples are exhibited in agreement with existing results.

A personal recollection of events that preceded the construction of Supergravity and of some subsequent developments.

We consider the dimensional reduction to D = 3 of four maximal-rank supergravities which preserve minimal supersymmetry in D = 11, 7, 5 and 4. Such "curious" theories were investigated some time ago, and the four-dimensional one corresponds to an N = 1 supergravity with 7 chiral multiplets spanning the seven-disk manifold. Recently, this latter theory provided cosmological models for alpha-attractors, which are based on the disk geometry with possible restrictions on the parameter alpha. A unified picture emerges in D = 3, where the Ehlers group of General Relativity merges with the S-, T- and U- dualities of the D = 4 parent theories.

We derive the mass formulae for ${\cal N}=1$, $D=4$ matter-coupled Supergravity for broken (and unbroken) Supersymmetry in curved space-time. These formulae are applicable to de Sitter configurations as is the case for inflation. For unbroken Supersymmetry in anti-de Sitter (AdS) one gets the mass relations modified by the AdS curvature. We compute the mass relations both for the potential and its derivative non-vanishing.

We prove that all inflationary models, including those with dark energy after the end of inflation, can be embedded in minimal supergravity with a single chiral superfield. Moreover, the amount of supersymmetry breaking is independently tunable due to a degeneracy in the choice for the superpotential. The inflaton is a scalar partner of the Goldstino in this set-up. We illustrate our general procedure with two examples that are favoured by the Planck data.

Abdus Salam was a true master of 20th Century Theoretical Physics. Not only was he a pioneer of the Standard Model (for which he shared the Nobel Prize with S. Glashow and S.Weinberg), but he also (co)authored many other outstanding contributions to the field of Fundamental Interactions and their unification. In particular, he was a major contributor to the development of supersymmetric theories, where he also coined the word "Supersymmetry" (replacing the earlier "Supergauges" drawn from String Theory). He also introduced the basic concept of "Superspace" and the notion of "Goldstone Fermion"(Goldstino). These concepts proved instrumental for the exploration of the ultraviolet properties and for the study of spontaneously broken phases of super Yang-Mills theories and Supergravity. They continue to play a key role in current developments in Early-Universe Cosmology. In this contribution we review models of inflation based on Supergravity with spontaneously broken local supersymmetry, with emphasis on the role of nilpotent superfields to describe a de Sitter phase of our Universe.

The c-map of four dimensional non-linear theories of electromagnetism is considered both in the rigid case and in its coupling to gravity. In this way theories with antisymmetric tensors and scalars are obtained, and the three non-linear representations of N=2 supersymmetry partially broken to N=1 related. The manifest $\mathrm{Sp}(2n)$ and $\mathrm{U}(n)$ covariance of these theories in their multifield extensions is also exhibited.

We elaborate on how to build, in a systematic fashion, two-field Abelian extensions of the Born-Infeld Lagrangian. These models realize the non-trivial duality groups that are allowed in this case, namely U(2), SU(2) and U(1)xU(1). For each class, we also construct an explicit example. They all involve an overall square root and reduce to the Born-Infeld model if the two fields are identified, but differ in quartic and higher interactions. The U(1)xU(1) and SU(2) examples recover some recent results obtained with different techniques, and we show that the U(1)xU(1) model admits an N=1 supersymmetric completion. The U(2) example includes some unusual terms that are not analytic at the origin of field space.

We study the application of a supersymmetric model with two constrained supermultiplets to inflationary cosmology. The first superfield S is a stabilizer chiral superfield satisfying a nilpotency condition of degree 2, S^2=0. The second superfield Phi is the inflaton chiral superfield, which can be combined into a real superfield B=(Phi-Phi*)/2i. The real superfield B is orthogonal to S, S B=0, and satisfies a nilpotency condition of degree 3, B^3=0. We show that these constraints remove from the spectrum the complex scalar sgoldstino, the real scalar inflaton partner (i.e. the "sinflaton"), and the fermionic inflatino. The corresponding supergravity model with de Sitter vacua describes a graviton, a massive gravitino, and one real scalar inflaton, with both the goldstino and inflatino being absent in unitary gauge. We also discuss relaxed superfield constraints where S^2=0 and S Phi* is chiral, which removes the sgoldstino and inflatino, but leaves the sinflaton in the spectrum. The cosmological model building in both of these inflatino-less models offers some advantages over existing constructions.

