We study gravity duals to a broad class of N = 2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector, finding a simple closed formula when the solution has U(1) × U(1) symmetry. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localization. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories with a large N limit, defined on a general class of background three-manifold geometries.

We study a general class of supersymmetric AdS 4 × Y 7 solutions of M-theory that have large N dual descriptions as N = 2 Chern-Simons-matter theories on S 3 . The Hamiltonian function h M for the M-theory circle, with respect to a certain contact structure on Y 7 , plays an important role in the duality. We show that an M2-brane wrapping the M-theory circle, giving a fundamental string in AdS 4 , is supersymmetric precisely at the critical points of h M , and moreover the value of this function at the critical point determines the M2-brane action. Such a configuration determines the holographic dual of a BPS Wilson loop for a Hopf circle in S 3 , and leads to an effective method for computing the Wilson loop on both sides of the correspondence in large classes of examples. We find agreement in all cases, including for several infinite families, and moreover we find that the image h M (Y 7 ) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of h M mapping to points where the derivative of the eigenvalue density is discontinuous.

We compute the large N limit of Wilson loop expectation values for a broad class of N = 2 supersymmetric gauge theories defined on a general class of background three-manifolds M 3 , diffeomorphic to S 3 . We find a simple closed formula which depends on the background geometry only through a certain supersymmetric Killing vector field. The supergravity dual of such a Wilson loop is an M2-brane wrapping the M-theory circle, together with a complex curve Σ 2 in a self-dual Einstein manifold M 4 , whose conformal boundary is M 3 . We show that the regularized action of this M2-brane also depends only on the supersymmetric Killing vector, precisely reproducing the large N field theory computation.

We study the geometry of the scalar manifolds emerging in the no-scale sector of Kähler moduli and matter fields in generic Calabi-Yau string compactifications, and describe its implications on scalar masses. We consider both heterotic and orientifold models and compare their characteristics. We start from a general formula for the Kähler potential as a function of the topological compactification data and study the structure of the curvature tensor. We then determine the conditions for the space to be symmetric and show that whenever this is the case the heterotic and the orientifold models give the same scalar manifold. We finally study the structure of scalar masses in this type of geometries, assuming that a generic superpotential triggers spontaneous supersymmetry breaking. We show in particular that their behavior crucially depends on the parameters controlling the departure of the geometry from the coset situation. We first investigate the average sGoldstino mass in the hidden sector and its sign, and study the implications on vacuum metastability and the mass of the lightest scalar. We next examine the soft scalar masses in the visible sector and their flavor structure, and study the possibility of realizing a mild form of sequestering relying on a global symmetry.