Non-perturbative Contributions from Complexified Solutions in $\mathbb{C}P^{N-1}$ Models

TL;DR

This paper investigates non-perturbative effects from real and complex saddle points in $ ext{CP}^{N-1}$ models using Lefschetz thimbles, revealing how bion solutions contribute to the resurgent structure and differ from sine-Gordon models.

Contribution

It introduces a detailed analysis of complexified instanton-antiinstanton solutions in $ ext{CP}^{N-1}$ models, extending the Lefschetz thimble formalism beyond Gaussian approximation.

Findings

Bion solutions stabilize in fermionic systems.

Non-perturbative contributions cancel in supersymmetric cases.

Differences between $ ext{CP}^{N-1}$ and sine-Gordon models are highlighted.

Abstract

We discuss the non-perturbative contributions from real and complex saddle point solutions in the $\mathbb{C}P^1$ quantum mechanics with fermionic degrees of freedom, using the Lefschetz thimble formalism beyond the gaussian approximation. We find bion solutions, which correspond to (complexified) instanton-antiinstanton configurations stabilized in the presence of the fermionic degrees of freedom. By computing the one-loop determinants in the bion backgrounds, we obtain the leading order contributions from both the real and complex bion solutions. To incorporate quasi zero modes which become nearly massless in a weak coupling limit, we regard the bion solutions as well-separated instanton-antiinstanton configurations and calculate a complexified quasi moduli integral based on the Lefschetz thimble formalism. The non-perturbative contributions from the real and complex bions are shown to cancel out in the supersymmetric case and give an (expected) ambiguity in the non-supersymmetric case, which plays a vital role in the resurgent trans-series. For nearly supersymmetric situation, evaluation of the Lefschetz thimble gives results in precise agreement with those of the direct evaluation of the Schr\"odinger equation. We also perform the same analysis for the sine-Gordon quantum mechanics and point out some important differences showing that the sine-Gordon quantum mechanics does not correctly describe the 1d limit of the $\mathbb{C}P^{N-1}$ field theory of $\mathbb{R} \times S^1$.