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

TL;DR

This work investigates SU($N$) Yang-Mills theory on $\mathbb{R}\times\mathbb{T}^3$ with a twisted $\mathbb{Z}_N$ center symmetry to cure infrared divergences and preserve confinement-like center symmetry at weak coupling. By compactifying on a torus and reducing to the 2d $\mathbb{CP}^{N-1}$ model, the authors establish a precise map between YM flat connections and CP($N-1$) vacua, with a one-form symmetry twist matching between the two theories. They show that there are $N$ classical vacua connected by fractional instantons whose tunneling dynamically restores center symmetry, providing a controlled setting for resurgence while highlighting manifold-dependent singularities on the Borel plane. The analysis connects 4d Yang-Mills to 2d non-linear sigma models, clarifying how nonperturbative effects in a compactified, IR-safe framework can inform IR questions and the structure of nonperturbative expansions in gauge theories. The results have implications for adiabatic continuity, confinement mechanisms, and the role of geometry in resurgence in quantum field theory.

Abstract

We study four-dimensional $\mathrm{SU}(N)$ Yang-Mills theory on $\mathbb{R} \times \mathbb{T}^3=\mathbb{R} \times S^1_A \times S^1_B \times S^1_C$, with a twisted boundary condition by a $\mathbb{Z}_N$ center symmetry imposed on $S^1_B \times S^1_C$. 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.

Figures (5)

• Figure 1: We consider Yang-Mills theory on the geometry $\mathbb{R} \times S^1_A\times S^1_B\times S^1_C$. When the torus $E=S^1_A\times S^1_B$ is small, we obtain the two-dimensional $\mathbb{CP}^{N-1}$-model in the remaining directions $\mathbb{R}\times S^1_C$. We include a unit 't Hooft magnetic flux (i.e., twisting by the center symmetry) along $S^1_B\times S^1_C$, which plays rather crucial roles in this paper.
• Figure 2: For the case $N=2$, the moduli space of flat connections ${\cal M}_{\rm flat}$ on $E=\mathbb{T}^2$ is $\mathbb{T}^2/\mathbb{Z}_2$, which as in this figure can be identified with $\mathbb{CP}^1$. The four fixed points of $\mathbb{T}^4/\mathbb{Z}_2$ are mapped into the four singular points on $\mathbb{CP}^1$, represented by black dots.
• Figure 3: The path $\gamma_i$ connects $p$ and $q_i$. Since this is a two-torus, the blue (red) edges are identified, and are the A-cycle and the B-cycle, respectively. The contour $C=ABA^{-1}B^{-1}$ can be deformed into the contour $C'$, which can be decomposed into small cycles surrounding $\gamma_i$'s.
• Figure 4: A fractional instanton for $\mathbb{CP}^1$-model interpolates between two different vacua, $0$ and $\infty$ (the black dots represents singular points, as discussed in Figure \ref{['fig:T4Z2']}). In cylindrical coordinate (so that the geometry is $\mathbb{R}\times S^1$) with fixed $x$, this moves along the $\mathbb{R}$-direction from past to future infinities, as we vary the parameter $t$. The full trajectory, with both $t$ and $x$ varied, is a hemisphere, and when the two such hemispheres are combined we obtain a full sphere. This is a manifestation of the fact that two fractional instantons combine into a single instanton.
• Figure 5: The geometry is more complicated for $N>2$ case than the $N=2$ case in Figure \ref{['fig:CP1fractional']}. However, the idea is the same: the classical moduli space $\mathbb{CP}^{N-1}$ is lifted by twisted boundary condition along $S^1_C$ into $N$ different points $P_1, \ldots P_N$, however tunneling between different these points, as given by fractional instantons, lifts the $N$-fold degeneracy, restoring the center symmetry dynamically.