Reading between the lines of four-dimensional gauge theories (1305.0318v5)

Abstract: Starting with a choice of a gauge group in four dimensions, there is often freedom in the choice of magnetic and dyonic line operators. Different consistent choices of these operators correspond to distinct physical theories, with the same correlation functions of local operators in R^4. In some cases these choices are permuted by shifting the theta-angle by 2pi. In other cases they are labeled by new discrete theta-like parameters. Using this understanding we gain new insight into the dynamics of four-dimensional gauge theories and their phases. The existence of these distinct theories clarifies a number of issues in electric/magnetic dualities of supersymmetric gauge theories, both for the conformal N=4 theories and for the low-energy dualities of N=1 theories.

• The paper shows that line operator selections can differentiate physical theories even when local observables appear equivalent.
• The authors develop a framework linking electric/magnetic duality with generalized discrete parameters such as the θ-angle.
• Compactification on nontrivial topologies reveals how non-local observables influence vacuum structures and gauge bundle configurations.

Insightful Overview of Line Operator Choices in Four-Dimensional Gauge Theories

The paper by Aharony, Seiberg, and Tachikawa provides an in-depth exploration of the choices and implications of line operators in four-dimensional gauge theories. This research directly enhances our understanding of the behavior of these theories, particularly regarding electric/magnetic dualities in supersymmetric contexts. The work focuses on developing a framework for understanding how line operator choices influence distinct physical theories, even when operating within the constraints of the same Lie algebra and correlation functions of local operators.

The authors start by decomposing gauge theories into those corresponding to various configurations of magnetic and dyonic line operators. A primary insight is that the selection of line operators prescribed by the gauge group $G$ can lead to distinct physical theories, despite seemingly equivalent local dynamics. These distinctions become apparent upon compactifying the theories on nontrivial topologies, such as $R^3 \times S^1$, where line operators manifest as local operators affecting the Witten index and vacuum structure.

One fundamental result centers on the fact that the permutations of line operator choices impose constraints and adjustments in four-dimensional theories and elucidate outstanding inconsistencies in electric/magnetic duality. This duality, explored here for $N=1$ and $N=4$ supersymmetric gauge theories, facilitates the identification of new dual phases differentiated by additional discrete parameters resembling the standard $\theta$-angle.

Particularly captivating is the way the paper details the effect of line operators in theories like $SU(N)/Z_N$ and SO(N) gauge groups, evidencing that their interplay defines the available gauge group bundles, influencing theories' dynamics significantly. These choices are manipulated through a generalization of the $\theta$-angle and discrete $\theta$-like parameters, which act as essential tools for mapping possible theories.

Strong numerical results underpin this analysis, such as the periodicity tests involving $\theta \to \theta + 2\pi$, leading to different configurations of line operators and their classifications based on their invariance or transformation under duality operations. These findings provide not only a robust mathematical framework but also predict distinct vacuum structures across various compactified topologies.

This exploration of four-dimensional gauge theories extends beyond typical considerations of local operators, shedding light on the implications of non-local observables, specifically line operators, within these contexts. The realization that these operators could introduce unbroken discrete gauge symmetries offers both theoretical and practical value. It opens pathways for advancements in constructing S-duality scenarios and analyzing anomaly cancellations, leading to deeper insights into quantum field theory.

Looking forward, this research presents potential directions for studying more intricate structures of surface operators and their relations to gauge groups' centers, projected in future work on discrete gauge field theories. The methodological approach developed here manages to bridge subtle topological features with physical observables, promising expansions in understanding beyond classical and surface-level interpretations of gauge theory behaviors.