Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB

TL;DR

This work provides explicit, order-by-order multi-instanton amplitudes for the sine-Gordon quantum mechanics and proves their exact agreement with uniform-WKB predictions, thereby supporting the resurgence framework in a concrete quantum-mechanical setting. By extending the Bogomolny–Zinn-Justin prescription to general multi-instanton configurations and employing a careful subtraction scheme, the authors show that imaginary ambiguities from instanton–anti-instanton sectors cancel the corresponding ambiguities in the large-order perturbative series. The results reproduce the energy corrections for the lowest band up to four instantons and demonstrate the consistency between semi-classical moduli integrals and nonperturbative boundary conditions. The analysis is further extended to neutral bions in the $CP^{N-1}$ model on $\mathbb{R}^{1}\times S^{1}$, where relative phase moduli modify the interaction and introduce quantitative corrections to sine-Gordon results, highlighting the rich structure of resurgent phenomena in compactified field theories. Collectively, the paper strengthens the link between perturbation theory, instanton physics, and Lefschetz-thimble perspectives in both quantum mechanics and field theory.

Abstract

We compute multi-instanton amplitudes in the sine-Gordon quantum mechanics (periodic cosine potential) by integrating out quasi-moduli parameters corresponding to separations of instantons and anti-instantons. We propose an extension of Bogomolnyi--Zinn-Justin prescription for multi-instanton configurations and an appropriate subtraction scheme. We obtain the multi-instanton contributions to the energy eigenvalue of the lowest band at the zeroth order of the coupling constant. For the configurations with only instantons (anti-instantons), we obtain unambiguous results. For those with both instantons and anti-instantons, we obtain results with imaginary parts, which depend on the path of analytic continuation. We show that the imaginary parts of the multi-instanton amplitudes precisely cancel the imaginary parts of the Borel resummation of the perturbation series, and verify that our results completely agree with those based on the uniform-WKB calculations, thus confirming the resurgence : divergent perturbation series combined with the nonperturbative multi-instanton contributions conspire to give unambiguous results. We also study the neutral bion contributions in the ${\mathbb C}P^{N-1}$ model on ${\mathbb R}^1\times S^{1}$ with a small circumference, taking account of the relative phase moduli between the fractional instanton and anti-instanton. We find that the sign of the interaction potential depends on the relative phase moduli, and that both the real and imaginary parts resulting from quasi-moduli integral of the neutral bion get quantitative corrections compared to the sine-Gordon quantum mechanics.

Figures (13)

• Figure 1: A schematic figure of an example of two instanton configurations $[\mathcal{I}\mathcal{I}]$. Each horizontal line stands for the vacuum in the sine-Gordon potential.
• Figure 2: A schematic figure of an example of one-instanton and one anti-instanton amplitude ($[\mathcal{I}\bar{\mathcal{I}}], [\bar{\mathcal{I}}\mathcal{I}]$). Each horizontal line stands for the vacuum in the sine-Gordon potential.
• Figure 3: A schematic figure of an example of three-instanton amplitudes ($[\mathcal{I}\mathcal{I}\mathcal{I}]$). Each horizontal line stands for the vacuum in the sine-Gordon potential.
• Figure 4: A schematic figure of an example of two-instanton and one--anti-instanton amplitudes ($[\mathcal{I} \bar{\mathcal{I}}\mathcal{I}], [\mathcal{I} \mathcal{I}\bar{\mathcal{I}}], [\bar{\mathcal{I}}\mathcal{I} \mathcal{I}]$). Each horizontal line stands for the vacuum in the sine-Gordon potential.
• Figure 5: A schematic figure of an example of the four-instanton amplitude $[\mathcal{I} \mathcal{I}\mathcal{I}\mathcal{I}]$. Each horizontal line stands for the vacuum in the sine-Gordon potential.