Abstract
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.
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Notes
These features are reminiscent of trigonometric Gaudin models and Gaudin with quasi-periodic boundary conditions [9], in which the diagonal symmetry is broken. In those cases reproduction produces one new solution, which describes the same eigenvector (up to proportionality) but with respect to a different Borel subalgebra.
References
Carpentier, S., De Sole, A., Kac, V.G.: Some algebraic properties of differential operators. J. Math. Phys. 53(6), 063501 (2012)
Carpentier, S., De Sole, A., Kac, V.G.: Rational matrix pseudodifferential operators. Sel. Math. (N.S.) 20(2), 403–419 (2014)
Cheng, S.-J., Wang, W.: Dualities and Representations of Lie Superalgebras. American Mathematical Society, Providence (2012)
Feigin, B., Frenkel, E., Reshetikhin, N.: Gaudin model, Bethe ansatz and critical level. Commun. Math. Phys. 166(1), 27–62 (1994)
Molev, A.I., Ragoucy, E.: The MacMahon Master theorem for right quantum superalgebras and higher Sugawara operators for \(\widehat{{\mathfrak{g}}{\mathfrak{l}}}(m|n)\). Mosc. Math. J. 14(1), 83–119 (2014)
Mukhin, E., Tarasov, V., Varchenko, A.: Schubert calculus and representations of the general linear group. J. Am. Math. Soc. 22(4), 909–940 (2009)
Mukhin, E., Varchenko, A.: Critical points of master functions and flag varieties. Commun. Contemp. Math. 6(1), 111–163 (2004)
Mukhin, E., Varchenko, A.: Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe ansatz conjecture. Trans. Am. Math. Soc. 359(11), 5383–5418 (2007)
Mukhin, E., Varchenko, A.: Quasi-polynomials and the Bethe ansatz. In: Groups, homotopy and configuration spaces. Geom. Topol. Monogr., vol. 13, pp. 385–420. Geometry and Topology Publications, Coventry (2008)
Mukhin, E., Vicedo, B., Young, C.: Gaudin models for \({\mathfrak{gl}}(m|n)\). J. Math. Phys. 56(5), 051704 (2015). 30 pp
Rybnikov, L.G.: A proof of the Gaudin Bethe Ansatz conjecture. Preprint arXiv:1608.04625
Acknowledgements
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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Huang, C., Mukhin, E., Vicedo, B. et al. The solutions of \({\mathfrak {g}}{\mathfrak {l}}_{M|N}\) Bethe ansatz equation and rational pseudodifferential operators. Sel. Math. New Ser. 25, 52 (2019). https://doi.org/10.1007/s00029-019-0498-3
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DOI: https://doi.org/10.1007/s00029-019-0498-3