Rational Q-systems, Higgsing and mirror symmetry

The rational Q-system is an efficient method to solve Bethe ansatz equations for quan-tum integrable spin chains. We construct the rational Q-systems for generic Bethe ansatz equations described by an Al-1 quiver, which include models with multiple momentum carrying nodes, generic inhomogeneities, generic diagonal twists and q-deformation. The rational Q-system thus constructed is specified by two partitions. Under Bethe /Gauge correspondence, the rational Q-system is in a one-to-one correspondence with a 3d N = 4 quiver gauge theory of the type Ti sigma [SU(n)], which is also specified by the same partitions. This shows that the rational Q-system is a natural language for the Bethe/Gauge correspondence, because known features of the Ti sigma [SU(n)] theories read-ily translate. For instance, we show that the Higgs and Coulomb branch Higgsing corre-spond to modifying one of the partitions in the rational Q-system while keeping the other untouched. Similarly, mirror symmetry is realized in terms of the rational Q-system by simply swapping the two partitions-exactly as for Ti sigma [SU(n)]. We exemplify the com-putational efficiency of the rational Q-system by evaluating topologically twisted indices for 3d N = 4 U(n) SQCD theories with n = 1, ... , 5.