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

TL;DR

This work investigates the resurgence structure of SUSY quantum mechanics under a SUSY-breaking deformation, focusing on two model classes: real multiplets on Riemannian manifolds and chiral multiplets on Kähler manifolds. By introducing a deformation parameter δϵ and expanding around SUSY or quasi-exact solvable points, the authors reveal all-orders multi-bion contributions and their intricate cancellations with non-Borel-summable perturbative series, analyzed via Lefschetz thimbles and localization. They demonstrate, with sine-Gordon QM and CP^{N−1} QM as representative examples, that semiclassical complex bion configurations reproduce the nonperturbative structure of the exact results, including imaginary ambiguities that cancel in the full trans-series. The work also uncovers how quasi-exact solvability constrains the resurgence pattern, and extends the discussion to squashed CP^1, where nontrivial cancellation mechanisms persist. Overall, the paper establishes a coherent framework linking complex saddles, localization, and resurgent trans-series in SUSY/QES quantum systems, with potential implications for higher-dimensional gauge theories and topological sectors.

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 $δε$, we derive exact results for the expansion coefficients. In the class of models described by real multiplets, the ${\mathcal O}(δε)$ 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 ${\mathcal O}(δε)$ ground state energy can be completely reconstructed from the semiclassical multi-bion contributions. Although the ${\mathcal O}(δε)$ ground state energy has trivial resurgence structure, a simple but rich resurgence structure appears at ${\mathcal O}(δε^{2})$. We show the complete cancellation between the ${\mathcal O}(δε^{2})$ 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.

Figures (8)

• Figure 1: The integration contour $C$ for the generating function $\langle 0 | 0 \rangle$ and the Lefschetz thimbles $\mathcal{J}_n$ associated with the saddle points $\theta = (n+1)\pi$. The thimble with $n=0$ jumps at $\arg g^2 = 0$ due to the Stokes phenomenon. The original integration contour $\mathcal{C}$ can be deformed and decomposed as $\mathcal{J}_0^+ - \mathcal{J}_1$ or $\mathcal{J}_0^- + \mathcal{J}_{-1}$ depending on $\arg g^2$. The ambiguous Borel resummation $I_0(z) \pm \frac{i}{\pi} K_0(z)$ corresponds to the integration along $\mathcal{J}_0^\pm$.
• Figure 2: Multi bion solution: $p=3$, $q=2$, $m=1$, $\epsilon=1$, $g^2=1/20000$, $\beta = 100$, $\tau_c=0$.
• Figure 3: Integration contour for $\sigma$. The poles of the integrand are located at $\sigma = -i k$ and $\sigma=-i(\epsilon+k)~(k \in \mathbb{Z}_{\geq 0})$.
• Figure 4: Singularities on the Borel plane for the perturbative Borel transform of $\mathbb{C} P^{N-1}$ QM.
• Figure 5: Examples of $(N-1)$ bion solutions for $N=4$: $m_1=1$, $m_2=2$, $m_3=3$, $g=10^{-4}$, $\epsilon=1$.