Thou shalt not tunnel: Complex instantons and tunneling suppression in deformed quantum mechanics

TL;DR

This paper explores deformed quantum mechanics derived from supersymmetric gauge theories, revealing a phase structure with tunneling suppression at specific parameter points and connections to wall-crossing phenomena.

Contribution

It uncovers the phase structure of deformed quantum models, linking tunneling behavior to super Yang-Mills theory and wall-crossing, using resurgent analysis.

Findings

Tunneling is suppressed at Toda lattice points in the weak coupling phase.

At the monopole point, non-perturbative amplitudes show anomalous scaling.

The phase transition reflects wall-crossing of the BPS spectrum.

Abstract

The quantization of the Seiberg-Witten curve of ${\cal N}=2$ super Yang-Mills theory leads to a deformation of one-dimensional quantum mechanics with unconventional behavior. Most notably, quantum tunneling is suppressed at special points in parameter space. In this paper we examine these deformed models in the case of double-well and cubic potentials, and we find that they have a rich phase structure. In what we call the strong coupling phase, the theory behaves like conventional quantum mechanics, instantons are real, and tunneling is not suppressed. In the weak coupling phase, the instantons responsible for tunneling become complex, and tunneling suppression takes place at the so-called Toda lattice points. At the critical point between the two phases, which corresponds to a monopole point in super Yang-Mills theory, the non-perturbative amplitudes display an anomalous scaling as a function of $\hbar$. This phase structure reflects the physics of the underlying super Yang-Mills theory and can be regarded as a physical manifestation of wall-crossing behavior of the BPS spectrum, which we determine in our problem by using resurgent techniques.