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Algebraic geometry and Bethe ansatz. I: The quotient ring for BAE. (English) Zbl 1388.81440
Summary: 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.

##### MSC:
 81T25 Quantum field theory on lattices 81R12 Groups and algebras in quantum theory and relations with integrable systems 81T13 Yang-Mills and other gauge theories in quantum field theory 81T60 Supersymmetric field theories in quantum mechanics
##### Software:
FGb; GBLA; GitHub; SINGULAR; slimgb
Full Text:
##### References:
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