Localization is a powerful tool to compute physical quantities such as partition functions, free energies and expectation values of certain operators exactly at any coupling in many supersymmetric theories. Due to this merit, the technique is able to provide highly nontrivial tests of AdS/CFT correspondence. We apply localization procedure to the most general three-dimensional N = 1 Chern-Simons matter theories, which are not studied in the previous localization literature, and show that they can also be formally localized. The other focus in this body of work is the study of an important aspect of high energy physics, the higher spin theories, and their conjectured CFT duals. Higher spin theory is a remarkable extension of Einstein gravity in which mass particles of all spin are described by self-consistent and fully nonlinear field equations. We perform tests of the duality between supersymmetric higher spin theories in AdS4 and the corresponding CFTs, by comparisons of the one loop free energies on both sides. We show that the mismatch between the free energies in the duality between Type-B higher spin theory/fermionic vector model cannot be solved by the introduction of supersymmetry. We then turn to another test of the HS/CFT correspondence, by comparing the tree-level three-point functions on both sides. We produce the full structures of three-point Witten diagrams for both parity-preserving and parity-violating bosonic HS theories, and show that they match perfectly with the corresponding ones on CFT side.
