Localization and its application to N = 1 Super Yang-Mills theory with matter on the 5-sphere

Abstract

Localization is a powerful mathematical tool to gain exact solutions for the partition function and observables on closed manifolds. The origin and validity of several index theorems will be discussed, specifically the Atiyah-Bott-Berline-Vergne theorem, and we will see how they are related to localization as introduced by E. Witten in 1988. A proof of the Poincaré-Hopf index theorem will be provided as an illustration of this method. Furthermore we will introduce a N=1 off-shell supersymmetric Yang-Mills theory with matter on the 5-sphere and show that the conditions for using localization can be satisfied to conclude by discussing how localization techniques have been used by K. Hosomichi e.a. (arXiv:1206.6008) and J.Källén e.a. (arXiv:1203.0371) to acquire exact results for the partition function.