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Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

  • Yunfeng Jiang
  • Yang Zhang
Open Access
Regular Article - Theoretical Physics

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.

Keywords

Bethe Ansatz Differential and Algebraic Geometry Lattice Integrable Models 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institut für Theoretische Physik, ETH ZürichZürichSwitzerland

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