Title:

Wilson loops and their gravity duals in AdS_4/CFT_3

In the first part of this thesis, we study the duality of Wilson loops and M2branes in AdS_{4}/CFT_{3}. We focus on supersymmetric Mtheory solutions on AdS_{4}xY_{7} that have a superconformal dual description on S^{3} = ?AdS^{4}. We will find that the Hamiltonian function h_{M} for the Mtheory circle plays an important role in the duality. We show that an M2brane wrapping the Mtheory circle is supersymmetric precisely at the critical points of h_{M}, and moreover the value of this function at those points determines the M2brane actions. Such a configuration determines the holographic dual of a Wilson loop for a Hopf circle in S^{3}. We find agreement in large classes of examples between the Wilson loop and its dual M2brane and also 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 will then move away from the threesphere and construct gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of threemanifold geometries. The gravity backgrounds are based on Euclidean selfdual solutions to fourdimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the fourball 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 threemanifold geometries. To further verify that our gravitational backgrounds are indeed dual to field theories on their boundaries, we compute Wilson loops and their M2brane 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 M2brane precisely reproduces this large N field theory result. This constitutes a further check of AdS_{4}/CFT_{3} for a very broad class of examples.
