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.

We study the classical geometry associated to fractional D3-branes of type IIB string theory on IR 4 /Z 2 which provide the gravitational dual for N = 2 super Yang-Mills theory in four dimensions. As one can expect from the lack of conformal invariance on the gauge theory side, the gravitational background displays a repulson-like singularity. It turns out however, that such singularity can be excised by an enhançon mechanism. The complete knowledge of the classical supergravity solution allows us to identify the coupling constant of the dual gauge theory in terms of the string parameters and to find a logarithmic running that is governed precisely by the β function of the N = 2 super Yang-Mills theory. 1 Work partially supported by the European Commission RTN programme HPRN-CT-2000-00131 and by MURST.Contents 1 Introduction 1 2 The geometry of fractional D3-branes 2 3 The enhançon and the dual gauge theory 10

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 engineer a class of quiver gauge theories with several interesting features by studying D-branes at a simple Calabi-Yau singularity. At weak 't Hooft coupling we argue using field theory techniques that these theories admit both supersymmetric vacua and meta-stable non-supersymmetric vacua, though the arguments indicating the existence of the supersymmetry breaking states are not decisive. At strong 't Hooft coupling we find simple candidate gravity dual descriptions for both sets of vacua. B. Toric geometry, (p, q) webs and complex deformations 35C. Type IIA description 39 2 In [28], it has been suggested that a meta-stable, supersymmetry breaking vacuum also exists for N f = N c . We discuss the difficulties in the analysis of this case in §4.3 Some references which are in some respect relevant to our work are [29,30,31,32]. 4 This question has been raised by the present authors, H. Ooguri, N. Seiberg, H. Verlinde and many others.