Rational $Q$-systems, Higgsing and mirror symmetry

Abstract

The rational QQ-system is an efficient method to solve Bethe ansatz equations for quantum integrable spin chains. We construct the rational QQ-systems for generic Bethe ansatz equations described by an A_{\ell-1}Aℓ−1 quiver, which include models with multiple momentum carrying nodes, generic inhomogeneities, generic diagonal twists and qq-deformation. The rational QQ-system thus constructed is specified by two partitions. Under Bethe/Gauge correspondence, the rational QQ-system is in a one-to-one correspondence with a 3d \mathcal{N}=4𝒩=4 quiver gauge theory of the type {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)]Tρσ[SU(n)], which is also specified by the same partitions. This shows that the rational QQ-system is a natural language for the Bethe/Gauge correspondence, because known features of the {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)]Tρσ[SU(n)] theories readily translate. For instance, we show that the Higgs and Coulomb branch Higgsing correspond to modifying one of the partitions in the rational QQ-system while keeping the other untouched. Similarly, mirror symmetry is realized in terms of the rational QQ-system by simply swapping the two partitions - exactly as for {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)]Tρσ[SU(n)]. We exemplify the computational efficiency of the rational QQ-system by evaluating topologically twisted indices for 3d \mathcal{N}=4𝒩=4\mathrm{U}(n)U(n) SQCD theories with n=1,\ldots,5n=1,…,5.