On the Global Structure of Deformed Yang-Mills Theory and QCD(adj) on $\mathbb R^3 \times \mathbb S^1$

TL;DR

This work analyzes how the global structure of non-Abelian gauge theories on ${\mathbb R}^3 \times {\mathbb S}^1$ is encoded in semiclassical dynamics, focusing on ${SU(N_c)}/{\mathbb Z}_k$ theories distinguished by discrete $\theta$-angles and one-form center symmetry gauging.By deriving a long-distance abelianized description with dual photons and a controlled hierarchy $m_W \gg m_{\pmb\phi} \gg m_{\pmb\sigma}$, the authors determine how center symmetry acts on electric and magnetic variables and how this action dictates the vacuum structure and line-operator content.The paper shows that different global structures yield distinct vacua and confinement patterns via a domain-wall/string dichotomy, with explicit vacua counts in dYM and QCD(adj across prime and non-prime $N_c$, and with detailed analysis for ${\mathfrak{su}}(2)$, ${\mathfrak{su}}(3)$, and ${\mathfrak{su}}(4)$ in QCD(adj).A novel thermal aspect reveals a Kramers-Wannier-like duality in the deconfined regime, linking theories with different global structure and drawing connections to Ising-like dualities and $S$-duality ideas in SUSY contexts.Overall, the results extend the Aharony et al. framework on global structure to a broad non-supersymmetric setting, providing a concrete, calculable map between gauging, vacua, and confinement phenomena in deformed YM and adjoint QCD.

Abstract

Spatial compactification on $\mathbb R^{3} \times \mathbb S^1_L$ at small $\mathbb S^1$-size $L$ often leads to a calculable vacuum structure, where various "topological molecules" are responsible for confinement and the realization of the center and discrete chiral symmetries. Within this semiclassically calculable framework, we study how distinct theories with the same $SU(N_c)/\mathbb Z_k$ gauge group (labeled by "discrete $θ$-angles") arise upon gauging of appropriate $\mathbb Z_k$ subgroups of the one-form global center symmetry of an $SU(N_c)$ gauge theory. We determine the possible $\mathbb Z_k$ actions on the local electric and magnetic effective degrees of freedom, find the ground states, and use domain walls and confining strings to give a physical picture of the vacuum structure of the different $SU(N_c)/\mathbb Z_k$ theories. Some of our results reproduce ones from earlier supersymmetric studies, but most are new and do not invoke supersymmetry. We also study a further finite-temperature compactification to $\mathbb R^{2}\times \mathbb S^1_β\times\mathbb S^1_L$. We argue that, in deformed Yang-Mills theory, the effective theory near the deconfinement temperature $β_c \gg L$ exhibits an emergent Kramers-Wannier duality and that it exchanges high- and low-temperature theories with different global structure, sharing features with both the Ising model and $S$-duality in ${\cal N}$$=$$4$ supersymmetric Yang-Mills theory.

Figures (11)

• Figure 1: The Weyl chamber of $\pmb\phi \over 2 \pi$ for $SU(3)$ is the shaded equilateral triangle between the two fundamental weights $\pmb w_1$ and $\pmb w_2$. The dot in the center of the triangle is the center symmetric point $\pmb \rho \over 3$ (\ref{['vacuumcenter']}). The global ${\mathbb Z}_3$ center transformation (\ref{['zenphi']}) acts as a counterclockwise $\pi/3$ rotation $\cal P$ (\ref{['calP']}) around the origin (the vectors ${\cal P} \pmb w_{1,2}$ are shown) followed by a $\pmb w_1$ shift. In effect, this produces a $\pi/3$ rotation of the shaded triangle around its center. In the $SU(3)/{\mathbb Z}_3$ theory, the ${\mathbb Z}_3$ rotation of the Weyl chamber around its center is a gauge symmetry and the Weyl chamber is correspondingly reduced.
• Figure 2: dYM: The $\pmb\sigma\over 2\pi$ plane for $su(3)$. The $SU(3)$ fundamental domain is $\Gamma_w$, spanned by $\pmb w_{1,2}$. A contour plot of the potential (\ref{['dympotential']}) is overlaid with the minima (\ref{['dymminima']}) of the potential for dYM indicated by the dark (red) circles. There is a single ground state for dYM at $\pmb\sigma = 0$ within the $SU(3)$ fundamental domain---but not within the larger domain, the root lattice $\Gamma_r$ spanned by $\pmb \alpha_{1,2}$, for $SU(3)/{\mathbb Z}_3$.
• Figure 3: QCD(adj): The $\pmb\sigma\over 2\pi$ plane for $su(3)$ and the minima of the potential---the extrema of (\ref{['superpotential1']})---indicated by dark (red) circles. There are three minima, at $\pmb \sigma = {2 \pi k \pmb \rho \over 3}$, $k=0,1,2$, within the $SU(3)$ fundamental domain, the weight lattice. As on Fig. \ref{['fig:dymsu3']}, there are three times as many minima within the root lattice (used later in finding the $[SU(3)/{\mathbb Z}_3]_p$ theory ground states).
• Figure 4: Two kinds of loops $C$. The Wilson and 't Hooft (dyonic) loop operators measure the magnetic ($\Phi_m$) or electric ($\Phi_e$) flux (a combination thereof), respectively, through surfaces $\Sigma$ spanning the contour $C$. The two kinds of surfaces shown give rise to the operators (\ref{['operators2']}), (\ref{['operators1']}), respectively.
• Figure 5: The identification of ground states by the action of $\hat{\gamma}^{(1,q)}$ for $[SU(3)/{\mathbb Z}_3]_q$. Left: The $q=0$ theory has three vacua within the unit cell of the root lattice (shown by the solid ovals indicated by different colors). There are true domain walls between them, consistent with the absence of confined local probes. Right: The $q=1 \;(2)$ theories have all three vacua within $\Gamma_r$ identified, the "domain walls" are now strings confining the $W H$ ($W^2 H$) local probes (and their powers).