Opers for Higher States of Quantum KdV Models

Counting monster potentials

“…Finally, let us mention that the ODE/IM correspondence exist also for the (generalised) Quantum g-KdV models, where g is an untwisted Kac-Moody algebra (the Quantum KdV model coincides with the case g = sl (1) 2 ), and massive deformations of these models [9,25]. In the massless case, the analogous of the monster potentials are called Quantum KdV JHEP02(2021)059 opers, they were introduced by Feigin and Frenkel in [18] and explicitly constructed in [26,27]. Of these opers very little is known and the analogous of the BLZ system exists, but it is too intricate to be manipulated by hand.…”

“…We recall that the BLZ conjecture was extended to the generalised Quantum g-KdV model, where g is an affine Kac-Moody algebra, in the seminal work by Feigin and Frenkel [18] (the Quantum KdV model coincides with the case g = sl (1) 2 ). Building on this work and using heavily the theory of opers, the analogue of the monster potentials and of the BLZ system (1.1) are explicitly described in [26] in the case g = g (1) , with g a simply-laced simple Lie algebra. The sub-case g = sl (1) 3 is treated with more elementary techniques in [27].…”

“…The BLZ conjecture is an instance of the celebrated ODE/IM -we recall here that the acronym ODE/IM stands for Ordinary Differential Equation(s)/Integrable Model(s)correspondence firstly discovered by Dorey and Tateo [16], and later generalised by many more authors, see e.g. [6,15,18,21,22,25,26,28,29,32,33].…”

Kondo line defects and affine Gaudin models

“…3. Solutions of the affine sl(2) Bethe equations can be used to produce PSU(2) λ-opers with singularities of trivial monodromy [20,21,[23][24][25][26][27]. We identify the spectrum of the transfer matrices with the Stokes data of the λ-opers.…”

$$ \mathrm{T}\overline{\mathrm{T}} $$-deformed nonlinear Schrödinger

“…soliton solutions, and the link with the ODE/IM correspondence [37][38][39][40]. Understanding, at a deeper level, the TT-like deformations of quantum spin-chains [41][42][43][44], possibly within the Quantum Spectral Curve framework of AdS/CFT [45][46][47][48][49][50], is also an unexplored avenue that might lead to unexpected and exciting discoveries.…”