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

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

Jun 16, 2016

54 pages

Published in:

Commun.Math.Phys. 362 (2018) 2, 361-414

Published: Jul 30, 2018

e-Print:

1606.05301 [math.QA]

DOI:

10.1007/s00220-018-3194-9

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Abstract: (Springer)

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).

Note:

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

algebra: affine
algebra: Kac-Moody
model: integrability
Hamiltonian
duality
category
Korteweg-de Vries equation
Bethe ansatz
background

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