We describe some recent investigation about the structure of generic D=4,5 theories obtained by generalized dimensional reduction of D=5,6 theories with eight supercharges. We relate the Scherk-Schwarz reduction to a special class of N=2 no-scale gauged supergravities.

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.

We consider the contraction of some nonlinear σ models which appear in effective supergravity theories. In particular we consider the contractions of maximally symmetric spaces corresponding to N=1 and N=2 theories, as they appear in certain low energy effective supergravity actions with mass deformations. The contraction procedure is shown to describe the integrating out of massive modes in the presence of interactions, as it happens in many supergravity models after spontaneous supersymmetry breaking.

The aim of this paper is to make progress in the understanding of the Scherk-Schwarz dimensional reduction in terms of a compactification in the presence of background fluxes and torsion. From the eleven dimensional supergravity point of view, we find that a general E-6(6) (S-S phase may be obtained by turning on an appropriate background torsion, together with suitable fluxes, some of which can be directly identified with certain components of the four-form field-strength. Furthermore, we introduce a novel (four dimensional) approach to the study of dualities between flux/torsion compactifications of type II/M-theory. This approach defines the action that duality should have on the background quantities, in order for the E)(7(7)) invariance of the field equations and Bianchi identities to be restored also in the presence of fluxes/torsion. This analysis further implies the interpretation of the torsion flux as the T-dual of the NS three-form flux.

We give the N = 2 gauged supergravity interpretation of a generic D = 4, N = 2 theory as it comes from the generalized Scherk-Schwarz reduction of D = 5, N = 2 (ungauged) supergravity. We focus on the geometric aspects of the D = 4 data such as the general form of the scalar potential and masses in terms of the gauging of a 'flat group'. Higgs and super-Higgs mechanisms are discussed in some detail.

We compute the extension of the lagrangian of N = 1 supersymmetric theories to the case in which some axion symmetries are gauged. It turns out that generalized Chern-Simons terms appear that were not considered in previous superspace formulations of general N = 1 theories. Such gaugings appear in supergravities arising from flux compactifications of superstrings, as well as from Scherk-Schwarz generalized dimensional reduction in M-theory. We also present the dual superspace formulation where axion chiral multiplets are dualized into linear multiplets.

We perform a generalized dimensional reduction of six dimensional supergravity theories to five dimensions. We consider the minimal (2,0) and the maximal (4,4) theories. In each case the reduction allows us to obtain gauged supergravities of no-scale type in dimension five with gauge groups that escape previous classifications. In the minimal case, the geometric data of the reduced theory correspond to particular cases of the D = 5 real special geometry. In the maximal case we find a four parameter solution which allows partial breaking of supersymmetry.

Non commutative superspaces can be introduced as the Moyal-Weyl quantization of a Poisson bracket for classical superfields. Different deformations are studied corresponding to constant background fields in string theory. Supersymmetric and non supersymmetric deformations can be defined, depending on the differential operators used to define the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N > 1), some new deformations can be defined, with no analogue in the N = 1 case.

We consider the interplay of duality symmetries and gauged isometries of supergravity models giving N-extended, spontaneously broken supergravity with a no-scale structure. Some examples motivated by superstring and M-theory compactifications are described.

We derive the scalar potential of the effective theory of type IIB orientifold with 3-form fluxes turned on in presence of non abelian brane coordinates. N=4 supergravity predicts a positive semidefinite potential with vanishing cosmological constant in the vacuum of commuting coordinates, with a classical moduli space given by three radial moduli and three RR scalars which complete three copies of the coset (U(1,1+n)/U(1)\otimes U(1+n)), together with 6n D3-branes coordinates, n being the rank of the gauge group G. Implications for the super Higgs mechanism are also discussed.

In the context of N = 2 supergravity we explain the occurrence of partial super-Higgs with vanishing vacuum energy and moduli stabilization in a model suggested by superstring compactificat ions on type-IIB orientifolds with 3-form fluxes. The gauging of axion symmetries of the quaternionic manifold, together with the use of degenerate symplectic sections for special geometry, are the essential ingredients of the construction.

