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We give a pedagogical review of the localization of supersymmetric gaugetheory on 5d toric Sasaki-Einstein manifolds. We construct the cohomologicalcomplex resulting from supersymmetry and consider its natural toricdeformations with all equivariant parameters turned on. We also give detaileddiscussion on how the Sasaki-Einstein geometry permeates every aspect of thecalculation, from Killing spinor, vanishing theorems to the index theorems.

Review of localization for 5d supersymmetric gauge theories