Quantum geometry and quiver gauge theories

We study macroscopically two dimensional N = (2 , 2) supersymmetric gauge theories constructed by compactifying the quiver gau ge theories with eight su- percharges on a product T d ? R 2 ? of a d -dimensional torus and a two dimensional cigar with ? -deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondenc e between the gauge the- ories, 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 ? ( g ? ) , quantum affine algebra U aff q ( g ? ) , or the quantum elliptic algebra U ell q,p ( g ? ) associated to Kac-Moody algebra g ? for quiver