Wilson loops and their gravity duals in AdS4/CFT3

In the first part of this thesis, we study the duality of Wilson loops and M2-branes in AdS4/CFT3. We focus on supersymmetric M-theory solutions on AdS4xY7 that have a superconformal dual description on S3 = ∂AdS4. We will find that the Hamiltonian function hM for the M-theory circle plays an important role in the duality. We show that an M2-brane wrapping the M-theory circle is supersymmetric precisely at the critical points of hM, and moreover the value of this function at those points determines the M2-brane actions. Such a configuration determines the holographic dual of a Wilson loop for a Hopf circle in S3. We find agreement in large classes of examples between the Wilson loop and its dual M2-brane and also that the image hM(Y7) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of hM mapping to points where the derivative of the eigenvalue density is discontinuous. We will then move away from the three-sphere and construct 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. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localisation. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories defined on a general class of background three-manifold geometries. To further verify that our gravitational backgrounds are indeed dual to field theories on their boundaries, we compute Wilson loops and their M2-brane duals in this general setting. We find that the Wilson loop is given by a simple closed formula which depends on the background geometry only through the supersymmetric Killing vector field. The supergravity dual M2-brane precisely reproduces this large N field theory result. This constitutes a further check of AdS4/CFT3 for a very broad class of examples.