Exact quantization conditions for the relativistic Toda lattice

Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic toda lattice of n particles. these conditions involve the nekrasov-shatashvili free energy, which resums the perturbative wkb expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an s-duality transformation of the planck constant. we test the quantization conditions against explicit calculations of the spectrum for n = 3. our proposal can be generalized to arbitrary toric calabi-yau manifolds and might solve the corresponding quantum integrable system of goncharov and kenyon.