This review is devoted to highlight some aspects of the relevance of Majorana fermions in rigid supersymmetry breaking in four spacetime dimensions. After introducing some basic facts on spinors, and on their symmetries and reality properties, we consider Goldstino actions describing partial breaking of rigid supersymmetry, then focussing on Born-Infeld non-linear theory, its duality symmetry, and its supersymmetric extensions, also including multi-field generalizations exhibiting doubly self-duality.

We build the minimal supergravity model where the nilpotent chiral goldstino superfield is coupled to a chiral matter superfield, realising a different non-linear representation through a mixed nilpotency constraint. The model describes the spontaneous breaking of local supersymmetry in the presence of a generically massive Majorana fermion, but in the absence of elementary scalars. The sign and the size of the cosmological constant, the spectrum and the four-fermion interactions are controlled by suitable parameters.

A scale invariant Goldstino theory coupled to Supergravity is obtained as a standard supergravity dual of a rigidly scale invariant higher--curvature Supergravity with a nilpotent chiral scalar curvature. The bosonic part of this theory describes a massless scalaron and a massive axion in a de Sitter Universe.

We construct Supergravity models where the goldstino multiplet has a gravitational origin, being dual to the chiral curvature superfield. Supersymmetry is nonlinearly realized due to a nilpotent constraint, while the goldstino arises from $\gamma$-traces of the gauge-invariant gravitino field strength. After duality transformations one recovers, as expected, the standard Volkov-Akulov Lagrangian coupled to Supergravity, but the gravitational origin of the goldstino multiplet restricts the available types of matter couplings. We also construct explicitly some inflationary models of this type, which contain both the inflaton and the nilpotent superfield.

This review is devoted to some aspects of non-linear Supersymmetry in four dimensions that can be efficiently described via nilpotent superfields, in both rigid and curved Superspace. Our focus is mainly on the partial breaking of rigid $N=2$ Supersymmetry and on a class of generalized Born-Infeld systems that originate from Special Geometry and on some prototype cosmological models, starting from the Supergravity embedding of Starobinsky inflation. However, as an aside we also review briefly some interesting two-field extensions of the Born-Infeld Lagrangian whose field equations enjoy extended duality symmetries.

We formulate $R^2$ pure supergravity as a scale invariant theory built only in terms of superfields describing the geometry of curved superspace. The standard supergravity duals are obtained in both "old" and "new" minimal formulations of auxiliary fields. These theories have massless fields in de Sitter space as they do in their non supersymmetric counterpart. Remarkably, the dual theory of $R^2$ supergravity in the new minimal formulation is an extension of the Freedman model, describing a massless gauge field and a massless chiral multiplet in de Sitter space, with inverse radius proportional to the Fayet-Iliopoulos term. This model can be interpreted as the "de-Higgsed" phase of the dual companion theory of $R+R^2$ supergravity.

The self-duality of the N=1 supersymmetric Born--Infeld action implies a double self-duality of the tensor multiplet square-root action when the scalar and the antisymmetric tensor are interchanged via Poincare' duality. We show how this phenomenon extends to D space-time dimensions for non-linear actions involving pairs of forms of rank p and D-p-2. As a byproduct, we construct a new two-field generalization of the Born-Infeld action whose equations of motion are invariant under a U(1) duality. In these systems, the introduction of Green-Schwarz terms results in explicit non-linear mass-like terms for dual massive pairs.

We consider massive dual pairs of p-forms and (D-p-1)-forms described by non-linear Lagrangians, where non-linear curvature terms in one theory translate into non-linear mass-like terms in the dual theory. In particular, for D=2p and p even the two non-linear structures coincide when the non-linear massless theory is self-dual. This state of affairs finds a natural realization in the four-dimensional massive N=1 supersymmetric Born-Infeld action, which describes either a massive vector multiplet or a massive linear (tensor) multiplet with a Born-Infeld mass-like term. These systems should play a role for the massive gravitino multiplet obtained from a partial super-Higgs in N=2 Supergravity.

We study the partial breaking of $N=2$ rigid supersymmetry for a generic rigid special geometry of $n$ abelian vector multiplets in the presence of Fayet-Iliopoulos terms induced by the Hyper-Kähler momentum map. By exhibiting the symplectic structure of the problem we give invariant conditions for the breaking to occur, which rely on a quartic invariant of the Fayet-Iliopoulos charges as well as on a modification of the $N=2$ rigid symmetry algebra by a vector central charge.

