Perfect Integrability and Gaudin Models

Kang Lu

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.

Détail

Keywords: Gaudin model, Bethe ansatz, Frobenius algebra.

Bibliographie

[1] Atiyah M. F., Macdonald I. G., Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass. – London – Don Mills, Ont., 1969 | MR | Zbl

[2] Chernyak D., Leurent S., Volin D., Completeness of Wronskian Bethe equations for rational $\mathfrak{gl}(m|n)$ spin chain, arXiv: 2004.02865

[3] Feigin B., Frenkel E., Reshetikhin N., “Gaudin model, Bethe ansatz and critical level”, Comm. Math. Phys., 166 (1994), 27–62, arXiv: hep-th/9402022 | DOI | MR | Zbl

[4] Feigin B., Frenkel E., Rybnikov L., “Opers with irregular singularity and spectra of the shift of argument subalgebra”, Duke Math. J., 155 (2010), 337–363, arXiv: 0712.1183 | DOI | MR | Zbl

[5] Feigin B., Frenkel E., Toledano Laredo V., “Gaudin models with irregular singularities”, Adv. Math., 223 (2010), 873–948, arXiv: math.QA/0612798 | DOI | MR | Zbl

[6] Frenkel E., Langlands correspondence for loop groups, Cambridge Studies in Advanced Mathematics, 103, Cambridge University Press, Cambridge, 2007 | MR | Zbl

[7] Ilin A., Rybnikov L., On classical limits of Bethe subalgebras in Yangians, arXiv: 2009.06934

[8] Lu K., “Lower bounds for numbers of real self-dual spaces in problems of Schubert calculus”, SIGMA, 14 (2018), 046, 15 pp., arXiv: 1710.06534 | DOI | MR | Zbl

[9] Lu K., Mukhin E., “On the Gaudin model of type $\rm G_2$”, Commun. Contemp. Math., 21 (2019), 1850012, 31 pp., arXiv: 1711.02511 | DOI | MR | Zbl

[10] Lu K., Mukhin E., On the supersymmetric XXX spin chains associated to $\mathfrak{gl}_{1|1}$, arXiv: 1910.13360

[11] Lu K., Mukhin E., Varchenko A., “Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, $\mathfrak{sp}_{2r}$, $\mathfrak{so}_{2r+1}$”, Pure Appl. Math. Q., 13 (2017), 291–335, arXiv: 1705.02048 | DOI | MR | Zbl

[12] Molev A. I., “Feigin–Frenkel center in types $B$, $C$ and $D$”, Invent. Math., 191 (2013), 1–34, arXiv: 1105.2341 | DOI | MR | Zbl

[13] Mukhin E., Tarasov V., “Lower bounds for numbers of real solutions in problems of Schubert calculus”, Acta Math., 217 (2016), 177–193, arXiv: 1404.7194 | DOI | MR | Zbl

[14] Mukhin E., Tarasov V., Varchenko A., “Bethe eigenvectors of higher transfer matrices”, J. Stat. Mech. Theory Exp., 2006 (2006), P08002, 44 pp., arXiv: math.QA/0605015 | DOI | MR | Zbl

[15] Mukhin E., Tarasov V., Varchenko A., “Spaces of quasi-exponentials and representations of $\mathfrak{gl}_N$”, J. Phys. A: Math. Theor., 41 (2008), 194017, 28 pp., arXiv: 0801.3120 | DOI | MR | Zbl

[16] Mukhin E., Tarasov V., Varchenko A., “Schubert calculus and representations of the general linear group”, J. Amer. Math. Soc., 22 (2009), 909–940, arXiv: 0711.4079 | DOI | MR | Zbl

[17] Mukhin E., Tarasov V., Varchenko A., “Spaces of quasi-exponentials and representations of the Yangian $Y(\mathfrak{gl}_N)$”, Transform. Groups, 19 (2014), 861–885, arXiv: 1303.1578 | DOI | MR | Zbl

[18] Rimányi R., Tarasov V., Varchenko A., “Trigonometric weight functions as $K$-theoretic stable envelope maps for the cotangent bundle of a flag variety”, J. Geom. Phys., 94 (2015), 81–119, arXiv: 1411.0478 | DOI | MR | Zbl

[19] Rybnikov L., “The argument shift method and the Gaudin model”, Funct. Anal. Appl., 40 (2006), 188–199, arXiv: math.RT/0606380 | DOI | MR | Zbl

[20] Rybnikov L., “Uniqueness of higher Gaudin Hamiltonians”, Rep. Math. Phys., 61 (2008), 247–252, arXiv: math.QA/0608588 | DOI | MR

[21] Rybnikov L., “Proof of the Gaudin Bethe ansatz conjecture”, Int. Math. Res. Not., 2020 (2020), 8766–8785, arXiv: 1608.04625 | DOI