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

TL;DR

This paper introduces algebraic geometry techniques, including Gr"obner bases and quotient rings, to analyze Bethe ansatz equations, enabling efficient solution counting and summation over solutions in integrable models.

Contribution

It develops novel algebraic geometry methods for studying Bethe ansatz equations, including solution counting and summation techniques, advancing the mathematical understanding of integrable models.

Findings

Efficient solution counting using Gr"obner basis and quotient ring.

Analytical method to sum over solutions without explicit solving.

Application to Heisenberg spin chain and super-Yang-Mills theory.

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 Gr\"obner 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.