A formal Bethe/gauge correspondence for classical gauge groups

Abstract

<jats:p> We study a formal algebraic Bethe/Gauge matching between Nekrasov-Shatashvili (NS)-limit vacuum equations extracted from four-dimensional <jats:inline-formula> <jats:alternatives> <jats:tex-math>\mathcal N=1</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="script">𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> localization on <jats:inline-formula> <jats:alternatives> <jats:tex-math>D^2× T^2</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and the homogeneous two-term Bethe equations of a special open XYZ spin-chain sector. The main new calculation is an explicit four-dimensional elliptic construction for the <jats:inline-formula> <jats:alternatives> <jats:tex-math>C_N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> gauge group. We also present the <jats:inline-formula> <jats:alternatives> <jats:tex-math>A_N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math>B_N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> , and <jats:inline-formula> <jats:alternatives> <jats:tex-math>D_N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> cases in the same normalization, thereby giving a uniform comparison across the classical series. The Bethe equations used in the matching are not the most general finite-size off-diagonal equations with an inhomogeneous term. We further observe that, after the NS scaling, the formal vacuum equations no longer depend explicitly on the chiral multiplet <jats:inline-formula> <jats:alternatives> <jats:tex-math>R</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>R</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -charge. The NS scaling should not be interpreted as a physical cancellation or bypassing of four-dimensional anomalies, but as a convenient way to derive the Bethe/Gauge correspondence. Outside the anomaly-free locus, the equations considered here are used only as formal semiclassical algebraic expressions to be compared with the Bethe equations. This formal matching is compatible with the standard dimensional hierarchy: $  N=(2,~2) $ <jats:inline-formula> <jats:alternatives> <jats:tex-math>↔</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>↔</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> <jats:inline-formula> <jats:alternatives> <jats:tex-math>\text{XXX spin chain,}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mtext mathvariant="normal">XXX spin chain,</mml:mtext> </mml:math> </jats:alternatives> </jats:inline-formula> <jats:inline-formula> <jats:alternatives> <jats:tex-math>\text{three-dimensional}\ \mathcal N=2\ \text{theory}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">three-dimensional</mml:mtext> <mml:mspace width="0.222em"/> <mml:mi mathvariant="script">𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mspace width="0.222em"/> <mml:mtext mathvariant="normal">theory</mml:mtext> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> <jats:inline-formula> <jats:alternatives> <jats:tex-math>↔</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>↔</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> $ $ <jats:inline-formula> <jats:alternatives> <jats:tex-math>\text{four-dimensional}\ \mathcal N=1\ \text{theory}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">four-dimensional</mml:mtext> <mml:mspace width="0.222em"/> <mml:mi mathvariant="script">𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mspace width="0.222em"/> <mml:mtext mathvariant="normal">theory</mml:mtext> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> <jats:inline-formula> <jats:alternatives> <jats:tex-math>↔</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>↔</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> <jats:inline-formula> <jats:alternatives> <jats:tex-math>\text{XYZ spin chain.}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mtext mathvariant="normal">XYZ spin chain.</mml:mtext> </mml:math> </jats:alternatives> </jats:inline-formula> </jats:p>