Rational $Q$-systems, Higgsing and mirror symmetry

Abstract

The rational Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -system is an efficient method to solve Bethe ansatz equations for quantum integrable spin chains. We construct the rational Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -systems for generic Bethe ansatz equations described by an A_{\ell-1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>ℓ</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> quiver, which include models with multiple momentum carrying nodes, generic inhomogeneities, generic diagonal twists and q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>q</mml:mi> </mml:math> -deformation. The rational Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -system thus constructed is specified by two partitions. Under Bethe/Gauge correspondence, the rational Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -system is in a one-to-one correspondence with a 3d \mathcal{N}=4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> quiver gauge theory of the type {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>ρ</mml:mi> <mml:mi>σ</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo stretchy="true" form="prefix">[</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo stretchy="true" form="postfix">]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which is also specified by the same partitions. This shows that the rational Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -system is a natural language for the Bethe/Gauge correspondence, because known features of the {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>ρ</mml:mi> <mml:mi>σ</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo stretchy="true" form="prefix">[</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo stretchy="true" form="postfix">]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -system while keeping the other untouched. Similarly, mirror symmetry is realized in terms of the rational Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -system by simply swapping the two partitions - exactly as for {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>ρ</mml:mi> <mml:mi>σ</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo stretchy="true" form="prefix">[</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow>

Topics & Concepts