Quantum geometry and quiver gauge theories

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 ${\mathbb{T}}^d\times {\mathbb{R}}_{\epsilon }^2$ 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 ${\mathbf{U}}_q^{\mathrm{aff}}({\mathfrak{g}}_{\Gamma })$, or the quantum elliptic algebra ${\mathbf{U}}_{q,p}^{\mathrm{ell}}({\mathfrak{g}}_{\Gamma })$ associated to Kac-Moody algebra ${\mathfrak{g}}_{\Gamma }$ for quiver $\Gamma$.

arXiv comment83 pages