We investigate $U(1)^{\,n}$ supersymmetric Born-Infeld Lagrangians with a second non-linearly realized supersymmetry. The resulting non-linear structure is more complex than the square root present in the standard Born-Infeld action, and nonetheless the quadratic constraints determining these models can be solved exactly in all cases containing three vector multiplets. The corresponding models are classified by cubic holomorphic prepotentials. Their symmetry structures are associated to projective cubic varieties.

We derive new types of $U(1)^n$ Born-Infeld actions based on N=2 special geometry in four dimensions. As in the single vector multiplet (n=1) case, the non--linear actions originate, in a particular limit, from quadratic expressions in the Maxwell fields. The dynamics is encoded in a set of coefficients $d_{ABC}$ related to the third derivative of the holomorphic prepotential and in an SU(2) triplet of N=2 Fayet-Iliopoulos charges, which must be suitably chosen to preserve a residual N=1 supersymmetry.

We discuss N=1 supergravity inflationary models based on two chiral multiplets, the inflaton and the goldstino superfield. Using superconformal methods for these models, we propose to replace the unconstrained chiral goldstino multiplet by the nilpotent one associated with non-linearly realized supersymmetry of the Volkov-Akulov type. In the new cosmological models, the sgoldstino is proportional to a bilinear combination of fermionic goldstinos. It does not acquire any vev, does nor require stabilization, and does not affect the cosmological evolution. We explain a universal relation of these new models to kappa-symmetric super-Dp-brane actions. This modification significantly simplifies a broad class of the presently existing inflationary models based on supergravity and string theory, including the simplest versions of chaotic inflation, the Starobinsky model, a broad class of cosmological attractors, the Higgs inflation, and much more. In particular, this is a step towards a fully supersymmetric version of the string theory axion monodromy inflation. The new construction serves as a simple and manifestly supersymmetric uplifting tool in the KKLT-type string theory landscape.

The supersymmetric extension of "Starobinsky" $R+\alpha R^2$ models of inflation is particularly simple in the "new minimal" formalism of supergravity, where the inflaton has no scalar superpartners. This paper is devoted to matter couplings in such supergravity models. We show how in the new minimal formalism matter coupling presents certain features absent in other formalisms. In particular, for the large class of matter couplings considered in this paper, matter must possess an R-symmetry, which is gauged by the vector field which becomes dynamical in the "new minimal" completion of the $R+\alpha R^2$ theory. Thus, in the dual formulation of the theory, where the gauge vector is part of a massive vector multiplet, the inflaton is the superpartner of the massive vector of a nonlinearly realized R-symmetry. The F-term potential of this theory is of no-scale type, while the inflaton potential is given by the D-term of the gauged R-symmetry. The absolute minimum of the potential is always exactly supersymmetric, so in this class of models if realistic vacua exist, they must be always metastable. We also briefly comment on possible generalizations of the examples discussed here and we exhibit some features of higher-curvature supergravity coupled to matter in the "old minimal" formalism.

In these lectures, after a short introduction to cosmology, we discuss the supergravity embedding of higher curvature models of inflation. The supergravity description of such models is presented for the two different formulations of minimal supergravity.

We elaborate on the predictions of the imaginary Starobinsky model of inflation coupled to matter, where the inflaton is identified with the imaginary part of the inflaton multiplet suggested by the Supergravity embedding of a pure R + R^2 gravity. In particular, we study the impact of higher-order curvature terms and show that, depending on the parameter range, one may find either a quadratic model of chaotic inflation or monomial models of chaotic inflation with fractional powers between 1 and 2.

We consider N=2 supergravity theories that have the same spectrum as the R+R^2 supergravity, as predicted from the off-shell counting of degrees of freedom. These theories describe standard N=2 supergravity coupled to one or two long massive vector multiplets. The central charge is not gauged in these models and they have a Minkowski vacuum with N=2 unbroken supersymmetry. The gauge symmetry, being non-compact, is always broken. $\alpha$-deformed inflaton potentials are obtained, in the case of a single massive vector multiplet, with $\alpha=1/3$ and $2/3$. The $\alpha=1$ potential (i.e. the Starobinsky potential) is also obtained, but only at the prize of having a single massive vector and a residual unbroken gauge symmetry. The inflaton corresponds to one of the Cartan fields of the non-compact quaternionic-Kähler cosets.

