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Spectra of Quantum KdV Hamiltonians, Langlands Duality, and Affine Opers

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We prove a system of relations in the Grothendieck ring of the category \({\mathcal{O}}\) of representations of the Borel subalgebra of an untwisted quantum affine algebra \({U_q(\widehat{\mathfrak{g}})}\) introduced in Hernandez and Jimbo (Compos Math 148:1593–1623, 2012). This system was discovered, under the name \({Q\widetilde{Q}}\)-system, in Masoero et al. (Commun Math Phys 344:719–750, 2016; Commun Math Phys 349:1063–1105, 2017), where it was shown that solutions of this system can be attached to certain \({^L\widehat{\mathfrak{g}}}\)-affine opers, introduced in Feigin and Frenkel (Adv Stud Pure Math 61:185–274, 2007), where \({^L\widehat{g}}\) is the Langlands dual affine Kac–Moody algebra of \({\widehat{\mathfrak{g}}}\). Together with the results of Bazhanov et al. (Commun Math Phys 200:297–324, 1999; Nucl Phys B 622:475–547 2002) which enable one to associate quantum \({\widehat{\mathfrak{g}}}\)-KdV Hamiltonians to representations from the category \({\mathcal{O}}\), this provides strong evidence for the conjecture of Feigin and Frenkel (Adv Stud Pure Math 61:185–274, 2007) linking the spectra of quantum \({\widehat{\mathfrak{g}}}\)-KdV Hamiltonians and \({^L\widehat{\mathfrak{g}}}\)-affine opers. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz equations for a large class of quantum integrable models associated to arbitrary untwisted quantum affine algebras, under a mild genericity condition. We also conjecture analogues of these results for the twisted quantum affine algebras and elucidate the notion of opers for twisted affine algebras, making a connection to twisted opers introduced in Frenkel and Gross (Ann Math 170:1469–1512, 2009).

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Acknowledgement

We are grateful to Davide Masoero and Andrea Raimondo for fruitful discussions and explanations about their work [MRV1,MRV2], which was the main motivation for this paper. We also thank them for raising a question about the meaning of our formula for \({^L\widehat{g}}\)-opers in the non-simply laced case, which helped us to formulate it more precisely (Sect. 8.6).We thank Bernard Leclerc for his comments on the first version of this paper, and the referees for their thorough reading of the paper and useful comments. E. Frenkel was supported by the NSF Grant DMS-1201335. D. Hernandez was supported in part by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement No. 647353 QAffine.

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Correspondence to Edward Frenkel.

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Communicated by N. Nekrasov

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Frenkel, E., Hernandez, D. Spectra of Quantum KdV Hamiltonians, Langlands Duality, and Affine Opers. Commun. Math. Phys. 362, 361–414 (2018). https://doi.org/10.1007/s00220-018-3194-9

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