Exact quantization conditions for the relativistic Toda lattice

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.

31pages, v2: misprints corrected, references added, v3: published version, v4: typo in eq.(2.49) fixed

References in corpus (7)

• The Refined Topological Vertex
• Extended Holomorphic Anomaly in Gauge Theory
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• New Exact Quantization Condition for Toric Calabi-Yau Geometries
• Topological Strings and Quantum Spectral Problems
• Seiberg-Witten theory and matrix models
• Non-Perturbative Quantum Geometry II

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