Supersymmetric Localization and Probe Branes in the AdS/CFT correspondence

Thesis (Ph.D.)--University of Washington, 2017-07In this thesis, a precise, rigorous test of probe brane holography will be constructed. Since its discovery, the AdS/CFT correspondence has provided a window into the strongly coupled dynamics of supersymmetric gauge theories. The ability to include degrees of freedom that provide analogs for the physics of heavy quarks via the probe brane paradigm has further expanded the utility of the duality. The deformation away from a strictly conformal theory by the addition of flavor degrees of freedom induces a Landau pole outside of the `t Hooft limit where $N_c\rar \infty$ and $\frac{N_f}{N_c}\ll 1$, which invites questions about the utility of the probe brane paradigm. Following from the recent application equivariant localization to massive supersymmetric gauge theories on curved backgrounds, a precise question can be formulated to compare, e.g., the free energy of a supersymmetric probe brane embedding and that of the localized dual field theory. This thesis will apply those concepts to the D3/D7 probe brane system dual to $N_f$ $\CN=2$ fundamental hypermultiplets on an S$^4$ and the D3/D5 probe brane system dual to $N_f$ $\CN=2$ fundamental hypermultiplets living on a co-dimension one defect-- an equatorial S$^3\subset$ S$^4$. In that framework, exact matching to the localization results are found