Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers

Abstract

We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking the spectra of quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine algebra. 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 [FG].

54 pages (v3: some examples added; opers for twisted affine algebras elucidated). Accepted for publication in Communications in Mathematical Physics

References in corpus (5)

• Baxter Q-Operators and Representations of Yangians
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• On minimal affinizations of representations of quantum groups
• Finite type modules and Bethe Ansatz for quantum toroidal gl(1)
• Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras

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