Rational $Q$-systems, Higgsing and Mirror Symmetry

Abstract

The rational $Q$-system is an efficient method to solve Bethe ansatz equations for quantum integrable spin chains. We construct the rational $Q$-systems for generic Bethe ansatz equations described by an $A_{\ell -1}$ 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 $\mathcal{N}=4$ quiver gauge theory of the type $T_{\mathbf{\rho }}^{\mathbf{\sigma }}[SU(n)]$, 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 $T_{\mathbf{\rho }}^{\mathbf{\sigma }}[SU(n)]$ theories readily translate. For instance, we show that the Higgs and Coulomb branch Higgsing correspond to modifying one of the partitions in the rational $Q$-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 $T_{\mathbf{\rho }}^{\mathbf{\sigma }}[SU(n)]$. We exemplify the computational efficiency of the rational $Q$-system by evaluating topologically twisted indices for 3d $\mathcal{N}=4$ $U(n)$ SQCD theories with $n=1,\dots ,5$.

arXiv commentv3: 53 pages + appendices, added clarifications, fixed typos

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• Orthosymplectic Quotient Quiver Subtraction
• Spin-$s$ Rational $Q$-system
• Quotient Quiver Subtraction
• Rational Q-systems at Root of Unity I. Closed Chains
• On Bethe eigenvectors and higher transfer matrices for supersymmetric spin chains
• Quiver Polymerisation
• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
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• Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries
• Spin-$s$ $Q$-systems: Twist and Open Boundaries
• Chiral Integrable Boundary States in the SU(4) Alternating Spin Chain