The recent detection by the BICEP2 collaboration of a high level of tensor modes seems to exclude the Starobinsky model of inflation. In this paper we show that this conclusion can be avoided: one can embed the Starobinsky model in supergravity and identify the inflaton field with the imaginary (instead of the real) part of the chiral scalaron multiplet in its formulation. Once coupled to matter, the Starobinsky model may then become the chaotic quadratic model with shift symmetry during inflation and is in good agreement with the current data.

We construct a supergravity model whose scalar degrees of freedom arise from a chiral superfield and are solely a scalaron and an axion that is very heavy during the inflationary phase. The model includes a second chiral superfield $X$, which is subject however to the constraint $X^2=0$ so that it describes only a Volkov - Akulov goldstino and an auxiliary field. We also construct the dual higher - derivative model, which rests on a chiral scalar curvature superfield ${\cal R}$ subject to the constraint ${\cal R}^2=0$, where the goldstino dual arises from the gauge - invariant gravitino field strength as $\gamma^{mn} {\cal D}_m \psi_n$. The final bosonic action is an $R+R^2$ theory involving an axial vector $A_m$ that only propagates a physical pseudoscalar mode.

We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in N-extended supergravity in various space-time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits-Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space-time dimension of the theory. These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.

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.

We consider global issues in minimal supergravity models where a single field inflaton potential emerges. In a particular case we reproduce the Starobinsky model and its description dual to a certain formulation of R+R^2 supergravity. For definiteness we confine our analysis to spaces at constant curvature, either vanishing or negative. Five distinct models arise, two flat models with respectively a quadratic and a quartic potential and three based on the SU(1,1)/U(1) space where its distinct isometries, elliptic, hyperbolic and parabolic are gauged. Fayet-Iliopoulos terms are introduced in a geometric way and they turn out to be a crucial ingredient in order to describe the de Sitter inflationary phase of the Starobinsky model.

We review equivalent formulations of nonlinear and higher derivatives theories of electromagnetism exhibiting electric-magnetic duality rotations symmetry. We study in particular on shell and off shell formulations of this symmetry, at the level of action functionals as well as of equations of motion. We prove the conjecture that the action functional leading to Born-Infeld nonlinear electromagnetism, that is duality rotation invariant off shell and that is known to be a root of an algebraic equation of fourth order, is a hypergeometric function.

We discuss an ${\cal R}+{\cal R}^n$ class of modified ${\cal N}=1$, D=4 supergravity models where the deformation is a monomial ${\cal R}^n\big|_F$ in the chiral scalar curvature multiplet ${\cal R}$ of the "old minimal" auxiliary field formulation. The scalaron and goldstino multiplets are dual to each other in this theory. Since one of them is not dynamical, this theory, as recently shown, cannot be used as the supersymmetric completion of $R+R^n$ gravity. This is confirmed by investigating the scalar potential and its critical points in the dual standard supergravity formulation with a single chiral multiplet with specific Kähler potential and superpotential. We study the vacuum structure of this dual theory and we find that there is always a supersymmetric Minkowski critical point which however is pathological for $n\geq 3$ as it corresponds to a corner ($n=3$) and a cusp ($n>3$) point of the potential. For $n>3$ an anti-de Sitter regular supersymmetric vacuum emerges. As a result, this class of models are not appropriate to describe inflation. We also find the mass spectrum and we provide a general formula for the masses of the scalars of a chiral multiplet around the anti-de Sitter critical point and their relation to $osp(1,4)$ unitary representations.

We revisit and clarify the supersymmetric versions of $R+ R^2$ gravity, in view of the renewed interest to these models in cosmology. We emphasize that the content of the dual standard supergravity theory in the old minimal formulation necessarily includes two massive chiral multiplets, that we call the inflaton and the goldstino. We point out that the presence of these multiplets is model independent in the old minimal formulation and therefore any theory that contains a single chiral multiplet fails to be a supersymmetric generalization of the $R+R^2$ gravity. The supergravity interactions of the two chiral multiplets are encoded in a superpotential mass term and an arbitrary Kahler potential for the goldstino multiplet. The implication for cosmology of the supersymmetric $R+R^2$ gravity is also discussed.

