Abstract
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-Unsal, 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.
Highlights
JHEP02(2014)084 with Ep(n) the expansion coefficients, while Enp refers to contributions to the energy of nonperturbative origin
The quantum mechanical perturbative energy arises from the ensemble free energy in a novel large N limit
The exact quantization condition of Zinn-Justin and Jentschura is equivalent to the Nekrasov-Shatashvili quantization condition on the level of the ensemble
Summary
2.1 Saddle-point approximation Consider the eigenvalue ensemble (1.6) with cubic potential. ± g√s βδ k/2 Sk±) and subsequent expansion for small gs, the partition function of the cubic can be turned into a sum of normalized gaussian correlators. This leads to a split of the partition function into three parts (cf., [14]), i.e.,. The perturbative factor Zpert is given by a sum of products of normalized gaussian correlators. Calculating Zpert order by order in gs and combining with the contributions (2.7) and (2.8) yields the partition function of the β-ensemble with potential (2.1), in the two cut phase
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