Abstract
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.
Highlights
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
In order to find a convergent series for the exact quantization condition, as in the 4d case, the resummation provided by the instanton partition function is not enough, and one needs the explicit non-perturbative contributions written down in (2.68)
Summary
We review some basic features of the relativistic Toda lattice. Our conventions are similar to those in [7], with some small modifications. We can mod out the motion of the center of mass by fixing the total momentum to be zero, so that HN = 1 In this way we have N − 1 non-trivial commuting Hamiltonians H1, · · · , HN−1. In order to formulate the spectral problem for the quantum relativistic Toda lattice, we eliminate the motion of the center of mass in favor of N − 1 coordinates ζ1, · · · , ζN−1. In the limit R → 0 we recover the spectral curve of the standard Toda lattice, eξ + e−ξ + (μ − μi) = 0. The method of separation of variables, when applied to the relativistic Toda lattice, implies that the eigenvalue problem (2.15) can be solved by considering the “quantum” version of the spectral curve [6, 7, 21]. Instead of solving this equation analytically, we will propose an exact quantization condition for the eigenvalues H1, · · · , HN−1, based on insights from [8] and on the recent progress in the quantization of mirror curves [22, 24, 25]
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