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

TL;DR

This work develops a nonperturbative analysis of ${\mathbb{C}}P^{1}$ quantum mechanics with fermions by incorporating real and complex bion saddles through the Lefschetz thimble framework. By calculating one-loop determinants around bion backgrounds and performing complexified quasi-moduli integrals, the authors show that real and complex bion contributions cancel in the supersymmetric limit and yield controlled, ambiguity-aware corrections in the nearly SUSY regime, consistent with exact Schrödinger results. The methodology is extended to sine-Gordon quantum mechanics for comparison, clarifying differences due to target-space topology and the role of phase moduli. Overall, the paper connects 1d resurgent trans-series to 2d ${\mathbb{C}}P^{N-1}$ dynamics, providing a rigorous path integral treatment of bions and outlining implications for higher-dimensional gauge theories via vortices and complexified saddles.

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 Schrödinger 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 $\mathbb{R} \times S^1$.

Figures (10)

• Figure 1: Kink-like profiles of (fractional) instanton configurations in ${\mathbb C}P^{3}$ model with ${\mathbb Z}_{4}$-twisted boundary conditions. Each horizontal dotted line corresponds to the value of $\Sigma$ in each vacuum. Note that $\Sigma$ is a periodic quantity with period $Nm=\frac{2\pi}{L}$ since it is shifted by the transformation \ref{['eq:equivalence']} with non-zero winding number $w~(V(x_2)=e^{2\pi i w \frac{x_2}{L}})$ as $\Sigma \rightarrow \Sigma + wNm$.
• Figure 2: Bion configuration is in ${\mathbb C}P^3$ model with ${\mathbb Z}_{4}$-twisted boundary conditions.
• Figure 3: The potential $V$ with the contribution of the fermion. The horizontal axis denotes the latitude $\theta \equiv 2 {\rm arctan} |\varphi|$ on $\mathbb{C} P^1 \cong S^2$.
• Figure 4: Kink profiles of for real and complex bions. The complex bion solution has singularities at which $\Sigma(\tau)$ diverges. Note that $\Sigma(\tau)$ can also be complex in the complexified model.
• Figure 5: Kink profile of regularized complex bion