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Rational $Q$systems, Higgsing and Mirror Symmetry
by Jie Gu, Yunfeng Jiang, Marcus Sperling
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Submission summary
Authors (as registered SciPost users):  Jie Gu · Marcus Sperling 
Submission information  

Preprint Link:  https://arxiv.org/abs/2208.10047v2 (pdf) 
Date accepted:  20221111 
Date submitted:  20221021 03:44 
Submitted by:  Sperling, Marcus 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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 $A_{\ell1}$ 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 onetoone correspondence with a 3d $\mathcal{N}=4$ quiver gauge theory of the type ${T}_{\boldsymbol{\rho}}^{\boldsymbol{\sigma}}[SU(n)]$, 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 ${T}_{\boldsymbol{\rho}}^{\boldsymbol{\sigma}}[SU(n)]$ 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 ${T}_{\boldsymbol{\rho}}^{\boldsymbol{\sigma}}[SU(n)]$. 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 $n=1,\ldots,5$.
List of changes
1) Added clarification on efficiency of Qsystem in Section 3.4; e.g. add reference for XXX spin chain case.
2) Added clarification on assumption that (2.23) does not vanish.
3) Added clarification on general mirror symmetry for 3d N=4 linear quiver theories labeled by two partitions in Section 7.1.
4) Added clarification on mirror map (6.9) of the parameters.
5) Added explanations on the role of parameters during Higgsing transitions in the rational Qsystem and the BAE. Specifically, Sections 5.2.1 and 5.4.1 as well as Appendix A.3 provide further details.
6) Rephrased and clarified the statement on the role of balance for Coulomb branch Higgsing in Section 5.3.
7) Some typos corrected.
Published as SciPost Phys. 14, 034 (2023)