From 4d Yang-Mills to 2d ${\mathbb{CP}}^{N-1}$ model: IR problem and confinement at weak coupling

Abstract

We study four-dimensional $\mathrm{SU}(N)$ Yang-Mills theory on $\mathbb{R}\times {\mathbb{T}}^3=\mathbb{R}\times S_A^1\times S_B^1\times S_C^1$, with a twisted boundary condition by a ${\mathbb{Z}}_N$ center symmetry imposed on $S_B^1\times S_C^1$. This setup has no IR zero modes and hence is free from IR divergences which could spoil trans-series expansion for physical observables. Moreover, we show that the center symmetry is preserved at weak coupling regime. This is shown by first reducing the theory on ${\mathbb{T}}^2=S_A\times S_B$, to connect the model to the two-dimensional ${\mathbb{CP}}^{N-1}$-model. Then, we prove that the twisted boundary condition by the center symmetry for the Yang-Mills is reduced to the twisted boundary condition by the ${\mathbb{Z}}_N$ global symmetry of ${\mathbb{CP}}^{N-1}$. There are $N$ classical vacua, and fractional instantons connecting those $N$ vacua dynamically restore the center symmetry. We also point out the presence of singularities on the Borel plane which depend on the shape of the compactification manifold, and comment on its implications.

37 pages, 5 figures; v2:references added

References in corpus (10)

• Generalized Global Symmetries
• Theta, Time Reversal, and Temperature
• Volume independence in large Nc QCD-like gauge theories
• Abelian duality, confinement, and chiral symmetry breaking in QCD(adj)
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Instanton constituents in the O(3) model at finite temperature
• Theta dependence of the vacuum energy in the SU(3) gauge theory from the lattice
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Notes on natural inflation

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• Modified instanton sum in QCD and higher-groups
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• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
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• Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching
• On exact-WKB analysis, resurgent structure, and quantization conditions
• Anomaly matching in QCD thermal phase transition
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• Parity and the modular bootstrap
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• Confinement-Deconfinement Crossover in the Lattice $\mathbb{C}{P}^{N-1}$ Model
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• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Strongly coupled QFT dynamics via TQFT coupling
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• Winding theta and destructive interference of instantons
• Confinement as Analytic Continuation Beyond Infinity
• Resurgence of one-point functions in a matrix model for 2D type IIA superstrings
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