Quantum geometry and quiver gauge theories
Nikita Nekrasov, Vasily Pestun, Samson Shatashvili
TL;DR
The paper advances the Bethe/gauge correspondence by analyzing macroscopic ${\rm N}=(2,2)$ theories obtained from 4d quiver theories under ${\Omega}$-deformation on ${\mathbb T}^d\times {\mathbb R}^2_\epsilon$, focusing on the universal part of the twisted superpotential ${\\mathcal W}$ and its quantum-group interpretation. It constructs a quantum current $h(x)$ in ${\\mathbf U}_\epsilon\mathfrak{g}_{\Gamma}(\mathsf{C}_x)$ whose $q$-character relations reproduce a set of functional equations ${\\chi}_i(h(x))=T_i(x)$, thereby linking gauge-theory observables to $q$-characters and, in higher dimensions, to ${\\mathbf U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})$ and ${\\mathbf U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})$. The work unifies 4d, 5d, and 6d cases via a dimension-uniform quantum-current framework, analyzes the limit $\oldsymbol{\epsilon}_2\to 0$ to extract the twisted superpotential from partition functions, and discusses the limit-shape/bethe-ansatz equations that govern dominant configurations, with explicit examples across finite and affine quivers. By connecting the universal part of the effective superpotential to representations and $q$-characters of quantum algebras, the paper provides a robust bridge between gauge theories on curved/Ω-deformed spaces and the representation theory of Yangians, quantum affine, and elliptic algebras, including their toroidal variants. The results have potential implications for generalized Bethe ansatz constructions, opers in 4d, and the broader interplay between BPS/CFT-type correspondences and quantum algebras in high-energy and mathematical physics.
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 $\mathbb{T}^{d} \times \mathbb{R}^{2}_ε$ 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 $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^{1+d}$, for $d=0,1,2$, and the Yangian $\mathbf{Y}_ε(\mathfrak{g}_Γ)$, quantum affine algebra $\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_Γ)$, or the quantum elliptic algebra $\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_Γ)$ associated to Kac-Moody algebra $\mathfrak{g}_Γ$ for quiver $Γ$.