We discuss the nonperturbative contributions from real and complex saddle point solutions in the CP1 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-anti-instanton 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 quasizero modes which become nearly massless in a weak coupling limit, we regard the bion solutions as well-separated instanton-anti-instanton configurations and calculate a complexified quasimoduli integral based on the Lefschetz thimble formalism. The nonperturbative contributions from the real and complex bions are shown to cancel out in the supersymmetric case and give an (expected) ambiguity in the nonsupersymmetric case, which plays a vital role in the resurgent trans-series. For nearly supersymmetric situations, 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 CPN-1 field theory of R×S1.
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)