In the first part of this thesis, we study the duality of Wilson loops and M2-branes in AdS<sub>4</sub>/CFT<sub>3</sub>. We focus on supersymmetric M-theory solutions on AdS<sub>4</sub>xY<sub>7</sub> that have a superconformal dual description on S<sup>3</sup> = ?AdS<sup>4</sup>. We will find that the Hamiltonian function h<sub>M</sub> 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 h<sub>M</sub>, 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 S<sup>3</sup>. We find agreement in large classes of examples between the Wilson loop and its dual M2-brane and also that the image h<sub>M</sub>(Y<sub>7</sub>) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of h<sub>M</sub> 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 AdS<sub>4</sub>/CFT<sub>3</sub> for a very broad class of examples.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:658520 |
| Date | January 2015 |
| Creators | Farquet, Daniel |
| Contributors | Sparks, James |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:0241dcf6-166d-473d-8528-c3b1e52d3bc1 |
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