Supersymmetric Tools in Yang-Mills Theories at Strong Coupling: the Beginning of a Long Journey

Development of holomorphy-based methods in super-Yang-Mills theories started in the early 1980s and lead to a number of breakthrough results. I review some results in which I participated. The discovery of Seiberg’s duality and the Seiberg-Witten solution of $\mathcal{N}=2$ Yang-Mills were the milestones in the long journey of which, I assume, much will be said in other talks. I will focus on the discovery (2003) of non-Abelian vortex strings with various degree of supersymmetry, supported in some four-dimensional Yang-Mills theories and some intriguing implications of this discovery. One of the recent results is the observation of a soliton string in the bulk $\mathcal{N}=2$ theory with the $U(2)$ gauge group and four flavors, which can become critical in a certain limit. This is the case of a “reverse holography,” with a very transparent physical meaning.