[Paper Review] Rational $Q$-systems, Higgsing and Mirror Symmetry

Jie Gu, Yunfeng Jiang+1 more|arXiv (Cornell University)|Aug 22, 2022

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces a rational $Q$-system formulation for generic $A_{\ell-1}$ quiver Bethe ansatz equations, including multiple momentum-carrying nodes, inhomogeneities, twists, and $q$-deformations. It establishes a one-to-one correspondence between the rational $Q$-system and 3d $\mathcal{N}=4$ quiver gauge theories of type $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$, where partitions $\bm{\rho}$ and $\bm{\sigma}$ encode the system. The framework realizes Higgs and Coulomb branch Higgsing via partition modifications and mirror symmetry via partition exchange, enabling efficient computation of topologically twisted indices for $\mathrm{U}(n)$ SQCD with $n=1,\dots,5$. The method eliminates non-physical solutions and streamlines the Bethe/Gauge correspondence.

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 ${T}_{\boldsymbolρ}^{\boldsymbolσ}[SU(n)]$, 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 ${T}_{\boldsymbolρ}^{\boldsymbolσ}[SU(n)]$ 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 ${T}_{\boldsymbolρ}^{\boldsymbolσ}[SU(n)]$. 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 $n=1,\ldots,5$.

Motivation & Objective

• To develop a systematic rational $Q$-system formulation for generic $A_{\ell-1}$ quiver Bethe ansatz equations with multiple momentum-carrying nodes, inhomogeneities, twists, and $q$-deformations.
• To establish a one-to-one correspondence between the rational $Q$-system and 3d $\mathcal{N}=4$ quiver gauge theories $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$, which are also labeled by two partitions.
• To demonstrate that the rational $Q$-system naturally encodes geometric operations in gauge theory—such as Higgsing and mirror symmetry—through simple modifications of the partitions.
• To apply the rational $Q$-system to efficiently compute topologically twisted indices for $\mathrm{U}(n)$ SQCD theories with $n=1,\dots,5$, validating its computational superiority over direct BAE solving.

Proposed method

• The rational $Q$-system is constructed on a Young tableau associated with partition $\bm{\rho}$, with the second partition $\bm{\sigma}$ encoding additional structure, generalizing previous formulations to include multiple momentum-carrying nodes and $q$-deformations.
• The method uses $QQ$-relations on the Young tableau to encode physical solutions of the Bethe ansatz equations, automatically excluding non-physical solutions such as repeated roots.
• Higgsing on the Higgs branch corresponds to modifying $\bm{\rho}$ while keeping $\bm{\sigma}$ fixed, and Higgsing on the Coulomb branch corresponds to modifying $\bm{\sigma}$ while keeping $\bm{\rho}$ fixed.
• Mirror symmetry is realized by simply swapping the two partitions $\bm{\rho}$ and $\bm{\sigma}$, mirroring the known duality in $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$ theories.
• The framework leverages the Bethe/Gauge correspondence, where solutions of the BAE correspond to supersymmetric vacua of the 3d $\mathcal{N}=4$ quiver theory compactified on $S^1$, and uses localization to compute topologically twisted indices.
• The topologically twisted index is computed via a contour integral over Bethe roots, which is re-expressed using the rational $Q$-system to avoid numerical instability and non-physical solutions.

Experimental results

Research questions

• RQ1 How can the rational $Q$-system be generalized to include multiple momentum-carrying nodes, generic inhomogeneities, diagonal twists, and $q$-deformations in $A_{\ell-1}$ quiver models?
• RQ2 What is the precise correspondence between the rational $Q$-system and 3d $\mathcal{N}=4$ quiver gauge theories $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$?
• RQ3 How are Higgs and Coulomb branch Higgsing operations encoded in the rational $Q$-system structure?
• RQ4 How is mirror symmetry realized within the rational $Q$-system framework, and does it match the known duality in $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$ theories?
• RQ5 Can the rational $Q$-system be used to efficiently compute topologically twisted indices for $\mathrm{U}(n)$ SQCD with $n=1,\dots,5$?

Key findings

• The rational $Q$-system is constructed for generic $A_{\ell-1}$ quiver Bethe ansatz equations with multiple momentum-carrying nodes, inhomogeneities, twists, and $q$-deformations, using two partitions $\bm{\rho}$ and $\bm{\sigma}$ to fully specify the system.
• The rational $Q$-system is in one-to-one correspondence with 3d $\mathcal{N}=4$ quiver gauge theories $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$, where $\bm{\rho}$ labels the Young tableau and $\bm{\sigma}$ encodes additional structure.
• Higgs branch Higgsing corresponds to modifying $\bm{\rho}$ while keeping $\bm{\sigma}$ fixed, and Coulomb branch Higgsing corresponds to modifying $\bm{\sigma}$ while keeping $\bm{\rho}$ fixed, in perfect agreement with gauge theory expectations.
• Mirror symmetry is realized by simply swapping the two partitions $\bm{\rho}$ and $\bm{\sigma}$, exactly as in the $T_{\bm{\rho}}^{\bm{\sigma}}[\mathrm{SU}(n)]$ theories.
• The rational $Q$-system enables efficient computation of topologically twisted indices for $\mathrm{U}(n)$ SQCD with $n=1,\dots,5$, avoiding numerical instabilities and non-physical solutions inherent in direct BAE solving.
• The method achieves computational efficiency by encoding the entire solution space of the BAE within the $QQ$-relations on the Young tableau, with physical solutions corresponding to the zeros of a single $Q$-function.