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

TL;DR

This paper establishes a deep connection between the spectra of quantum KdV Hamiltonians, affine opers, and Langlands duality, providing new proofs and conjectures in the theory of quantum integrable systems.

Contribution

It proves relations in the Grothendieck ring of category O for quantum affine algebras and links these to affine opers, advancing the understanding of quantum KdV spectra and Bethe Ansatz equations.

Findings

Relations in the Grothendieck ring are proven.

Spectra of quantum g^-KdV Hamiltonians are linked to affine opers.

A uniform proof of Bethe Ansatz equations is provided.

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].