Review of localization for 5D supersymmetric gauge theories

Authors: Jian Qiu and Maxim Zabzine Preprint Number: UUITP-09/16 We give a pedagogical review of the localization of supersymmetric gauge theory on 5D toric Sasaki-Einstein manifolds. We construct  the cohomological complex resulting from supersymmetry  and consider its natural toric deformations with all equivariant parameters turned on. We also give detailed discussion on how the Sasaki-Einstein geometry permeates every aspect of the calculation, from Killing spinor, vanishing theorems to the index theorems.