We discuss a supersymmetry breaking mechanism for N = 1 theories triggered by higher dimensional op- erators. We consider such operators for real linear and chiral spinor superfields that break superymmetry and reduce to the Volkov-Akulov action. We also consider supersymmetry breaking induced by a higher dimensional operator of a nonminimal scalar (complex linear) multiplet. The latter differs from the stan- dard chiral multiplet in its auxiliary sector, which contains, in addition to the complex scalar auxiliary of a chiral superfield, a complex vector and two spinors auxiliaries. By adding an appropriate higher di- mension operator, the scalar auxiliary may acquire a nonzero vev triggering spontaneous supersymmetry breaking. We find that the spectrum of the theory in the supersymmetry breaking vacuum consists of a free chiral multiplet and a constraint chiral superfield describing the goldstino. Interestingly, the latter turns out to be one of the auxiliary fermions, which becomes dynamical in the supersymmetry breaking vacuum. In all cases we are considering here, there is no sgoldstino mode and thus the goldstino does not have a superpartner. The sgoldstino is decoupled since the goldstino is one of the auxiliaries, which is propagating only in the supersymmetry breaking vacuum. We also point out how higher dimension operators introduce a potential for the propagating scalar of the theory.

We study higher order corrections in new minimal supergravity models of a single scalar field inflation. The gauging in these models leads to a massive vector multiplet and the D-term potential for the inflaton field with a coupling g^2 ~ 10^-10. In the de-Higgsed phase with vanishing g^2, the chiral and vector multiplets are non-interacting, and the potential vanishes. We present generic manifestly supersymmetric higher order corrections for these models. In particular, for a supersymmetric gravity model -R+ R^2 we derive manifestly supersymmetric corrections corresponding to R^n. The dual version corresponds to a standard supergravity model with a single scalar and a massive vector. It includes, in addition, higher Maxwell curvature/scalar interaction terms of the Born-Infeld type and a modified D-term scalar field potential. We use the dual version of the model to argue that higher order corrections do not affect the last 60 e-foldings of inflation; for example the \xi R^4 correction is irrelevant as long as \xi< 10^24.

We present a superconformal master action for a class of supergravity models with one arbitrary function defining the Jordan frame. It leads to a gauge-invariant action for a real vector multiplet, which upon gauge fixing describes a massive vector multiplet, or to a dual formulation with a linear multiplet and a massive tensor field. In both cases the models have one real scalar, the inflaton, naturally suited for single-field inflation. Vectors and tensors required by supersymmetry to complement a single real scalar do not acquire vev's during inflation, so there is no need to stabilize the extra scalars which are always present in the theories with chiral matter multiplets. The new class of models can describe any inflaton potential which vanishes at its minimum and grows monotonically away from the minimum. In this class of supergravity models one can fit any desirable choice of inflationary parameters n_s and r.

In 4-dimensional supergravity theories, covariant under symplectic electric-magnetic duality rotations, a significant role is played by the symplectic matrix M(\phi), related to the coupling of scalars \phi to vector field-strengths. In particular, this matrix enters the twisted self-duality condition for 2-form field strengths in the symplectic formulation of generalized Maxwell equations in the presence of scalar fields. In this investigation, we compute several properties of this matrix in relation to the attractor mechanism of extremal (asymptotically flat) black holes. At the attractor points with no flat directions (as in the N = 2 BPS case), this matrix enjoys a universal form in terms of the dyonic charge vector Q and the invariants of the corresponding symplectic representation RQ of the duality group G, whenever the scalar manifold is a symmetric space with G simple and non-degenerate of type E7. At attractors with flat directions, M still depends on flat directions, but not MQ, defining the so-called Freudenthal dual of Q itself. This allows for a universal expression of the symplectic vector field strengths in terms of Q, in the near-horizon Bertotti-Robinson black hole geometry.

We present a systematic study of nonlinear and higher derivatives extensions of electromagnetism. We clarify when action functionals S[F] can be explicitly obtained from arbitrary (not necessarily self-dual) nonlinear equations of motion. We show that the "Deformed twisted self-duality condition" proposal originated in the context of supergravity counterterms is actually the general framework needed to discuss self-dual theories starting from a variational principle. We generalize to nonlinear and higher derivatives theories Schroedinger formulation of Born-Infeld theory, and for the latter, and more in general for nonlinear theories, we derive a closed form expression of the corresponding deformed twisted self-duality conditions. This implies that the hypergeometric expression entering these duality conditions and leading to Born-Infeld theory satisfies a hidden quartic equation.

