Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics

Abstract

We investigate the resurgence structure in quantum mechanical models originating in 2d non-linear sigma models with emphasis on nearly supersymmetric and quasi-exactly solvable parameter regimes. By expanding the ground state energy in powers of a supersymmetry-breaking deformation parameter $\delta \epsilon$, we derive exact results for the expansion coefficients. In the class of models described by real multiplets, the ${\mathcal O}(\delta\epsilon)$ ground state energy has a non-Borel summable asymptotic series, which gives rise to imaginary ambiguities leading to rich resurgence structure. We discuss the sine-Gordon quantum mechanics (QM) as an example and show that the semiclassical contributions from complex multi-bion solutions correctly reproduce the corresponding part in the exact result including the imaginary ambiguities. As a typical model described by chiral multiplets, we discuss the $\mathbb C P^{N-1}$ QM and show that the exact ${\mathcal O}(\delta \epsilon)$ ground state energy can be completely reconstructed from the semiclassical multi-bion contributions. Although the ${\mathcal O}(\delta \epsilon)$ ground state energy has trivial resurgence structure, a simple but rich resurgence structure appears at ${\mathcal O}(\delta \epsilon^{2})$. We show the complete cancellation between the ${\mathcal O}(\delta \epsilon^{2})$ imaginary ambiguities arising from the non-Borel summable perturbation series and those in the semiclassical contributions of $N-1$ complex bion solutions. We also discuss the resurgence structure of a squashed ${\mathbb C}P^1$ QM.