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q-opers, QQ-systems, and Bethe Ansatz II: Generalized minors

  • Peter Koroteev ORCID logo and Anton M. Zeitlin EMAIL logo

Abstract

In this paper, we describe a certain kind of q-connections on a projective line, namely Z-twisted ( G , q ) -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q-connections and 𝑄𝑄 -systems/Bethe Ansatz equations. Here we associate to a Z-twisted ( G , q ) -oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as ( G , q ) -Wronskians. Among other things, we show that the 𝑄𝑄 -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.

Award Identifier / Grant number: DMS-2203823

Funding source: Simons Foundation

Award Identifier / Grant number: 578501

Funding statement: Peter Koroteev is partially supported by AMS Simons Travel Grant. Anton M. Zeitlin is partially supported by Simons Collaboration Grant, Award ID: 578501 and NSF grant DMS-2203823.

Acknowledgements

We are grateful to Edward Frenkel for his support and valuable comments.

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Received: 2021-08-16
Revised: 2022-10-09
Published Online: 2023-01-27
Published in Print: 2023-02-01

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