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

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

49 pages, 10 figures; Corrected Fig.4 and Fig.5

References in corpus (48)

• From weak to strong coupling in ABJM theory
• Vortices, Instantons and Branes
• Nonabelian Superconductors: Vortices and Confinement in $\mathcal{N}=2$ SQCD
• Solitons in the Higgs phase -- the moduli matrix approach --
• Analytic Continuation Of Chern-Simons Theory
• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model
• Moduli Space of Non-Abelian Vortices
• Non-perturbative effects and the refined topological string
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
• Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
• Uniform WKB, Multi-instantons, and Resurgent Trans-Series
• Instantons in the Higgs Phase
• Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
• Construction of Non-Abelian Walls and Their Complete Moduli Space
• Generating Non-perturbative Physics from Perturbation Theory
• A semiclassical realization of infrared renormalons
• Nonperturbative Ambiguities and the Reality of Resurgent Transseries
• Continuity and Resurgence: towards a continuum definition of the CP(N-1) model
• Resurgence in QFT: Unitons, Fractons and Renormalons in the Principal Chiral Model
• Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations
• Non-Abelian Walls in Supersymmetric Gauge Theories
• New Methods in QFT and QCD: From Large-N Orbifold Equivalence to Bions and Resurgence
• Non-Abelian Vortices on Cylinder -- Duality between vortices and walls
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators
• Complexified path integrals, exact saddles and supersymmetry
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• Hidden topological angles and Lefschetz thimbles
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
• Multi-Instantons and Multi-Cuts
• Resumming the string perturbation series
• Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Borel summability of perturbative series in 4d N=2 and 5d N=1 theories
• Resurgence of the Cusp Anomalous Dimension
• Fractional instantons and bions in the O(N) model with twisted boundary conditions
• The Resurgence of the Cusp Anomalous Dimension
• Remark on the Dunne-Unsal relation in exact semi-classics
• Finite N from Resurgent Large N
• How to resum perturbative series in 3d N=2 Chern-Simons matter theories
• Non-BPS exact solutions and their relation to bions in $\mathbb{C}{P}^{N-1}$ models
• Fractional instantons and bions in the principal chiral model on ${\mathbb{R}}^2\times S^1$ with twisted boundary conditions
• What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles
• Neutral bions in the CP^{N-1} model for resurgence

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