Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case

Abstract

We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $\Psi$-system among subdominant solutions. We then use the $\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.

37 pages, 1 figure. Continuation of arXiv:1501.07421. Minor change in the title. New subsection 5.1 on the action of the Weyl group on the Bethe Ansatz solutions

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