[Paper Review] On the Global Structure of Deformed Yang-Mills Theory and QCD(adj) on R^3XS^1

Mohamed M. Anber, Erich Poppitz|arXiv (Cornell University)|Aug 4, 2015

Black Holes and Theoretical Physics8 citations

TL;DR

This paper investigates the vacuum structure of deformed Yang-Mills theory and QCD(adj) on $\mathbb{R}^3 \times S^1$ at small compactification radius $L$, showing how distinct $SU(N_c)/\mathbb{Z}_k$ gauge theories arise via gauging $\mathbb{Z}_k$ subgroups of the one-form center symmetry. It identifies ground states, domain walls, and confining strings, and reveals an emergent Kramers-Wannier duality near the deconfinement transition in finite-temperature compactifications, extending results beyond supersymmetric frameworks.

ABSTRACT

Spatial compactification on $\mathbb R^{3} imes \mathbb S^1_L$ 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 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 $ heta$-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 $\mathbb R^{2} imes \mathbb S^1_\beta imes\mathbb S^1_L$. We argue that, in deformed Yang-Mills theory, the effective theory near the deconfinement temperature $\beta_c \gg L$ 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.

Motivation & Objective

• To understand how different $SU(N_c)/\mathbb{Z}_k$ gauge theories emerge from a common $SU(N_c)$ theory via gauging of $\mathbb{Z}_k$ subgroups of the one-form center symmetry.
• To determine the global structure of the vacuum in these theories using semiclassical methods at small $L$.
• To characterize the effective degrees of freedom—electric and magnetic—under $\mathbb{Z}_k$ actions and identify the resulting ground states.
• To analyze the role of domain walls and confining strings in realizing the vacuum structure and symmetry patterns.
• To study finite-temperature compactification to $\mathbb{R}^2 \times S^1_\beta \times S^1_L$ and identify emergent duality near the deconfinement transition.

Proposed method

• Analyzing the $\mathbb{Z}_k$-gauging of the one-form center symmetry of an $SU(N_c)$ gauge theory to construct distinct $SU(N_c)/\mathbb{Z}_k$ theories.
• Mapping the action of $\mathbb{Z}_k$ on local electric and magnetic degrees of freedom in the effective theory at small $L$.
• Using domain walls and confining strings to classify and visualize the vacuum structure and topological order.
• Constructing the effective field theory near the deconfinement temperature $\beta_c \gg L$ in finite-temperature compactification.
• Identifying an emergent Kramers-Wannier duality in the effective theory that exchanges high- and low-temperature regimes.
• Drawing parallels with the Ising model and $S$-duality in $\mathcal{N}=4$ SYM, while working in a non-supersymmetric context.

Experimental results

Research questions

• RQ1 How do different $SU(N_c)/\mathbb{Z}_k$ gauge theories arise from a single $SU(N_c)$ theory via gauging of $\mathbb{Z}_k$ subgroups of the one-form center symmetry?
• RQ2 What is the structure of the vacuum in these $SU(N_c)/\mathbb{Z}_k$ theories, and how are the ground states classified?
• RQ3 How do domain walls and confining strings mediate the realization of center and chiral symmetries in the vacuum?
• RQ4 What emergent duality arises in the effective theory near the deconfinement transition in $\mathbb{R}^2 \times S^1_\beta \times S^1_L$ compactification?
• RQ5 How does the emergent duality relate to known dualities in the Ising model and $\mathcal{N}=4$ SYM, despite the absence of supersymmetry?

Key findings

• Distinct $SU(N_c)/\mathbb{Z}_k$ theories arise by gauging $\mathbb{Z}_k$ subgroups of the one-form center symmetry of an $SU(N_c)$ theory, with different $\theta$-angles labeling the resulting phases.
• The effective degrees of freedom—electric and magnetic—transform under specific $\mathbb{Z}_k$ actions, which determine the structure of the vacuum and topological excitations.
• The ground states of the $SU(N_c)/\mathbb{Z}_k$ theories are classified by the interplay of domain walls and confining strings, which realize the global symmetry patterns.
• In finite-temperature compactification to $\mathbb{R}^2 \times S^1_\beta \times S^1_L$, an emergent Kramers-Wannier duality appears in the effective theory near $\beta_c \gg L$, exchanging high- and low-temperature regimes.
• This duality shares structural features with both the Ising model and $S$-duality in $\mathcal{N}=4$ SYM, but is derived in a non-supersymmetric context.
• Several results reproduce earlier supersymmetric findings, but the paper establishes new, non-supersymmetric insights into the vacuum structure and duality in deformed Yang-Mills and QCD(adj).