Abstract. We prove conjecture due to Erickson-Semenoff-Zarembo and DrukkerGross which relates supersymmetric circular Wilson loop operators in the N = 4 supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N = 2 and the N = 2 * supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N = 2 superconformal gauge theory is treated similarly.

This is the foreword to the special volume on localization techniques in quantum field theory. The summary of individual chapters is given and their interrelation is discussed.

We localize the four-dimensional N = 4 super Yang-Mills theory on S 4 to the two-dimensional constrained Hitchin/Higgs-Yang-Mills (cHYM) theory on S 2 . We show that expectation values of certain 1/8 BPS supersymmetric Wilson loops on S 2 in the 4d N = 4 SYM is captured by the 2d cHYM theory. We further argue that expectation values of Wilson loops in the cHYM theory agree with the prescription "two-dimensional bosonic Yang-Mills excluding instanton contributions". Hence, we support the recent conjecture by Drukker, Giombi, Ricci and Trancanelli on the 1/8 BPS Wilson loops on S 2 in the 4d N = 4 SYM. * On leave of absence from ITEP, 117218, Moscow, Russia

We study macroscopically two dimensional N = (2, 2) supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product T d × R 2 ǫ of a d-dimensional torus and a two dimensional cigar with Ω-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on R 2−d × T 2+d and singular monopoles on R 2−d × T 1+d , for d = 0, 1, 2, and the Yangian Y ǫ (g Γ ), quantum affine algebra U aff q (g Γ ), or the quantum elliptic algebra U ell q,p (g Γ ) associated to Kac-Moody algebra g Γ for quiver Γ.

The path integral of a general N = 2 supersymmetric gauge theory on S 4 is exactly evaluated in the presence of a supersymmetric 't Hooft loop operator. The result we find -obtained using localization techniques -captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S 4 . As a by-product, our gauge theory calculations successfully confirm the predictions made for 't Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.