The global U-dualities of extended supergravity have played a central role in differentiating the distinct classes of extremal black hole solutions. When the U-duality group satisfies certain algebraic conditions, as is the case for a broad class of supergravities, the extremal black holes enjoy a further symmetry known as Freudenthal duality (F-duality), which although distinct from U-duality preserves the Bekenstein-Hawking entropy. Here it is shown that, by adopting the doubled Lagrangian formalism, F-duality, defined on the doubled field strengths, is not only a symmetry of the black hole solutions, but also of the equations of motion themselves. A further role for F-duality is introduced in the context of world-sheet actions. The Nambu-Goto world-sheet action in any (t, s) signature spacetime can be written in terms of the F-dual. The corresponding field equations and Bianchi identities are then related by F-duality allowing for an F-dual formulation of Gaillard-Zumino duality on the world-sheet. An equivalent polynomial "Polyakov- type" action is introduced using the so-called black hole potential. Such a construction allows for actions invariant under all groups of type E7, including E7 itself, although in this case the stringy interpretation is less clear.

We propose a first order formalism for multi-centered black holes with flat tree-dimensional base-space, within the stu model of N=2, D=4 ungauged Maxwell-Einstein supergravity. This provides a unified description of first order flows of this universal sector of all models with a symmetric scalar manifold which can be obtained by dimensional reduction from five dimensions. We develop a D=3 Cartesian formalism which suitably extends the definition of central and matter charges, as well as of black hole effective potential and first order "fake" superpotential, in order to deal with not necessarily axisimmetric solutions, and thus with multi-centered and/or (under-)rotating extremal black holes. We derive general first order flow equations for composite non-BPS and almost BPS classes, and we analyze some of their solutions, retrieving various single-centered (static or under-rotating) and multi-centered known systems. As in the t^3 model, the almost BPS class turns out to split into two general branches, and the well known almost BPS system is shown to be a particular solution of the second branch.

We analyze some properties of the four dimensional supergravity theories which originate from five dimensions upon reduction. They generalize to N>2 extended supersymmetries the d-geometries with cubic prepotentials, familiar from N=2 special Kähler geometry. We emphasize the role of a suitable parametrization of the scalar fields and the corresponding triangular symplectic basis. We also consider applications to the first order flow equations for non-BPS extremal black holes.

We argue that the observed UV finiteness of the 3-loop extended supergravities may be a manifestation of a hidden local superconformal symmetry of supergravity. We focus on the SU(2,2|4) dimensionless superconformal model. In Poincare gauge where the compensators are fixed to phi^2= 6 M_P^2 this model becomes a pure classical N=4 Einstein supergravity. We argue that in N=4 the higher-derivative superconformal invariants like phi^-4W^2 \bar W^2 and the consistent local anomaly delta (ln phi W^2) are not available. This conjecture on hidden local N=4 superconformal symmetry of Poincare supergravity may be supported by subsequent loop computations.

We study N=2 supergravity deformed by a genuine supersymmetric completion of the $\lambda R^4$ term, using the underlying off shell N=2 superconformal framework. The gauge-fixed superconformal model has unbroken local supersymmetry of N=2 supergravity with higher derivative deformation. Elimination of auxiliary fields leads to the deformation of the supersymmetry rules as well as to the deformation of the action, which becomes a Born-Infeld with higher derivative type action. We find that the gravitino supersymmetry deformation starts from $\lambda \, \pa^4 {\cal F}^3$ and has higher graviphoton couplings. In the action there are terms $\lambda^2 \pa^8 {\cal F}^{6}$ and higher, in addition to original on shell counterterm deformation. These deformations are absent in the on shell superspace and in the candidate on shell counterterms of N=4,~8 supergravities, truncated down to N=2. We conclude therefore that the undeformed on shell superspace candidate counterterms break the N=2 part of local supersymmetry.

By exploiting the Jordan pair structure of U-duality Lie algebras in D = 3 and the relation to the super-Ehlers symmetry in D = 5, we elucidate the massless multiplet structure of the spectrum of a broad class of D = 5 supergravity theories. Both simple and semi-simple, Euclidean rank-3 Jordan algebras are considered. Theories sharing the same bosonic sector but with different supersymmetrizations are also analyzed.

