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The solutions of \({\mathfrak {g}}{\mathfrak {l}}_{M|N}\) Bethe ansatz equation and rational pseudodifferential operators

  • Chenliang HuangEmail author
  • Evgeny Mukhin
  • Benoît Vicedo
  • Charles Young


We describe a reproduction procedure which, given a solution of the \({\mathfrak {g}}{\mathfrak {l}}_{M|N}\) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all \({\mathfrak {g}}{\mathfrak {l}}_{M|N}\) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.

Mathematics Subject Classification

82B23 17B67 



The research of EM is partially supported by a grant from the Simons Foundation #353831. CY is grateful to the Department of Mathematical Sciences, IUPUI, for hospitality during his visit in September 2017 when part of this work was completed.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Chenliang Huang
    • 1
    Email author
  • Evgeny Mukhin
    • 1
  • Benoît Vicedo
    • 2
  • Charles Young
    • 3
  1. 1.Department of Mathematical SciencesIUPUIIndianapolisUSA
  2. 2.Department of MathematicsUniversity of YorkYorkUK
  3. 3.School of Physics, Astronomy and MathematicsUniversity of HertfordshireHatfieldUK

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