Opers for higher states of quantum KdV models

Abstract

We study the ODE/IM correspondence for all states of the quantum $\hat{\mathfrak{g}}$-KdV model, where $\hat{\mathfrak{g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak{g}$. We construct quantum $\hat{\mathfrak{g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\hat{\mathfrak{g}}$-KdV opers satisfy the Bethe Ansatz equations of the quantum $\hat{\mathfrak{g}}$-KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.

A new Abstract and a better Introduction. Typos corrected. Minor changes in the bibliography. 66 pages

References in corpus (7)

• The ODE/IM Correspondence
• TBA equations and resurgent Quantum Mechanics
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• The Baxter's Q-operator for the W-algebra $W_N$
• Y-System and Deformed Thermodynamic Bethe Ansatz
• Opers for higher states of the quantum Boussinesq model
• Poisson pencils: reduction, exactness, and invariants

Cited by in corpus (7)

• Counting monster potentials
• On solutions of the Bethe Ansatz for the Quantum KdV model
• WKB analysis of the linear problem for modified affine Toda field equations
• Feigin-Frenkel-Hernandez Opers and the QQ-system
• Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: a Levinson type theorem on complex domains, and applications
• Affine opers and conformal affine Toda
• Anisotropic Kondo line defect and ODE/IM correspondence