Quantum geometry and quiver gauge theories (1312.6689v1)

Abstract: We 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, quantization of the moduli spaces of instantons on $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^{1+d}$, for $d=0,1,2$, 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 $\Gamma$.