Algebraic geometry and Bethe ansatz (I) the quotient ring for BAE

Abstract

In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar $\mathcal{N}=4$ super-Yang-Mills theory.

minor updates

References in corpus (5)

• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Fast analytic solver of rational Bethe equations
• The Polynomial Form of the Scattering Equations is an H-Basis
• Singular Solutions to the Bethe Ansatz Equations and Rigged Configurations
• Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM

Cited by in corpus (16)

• Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
• Two-Loop Master Integrals for Leading-Color $pp\to t\bar{t}H$ Amplitudes with a Light-Quark Loop
• Rational $Q$-systems, Higgsing and Mirror Symmetry
• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• On Generalized $Q$-systems
• Twisted indices, Bethe ideals and 3d $\mathcal{N}=2$ infrared dualities
• One-Loop Non-Planar Anomalous Dimensions in Super Yang-Mills Theory
• Bethe-State Counting and the Witten Index
• Cylinder partition function of the 6-vertex model from algebraic geometry
• Spin-$s$ Rational $Q$-system
• The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz
• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
• ${\text{AdS}}_4$ Holography and the Hilbert Scheme
• Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries
• General One-loop Reduction in Generalized Feynman Parametrization Form
• Spin-$s$ $Q$-systems: Twist and Open Boundaries