Resurgence and Dynamics of O(N) and Grassmannian Sigma Models

Abstract

We study the non-perturbative dynamics of the two dimensional $O(N)$ and Grassmannian sigma models by using compactification with twisted boundary conditions on $\mathbb{R}\times S^1$, semi-classical techniques and resurgence. While the $O(N)$ model has no instantons for $N>3$, it has (non-instanton) saddles on ${\mathbb{R}}^2$, which we call 2d-saddles. On $\mathbb{R}\times S^1$, the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the $\mathfrak{o}(N)$ algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in $O(N)$ models in compactified space are in one-to-one correspondence with monopole-instanton saddles in $SO(N)$ gauge theory on ${\mathbb{R}}^3\times S^1$. The Grassmannian sigma models $\mathrm{Gr}(N,M)$ have 2d instantons, which fractionalize into $N$ kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the $O(3)$ model and $\mathrm{Gr}(N,M)$, and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes.

arXiv comment32 pages, 6 figures

References in corpus (42)

• Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation
• Theta Dependence In The Large N Limit Of Four-Dimensional Gauge Theories
• 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
• Theta dependence of SU(N) gauge theories in the presence of a topological term
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
• Monopoles and Instantons on Partially Compactified D-Branes
• Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling
• Uniform WKB, Multi-instantons, and Resurgent Trans-Series
• Generating Non-perturbative Physics from Perturbation Theory
• A semiclassical realization of infrared renormalons
• 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
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Non-Abelian Vortices on Cylinder -- Duality between vortices and walls
• Theta dependence, sign problems and topological interference
• Structure of Lefschetz thimbles in simple fermionic systems
• Hidden topological angles and Lefschetz thimbles
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• Non-perturbative equivalences among large N gauge theories with adjoint and bifundamental matter fields
• Instanton constituents in the O(3) model at finite temperature
• Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups
• Embedding nonlinear O(6) sigma model into N=4 super-Yang-Mills theory
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Instanton constituents and fermionic zero modes in twisted CP(n) models
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• The Operator Product Expansion, Non-perturbative Couplings and the Landau Pole: Lessons from the O(N) $\sigma$-Model
• From the mass gap in O(N) to the non-Borel-summability in O(3) and O(4) sigma-models
• Instanton effects and quantum spectral curves
• Introduction to 1-summability and resurgence
• Fractional instantons and bions in the O(N) model with twisted boundary conditions
• Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential
• Three-loop Correction to the Instanton Density. II. The Sine-Gordon potential
• Theta dependence of the deconfining phase transition in pure SU(N_c) Yang-Mills theories
• Metastable vacuum decay and $\theta$ dependence in gauge theory. Deformed QCD as a toy model
• Nonlinear sigma models at nonzero chemical potential: breaking up instantons and the phase diagram
• Fractionalized Non-Self-Dual Solutions in the CP(N-1) Model
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles
• Clustering and decomposition for non BPS solutions of the ${\mathbb{CP}}^{N-1}$ models
• The spectral problem of the ABJ Fermi gas
• Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories
• New and Old about Renormalons: in Memoriam Kolya Uraltsev

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