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

Algebraic geometry and Bethe ansatz. Part 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...

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Main Authors: Yunfeng Jiang, Yang Zhang
Format: Article
Language:English
Published: Springer 2018-03-01
Series:Journal of High Energy Physics
Subjects:
Bethe Ansatz
Differential and Algebraic Geometry
Lattice Integrable Models
Online Access:http://link.springer.com/article/10.1007/JHEP03(2018)087
id doaj-art-7d9f1f80aa4147d5a286690528417fe4
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spelling doaj-art-7d9f1f80aa4147d5a286690528417fe42018-08-15T21:55:04ZengSpringerJournal of High Energy Physics1029-84792018-03-012018314010.1007/JHEP03(2018)087Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAEYunfeng Jiang0Yang Zhang1Institut für Theoretische Physik, ETH ZürichInstitut für Theoretische Physik, ETH ZürichAbstract 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 N=4 $$ \mathcal{N}=4 $$ super-Yang-Mills theory.http://link.springer.com/article/10.1007/JHEP03(2018)087Bethe AnsatzDifferential and Algebraic GeometryLattice Integrable Models
institution Open Data Bank
collection Open Access Journals
building Directory of Open Access Journals
language English
format Article
author Yunfeng Jiang
Yang Zhang
spellingShingle Yunfeng Jiang
Yang Zhang
Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE
Journal of High Energy Physics
Bethe Ansatz
Differential and Algebraic Geometry
Lattice Integrable Models
author_facet Yunfeng Jiang
Yang Zhang
author_sort Yunfeng Jiang
title Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE
title_short Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE
title_full Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE
title_fullStr Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE
title_full_unstemmed Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE
title_sort algebraic geometry and bethe ansatz. part i. the quotient ring for bae
publisher Springer
series Journal of High Energy Physics
issn 1029-8479
publishDate 2018-03-01
description 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 N=4 $$ \mathcal{N}=4 $$ super-Yang-Mills theory.
topic Bethe Ansatz
Differential and Algebraic Geometry
Lattice Integrable Models
url http://link.springer.com/article/10.1007/JHEP03(2018)087
_version_ 1612704918971351040

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