Supersymmetric Localization: Review and Recent Progress

Supersymmetric localization is a powerful tool that can provide us with some exact non-perturbative results of quantum field theories. In the first half of this talk we will review its basic idea and some important applications. The results of localization can be used to test some conjectured dualities, for instance the Seiberg-like dualities in various dimensions, and also to help set up new dualities. We will demonstrate these aspects with some examples. In the second half of this talk, we will discuss some recent progress on the localization of 4d N=1 gauge theories. We consider 4d N=1 gauge theories on S2*R2 and compute their partition functions. The results are related to the partition functions of 4d N=1 gauge theories on the Omega background with two epsilon parameters, which can be viewed as building blocks of the Nekrasov partition functions of the 4d N=2 theories on the Omega background. As an application, we use the N=1 partition functions to test various dualities such as the 4d Seiberg duality. Further applications to the 4d Argyres-Douglas theories and the generalized AGT relation will also be discussed. This talk is based on a few papers 1309.3266, 1411.4694, 1505.06207, 1705.01896 and some work in progress.