Holomorphic Chern–Simons theory and noncommutative geometry

In this talk I will explain how quiver gauge theories in even dimensions (up to ten) have holomorphic Chern–Simons theory as their common origin. The quiver gauge theories realize non-commutative resolutions of CY n-folds. Finally, I will explain the relationship between supersymmetric indices of the field theories and negative cyclic homology of the CY n-folds. For four dimensional quiver guage theories in the large-N limit, the relationship between the supersymmetric index of the quiver theory and negative cyclic homology gives a mathematical proof of the matching of local operators between the field theory and its holographic dual under the AdS/CFT correspondence.