We analyze N = 2;1;0 vacua of type-IIB string theory on K3£T 2 =Z2 ori- entifold in presence of three-form ∞uxes from a four dimensional supergravity viewpoint. The quaternionic geometry of the K3 moduli space together with the special geometry of the NS and R-R dilatons and of the T 2 -complex structure moduli play a crucial role in the analysis. The introduction of ∞uxes corresponds to a particular gauging of N = 2, D = 4 supergravity. Our results agree with a recent work of Tripathy and Trivedi. The present formulation shows the power of supergravity in the study of efiective theories with broken supersymmetry.

We show that N=8 spontaneously broken supergravity in four dimensions obtained by Scherk-Schwarz generalized dimensional reduction can be obtained from a pure four dimensional perspective by gauging a suitable electric subgroup of E_{7,7}. Owing to the fact that there are non isomorphic choices of maximal electric subgroups of the U-duality group their gaugings give rise to inequivalent theories. This in particular shows that the Scherk-Schwarz gaugings do not fall in previous classifications of possible gauged N=8 supergravities. Gauging of flat groups appear in many examples of string compactifications in presence of brane fluxes.

We make a short review on the Moyal–Weyl star products with emphasis in its convergence. We consider field theories on a deformed superspace and analyze the compatibility of such products with supersymmetry. We investigate some aspects of the Wess–Zumino model, super-Yang–Mills theories and analyze the correspondence of the latter with the supersymmetric Born–Infeld action.

In this paper we study a family of algebraic deformations of regular coadjoint orbits of compact semisimple Lie groups with the Kirillov Poisson bracket. The deformations are restrictions of deformations on the dual of the Lie algebra. We prove that there are non isomorphic deformations in the family. The star products are not differential, unlike the star products considered in other approaches. We make a comparison with the differential star product canonically defined by Kontsevich's map.

We consider simple superalgebras which are a supersymmetric extension of the spin algebra in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincare algebra as a contraction and another as a subalgebra. Because of the contraction property, some of these algebras can be interpreted as de Sitter or anti de Sitter superalgebras. However, the number of odd generators present in the contraction is not always minimal due to the different splitting properties of the spinor representations under a subalgebra. We consider the general case, with arbitrary dimension and signature, and examine in detail particular examples with physical implications in dimensions d = 10 and d = 4.

In this paper, an algebraic and a differential star product defined on a regular coadjoint orbit of a compact semisimple group are compared. It has been proved that there is an injective algebra homomorphism between the algebra of polynomials with the algebraic star product and the algebra of differential functions with the differential star product structure.

It is shown that the super Higgs mechanism that occurs in a wide class of models with vanishing cosmological constant (at the classical level) is obtained by the gauging of a flat group which must be an electric subgroup of the duality group. If the residual massive gravitinos which occur in the partial supersymmetry breaking are BPS saturated, then the flat group is nonabelian. This is so for all the models obtained by a Scherk–Schwarz supersymmetry breaking mechanism. If gravitinos occur in long multiplets, then the flat groups may be abelian. This is the case of supersymmetry breaking by string compactifications on an orientifold T6/Z2 with nontrivial brane fluxes.

We consider the reduction of supersymmetry in N-extended four dimensional supergravity via the super Higgs mechanism in theories without cosmological constant. We provide an analysis largely based on the properties of long and short multiplets of Poincare' supersymmetry. Examples of the super Higgs phenomenon are realized in spontaneously broken N=8 supergravity through the Scherk-Schwarz mechanism and in superstring compactification in presence of brane fluxes. In many models the massive vectors count the difference in number of the translation isometries of the scalar sigma-model geometries in the broken and unbroken phase.

In this paper we consider the problem of deformation quantization of the algebra of polynomial functions on coadjoint orbits of semisimple lie groups. The deformation of an orbit is realized by taking the quotient of the universal enveloping algebra of the Lie algebra of the given Lie group, by a suitable ideal. A comparison with geometric quantization in the case of SU(2) is done where both methods agree.