Quantum geometry and quiver gauge theories

International audienceWe study macroscopically two dimensional ${\mathcal{N}=(2,2)}$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product ${\mathbb{T}^{d} \times \mathbb{R}^{2\epsilon}}$ of a d-dimensional torus and a two dimensional cigar with ${\Omega}$ -deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories and the Yangian ${\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})}$ , quantum affine algebra ${\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})}$ , or the quantum elliptic algebra ${\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})}$ associated to Kac–Moody algebra ${\mathfrak{g}_{\Gamma}}$ for quiver ${\Gamma}$