Derives weakest quantization conditions in terms of monodromy data for higher-order DEs tied to quantum Toda chain and proves duality predictions for deformed Schrödinger operators.
Higher-Rank Mathieu Opers, Toda Chain, and Analytic Langlands Correspondence
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abstract
We study the Riemann-Hilbert problem associated to flat sections of oper connections of arbitrary rank on the twice-punctured Riemann sphere with irregular singularities of the mildest type. We construct the solutions in terms of the solutions to a single non-linear integral equation. It follows from this construction that the generating function of the submanifold of opers coincides with the Yang-Yang function of the quantum Toda chain, proving a conjecture by Nekrasov, Rosly and Shatashvili. In this way we may furthermore reformulate the quantization conditions of the Toda chain in terms of the connection problem, for which we also provide a solution. We finally interpret our results as a variant of the Analytic Langlands Correspondence for the real version of the Hitchin system corresponding to the Toda chain.
years
2026 2verdicts
UNVERDICTED 2representative citing papers
Deformed quantum mechanics from Seiberg-Witten curves shows phases with real or complex instantons, leading to tunneling suppression at Toda points and anomalous scaling at critical monopole points.
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Higher-Rank Connections and Deformed Schr\"odinger Operators
Derives weakest quantization conditions in terms of monodromy data for higher-order DEs tied to quantum Toda chain and proves duality predictions for deformed Schrödinger operators.
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Thou shalt not tunnel: Complex instantons and tunneling suppression in deformed quantum mechanics
Deformed quantum mechanics from Seiberg-Witten curves shows phases with real or complex instantons, leading to tunneling suppression at Toda points and anomalous scaling at critical monopole points.