After a brief introduction to the Attractor Mechanism, we review the appearance of groups of type E7 as generalized electric-magnetic duality symmetries in locally supersymmetric theories of gravity, with particular emphasis on the symplectic structure of fluxes in the background of extremal black hole solutions, with one or two centers. In the latter case, the role of an "horizontal" symmetry SL(2,R) is elucidated by presenting a set of two-centered relations governing the structure of two-centered invariant polynomials.

We classify the enhanced helicity symmetry of the Ehlers group to extended supergravity theories in any dimension. The vanishing character of the pseudo-Riemannian cosets occurring in this analysis is explained in terms of Poincaré duality. The latter resides in the nature of regularly embedded quotient subgroups which are non-compact rank preserving.

We study properties of D = 4, N >1 extended supergravities (and related compactifications of superstring theory) and their consistent truncation to the phenomenologically interesting models of N = 1 supergravity. This involves a detailed classification of the "degenerations" of the duality groups of type E7, when the corresponding quartic invariant polynomial built from the symplectic irreducible representation of G4 "degenerates" into a perfect square. With regard to cosmological applications, we conclude that the consistent truncation to N = 1 from higher-dimensional or higher-N theory gives a zero measure minimal coupling of vectors. A non-minimal coupling involving vectors coupled to scalars and axions is generic. These features of supergravity, following from the electric-magnetic duality, may be useful in other applications, like stabilization of moduli, and in studies of non-perturbative black-hole solutions of supergravity/string theory.

Understanding the consequences of the E_7(7) duality on the UV properties of N=8 supergravity requires unravelling when and how duality-covariant actions can be constructed so as to accommodate duality-invariant counter-terms. For non-supersymmetric abelian gauge theories exhibiting U(1)-duality, with and without derivative couplings, it was shown that such a covariant construction is always possible. In this paper we describe a similar procedure for the construction of covariant non-linear deformations of U(1)-duality invariant theories in the presence of rigid N=2 supersymmetry. This is a concrete step towards studying the interplay of duality and extended supersymmetry.

We report on some old and new results on the quantum aspects of four-dimensional maximal supergravity, and its hypothetical ultraviolet finiteness.

Some aspects of black holes in supersymmetric theories of gravity are reviewed and some recent results outlined.

We report some results on the relation between extremal black holes in locally supersymmetric theories of gravity and groups of type E7, appearing as generalized electric-magnetic duality symmetries in such theories. Some basics on the covariant approach to the stratification of the relevant symplectic representation are reviewed, along with a connection between special Kaehler geometry and a "generalization" of groups of type E7.

We study "minimal degree" complete bases of duality- and "horizontal"- invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D=4 Einstein supergravity models with symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL(2,R) "horizontal" symmetry. The first class encompasses N=2 and N=4 matter-coupled theories, with semi-simple U-duality given by SL(2,R) x SO(m,n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial "horizontal" stabilizer SO(2), includes N=2 minimally coupled and N=3 matter coupled theories, with U-duality given by the pseudo-unitary group U(r,s) (related to complex flux representations). Finally, we comment on the formulation of special Kaehler geometry in terms of "generalized" groups of type E7.

We relate the mechanism of matter creation in the universe after inflation to a simple and universal mathematical property of extended N > 1 supergravities and related compactifications of superstring theory. We show that in all such models, the inflaton field may decay into vector fields due to a nonminimal scalar-vector coupling. This coupling is compulsory for all scalars except N=2 hyperscalars. The proof is based on the fact that all extended supergravities described by symmetric coset spaces G/H have duality groups G of type E7, with exception of U(p,n) models. For N=2 we prove separately that special geometry requires a non-minimal scalar-vector coupling. Upon truncation to N=1 supergravity, extended models generically preserve the non-minimal scalar-vector coupling, with exception of U(p,n) models and hyperscalars. For some string theory/supergravity inflationary models, this coupling provides the only way to complete the process of creation of matter in the early universe.

We find general relations between the on-shell gravitational trace anomaly A_N, and the logarithmic correction Delta S_N to the entropy of "large" BPS extremal black holes in N>1 supergravity theories in D=4 space-time dimensions (recently computed by Sen [arXiv:1108.3842]). For (generalized) self-mirror theories (all having A_N = 0), we obtain the result DeltaS_N = - Delta S_(8-N) = 2 - N/2, whereas for generic theories the trace anomaly tildeA_N of the fully dualized theory turns out to coincide with 2Delta S_N, up to a model-independent shift: tildeA_N = 2Delta S_N - 1. We also speculate on N=1 theories displaying "large" extremal black hole solutions.