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Non-perturbative quantum geometry. (English) Zbl 1333.81346

Summary: The \(\beta\)-ensemble with cubic potential can be used to study a quantum particle in a double-well potential with symmetry breaking term. The quantum mechanical perturbative energy arises from the ensemble free energy in a novel large \(N\) limit. A relation between the generating functions of the exact non-perturbative energy, similar in spirit to the one of Dunne-Ünsal, is found. The exact quantization condition of Zinn-Justin and Jentschura is equivalent to the Nekrasov-Shatashvili quantization condition on the level of the ensemble. Refined topological string theory in the Nekrasov-Shatashvili limit arises as a large \(N\) limit of quantum mechanics.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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