Exact Resurgent Trans-series and Multi-Bion Contributions to All Orders

TL;DR

This work addresses the verification of the full resurgent trans-series in near-SUSY CP^1 quantum mechanics by combining exact semi-classical multi-bion analysis with perturbation theory. The authors derive exact δε-order expansion coefficients of the ground-state energy, construct infinite families of complex multi-bion saddles, and compute their contributions via Lefschetz thimbles, showing precise cancellation of imaginary ambiguities with perturbative data. They demonstrate that the multi-bion sector alone suffices to reconstruct the entire trans-series to all orders in the nonperturbative exponential, at least in the near-SUSY regime, and confirm this against the Bender-Wu perturbative expansion on the trivial vacuum. The results provide a concrete all-orders validation of resurgence in a quantum-mechanical model and offer a framework potentially extensible to CP^{N-1} systems and quantum field theories, including complex saddles and localization techniques.

Abstract

The full resurgent trans-series is found exactly in near-supersymmetric $\mathbb C P^1$ quantum mechanics. By expanding in powers of the SUSY breaking deformation parameter, we obtain the first and second expansion coefficients of the ground state energy. They are absolutely convergent series of nonperturbative exponentials corresponding to multi-bions with perturbation series on those background. We obtain all multi-bion exact solutions for finite time interval in the complexified theory. We sum the classical multi-bion contributions that reproduce the exact result supporting the resurgence to all orders. This is the first result in the quantum mechanical model where the resurgent trans-series structure is verified to all orders in nonperturbative multi-bion contributions.

Figures (3)

• Figure 1: Multi-bion solution: kink profile of $\Sigma(\tau)=(1-\varphi \tilde{\varphi})/(1+\varphi \tilde{\varphi})$ for $(p,q)=(3,1)$, $\epsilon=1$, $m=1$, $g = 1/200$, $\beta=100$ and $\tau_{c}=0$. $\Sigma=\pm1$ (dashed lines) correspond to north and south poles (global and local minima) of $\mathbb{C} P^1$.
• Figure 2: Multi-bion solution: $\theta=-2 \arctan|\varphi|$ for $(p,q)=(2,0)$ (top) and for $(p,q)=(2,1)$ (bottom). The other parameters are the same as those in Fig. \ref{['fig:multi_bion']}. $\theta=0,2\pi,...$ and $\theta=\pi,3\pi,...$ correspond to north and south poles of $\mathbb{C} P^1$.
• Figure 3: The asymptotic behavior of the ratio $A_l/\left[ \frac{1}{2^{l-1}} \frac{\Gamma(l+2(1-\epsilon))}{\Gamma(1-\epsilon)^2}\right]~(l \leq 100)$ for $0 \leq \epsilon \leq 2$ ($\epsilon = n/5,~n=0,\cdots,10$). $\delta$ is a regularization parameter ($\delta = 10^{-10}$).