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

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.

44 pages, 13 figures, typos corrected, references updated

References in corpus (33)

• Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions
• Abelian duality, confinement, and chiral symmetry breaking in QCD(adj)
• Instantons in the Higgs Phase
• Construction of Non-Abelian Walls and Their Complete Moduli Space
• Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
• Non-Abelian Walls in Supersymmetric Gauge Theories
• Conformality or confinement: (IR)relevance of topological excitations
• Structure of Lefschetz thimbles in simple fermionic systems
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• Hidden topological angles and Lefschetz thimbles
• D-brane Construction for Non-Abelian Walls
• Josephson vortices and the Atiyah-Manton construction
• Adjoint QCD on ${\mathbb{R}}^3\times S^1$ with twisted fermionic boundary conditions
• Resumming the string perturbation series
• Instanton constituents in the O(3) model at finite temperature
• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Global Structure of Moduli Space for BPS Walls
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• Statistical Mechanics of Vortices from D-branes and T-duality
• Instanton Effects in Orbifold ABJM Theory
• Introduction to 1-summability and resurgence
• Fractional instantons and bions in the O(N) model with twisted boundary conditions
• Finite N from Resurgent Large N
• Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential
• Fractional instantons and bions in the principal chiral model on ${\mathbb{R}}^2\times S^1$ with twisted boundary conditions
• On Perturbation theory improved by Strong coupling expansion
• Neutral bions in the CP^{N-1} model for resurgence
• Nonlinear sigma models at nonzero chemical potential: breaking up instantons and the phase diagram
• Interpolating function and Stokes Phenomena
• Resummations and Non-Perturbative Corrections
• Resurgence and Dynamics of O(N) and Grassmannian Sigma Models

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