The general form of N = 2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Lagrangians for rigid theories are also written in this general setting and the connection with local theories elucidated. The derivation of these results using geometrical techniques is briefly summarized.

All rights reserved. This boolc, or parts tlwreof, may 1101 be reproduced in any form orhyanymeans,electronicormechanit:lll,incllldifllphatocopying,recordingorany information storage and retrieWJI system IIOW ki!Own or to be invented, without written pcrmissiOfl from tlw Publislwr.ISBN 9971-S0-037-X (set) Printed in Singapore by Loi Printing Pte. Ltd. v PREFACEIn our hopes the present book is a self-contained account of the theory of supergravity and of the theory of superstrings. It is meant to be both introductory and advanced, this feature explaining its considerable length.The authors' views on the relevance of the whole subject can be related in a few words. On one hand we feel that reconciling quantum mechanics with general relativity is a logical necessity one cannot overlook, while trying to explain the structure of the other interactions. As far as we know the only serious candidate for a quantum theory of the graviton is superstring theory, whose low-energy approximation is supergravity. On the other hand, provided Higgs fields are necessary to explain spontaneous symmetry breaking, the only satisfactory solution of the gauge hierarchy problem seems to be given by spontaneously broken local supersymmetry. From the two opposite sides ofthe energy scale we come to the same suggestion: particle physics phenomenology should be described in terms of an effective supergravity model.Whether such views are supported by experimental evidence is a question that might be answered in the near future as soon as the LEP machine becomes operational.In any case it should be stressed that supersymmetry is a profound symmetry principle, with far reaching implications: it has the same standpoint as the principle of general covariance and, similarly, it provides an extremely elegant framework for the formulation of the laws of Nature.Furthermore, the structure of these theories encompasses so many different aspects and it is so ramified that they will continue to be interesting theoretical laboratories in many respects.We have chosen to present the whole subject in a systematic way, aiming more at the basic principles than at the specific applications: however the eventual use of the theory for the construction of a realistic model, describing particle phenomenology, has been our constant motivation directing our choices.We have tried to be exhaustive in the discussion of the different mechanisms, and models, but not in covering the various formalisms that have been historically utilized to derive the various results. Indeed we have presented everything in a unified language emphasizing the underlying geometrical structure. Furthermore, we have included several mathematical chapters, explaining, at each stage of the theory development, the mathematics involved in the construction.The book is divided into three volumes and six parts. The first volume introduces the geometric and algebraic foundations of supergravity (Parts One and Two, respectively). The second volume is devoted to the construction of supergravity ...

In this paper we discuss candidate superconformal N = 2 gauge theories that realize the AdS/CFT correspondence with M-theory compactified on the homogeneous Sasakian 7-manifolds M 7 that were classified long ago. In particular we focus on the two cases M 7 = Q 1,1,1 and M 7 = M 1,1,1 , for the latter the Kaluza Klein spectrum being completely known. We show how the toric description of M 7 suggests the gauge group and the supersingleton fields. The conformal dimensions of the latter can be independently calculated by comparison with the mass of baryonic operators that correspond to 5-branes wrapped on supersymmetric 5-cycles and are charged with respect to the Betti multiplets. The entire Kaluza Klein spectrum of short multiplets agrees with these dimensions. Furthermore, the metric cone over the Sasakian manifold is a conifold algebraically embedded in some C p . The ring of chiral primary fields is defined as the coordinate ring of C p modded by the ideal generated by the embedding equations; this ideal has a nice characterization by means of representation theory. The entire Kaluza Klein spectrum is explained in terms of these vanishing relations. We give the superfield interpretation of all short multiplets and we point out the existence of many long multiplets with rational protected dimensions, whose presence and pattern were already noticed in other compactifications and seem to be universal. * Supported in part by EEC under TMR contract ERBFMRX-CT96-0045 and by GNFM.

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We consider the group theoretical properties of R-R scalars of string theories in the low-energy supergravity limit and relate them to the solvable Lie subalgebra G s ⊂ U of the U-duality algebra that generates the scalar manifold of the theory: exp[G s ] = U/H. Peccei-Quinn symmetries are naturally related with the maximal abelian ideal A ⊂ G s of the solvable Lie algebra. The solvable algebras of maximal rank occurring in maximal supergravities in diverse dimensions are described in some detail. A particular example of a solvable Lie algebra is a rank one, 2(h 2,1 + 2)-dimensional algebra displayed by the classical quaternionic spaces that are obtained via c-map from the special Kählerian moduli spaces of Calabi-Yau threefolds.

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The general form of N = 2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Our results can be used to explore properties of theories admitting N = 2 supergravity as low energy limit.

We build a number of integrable one-scalar spatially flat cosmologies, which play a natural role in inflationary scenarios, examine their behavior in several cases and draw from them some general lessons on this type of systems, whose potentials involve combinations of exponential functions, and on similar non-integrable ones. These include the impossibility for the scalar to emerge from the initial singularity descending along asymptotically exponential potentials with logarithmic slopes exceeding a critical value ("climbing phenomenon") and the inevitable collapse in a big Crunch whenever the scalar tries to settle at negative extrema of the potential. We also elaborate on the links between these types of potentials and "brane supersymmetry breaking", a mechanism that ties together string scale and scale of supersymmetry breaking in a class of orientifold models.