Time-Reversal Breaking in QCD$_4$, Walls, and Dualities in 2+1 Dimensions

Abstract: We study $SU(N)$ Quantum Chromodynamics (QCD) in 3+1 dimensions with $N_f$ degenerate fundamental quarks with mass $m$ and a $\theta$-parameter. For generic $m$ and $\theta$ the theory has a single gapped vacuum. However, as $\theta$ is varied through $\theta=\pi$ for large $m$ there is a first order transition. For $N_f=1$ the first order transition line ends at a point with a massless $\eta'$ particle (for all $N$) and for $N_f>1$ the first order transition ends at $m=0$, where, depending on the value of $N_f$, the IR theory has free Nambu-Goldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the $4d$ bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different $3d$ phases. These turn out to be the phases of the recently studied $3d$ Chern-Simons matter theories, thus relating the dynamics of QCD$_4$ and QCD$_3$, and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless $4d$ theory has an $SU(N_f)$ sigma model, the domain wall theory at low (nonzero) mass supports a $3d$ massless $CP^{N_f-1}$ nonlinear $\sigma$-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.

Analyzing Time-Reversal Breaking in Quantum Chromodynamics with Domain Walls and 3D Dualities

The paper explores intricate dynamics in SU(N) Quantum Chromodynamics (QCD) within various spatial dimensions. It centers on the concept of time-reversal symmetry breaking, particularly in relation to domain walls and dualities in different dimensional frameworks. Through a detailed examination, the authors connect the behavior in 4D QCD to phenomena in 3D Chern-Simons matter theories, contributing to discussions of dualities in 2+1 dimensions.

The study explores QCD with N degenerate quarks of mass m, governed by a θ-parameter. A key result is the identification of a first-order phase transition at θ = π for large m, attributed to the spontaneous breaking of time-reversal symmetry. This finding persists into the massless regime (m = 0), depending on the number of flavors, $N_f$. For $N_f = 1$, the transition leads to features consistent with the emergent properties of a massless η particle. With \N_f > 1\, however, the post-transition phases exhibit varying properties: from free Nambu-Goldstone bosons to interacting conformal field theories or free gauge theories.

The role of domain walls and interfaces emerges as particularly significant. These structures delineate distinct 3D phases within the bulk 4D QCD, mirroring the phase structure in 3D Chern-Simons matter theories. Such theoretical predictions are grounded by the correspondences between 4D and 3D frameworks, notably aligning with duality conjectures in 2+1 dimensions. For instance, a massless 4D theory featuring an SU(\N_f) sigma model leads to a domain wall theory that upholds a 3D massless CP\N_f-1\ non-linear σ-model, underpinning a congruence with 2+1 dimensional dynamics.

The narrative of the paper also encompasses considerations around anomalies and their implications in the dynamics of QCD. Specifically, the presence of a ’t Hooft anomaly between time-reversal symmetry and a one-form global symmetry sheds light on the need for non-trivial dynamics within any ensuing domain wall theory, ensuring these anomalies are properly respected at lower energies.

The paper leads to several implications for the understanding of strong interaction dynamics in modern theoretical physics:

• By linking QCD's 4D dynamics to 2+1D dualities within Chern-Simons matter theories, the study unlocks new pathways for exploring emergent quantum field theories.
• It elucidates scenarios where symmetry restoration and breakdown manifest with different degrees of freedom, providing a foundation to predict phenomena in emergent systems.
• In the gauge theory landscape, the work offers insights on tailoring domain wall properties, integral to understanding confinement and dualities in non-Abelian gauge theories.

Future investigations following from this research could amplify interpretations of interface dynamics in QCD, possibly extending to other gauge groups or matter content, and refining the comprehension of quantum anomalies in gauge theories. Moreover, as the study sketches transitions within domain wall theories, a detailed inquiry into these phase transitions may unravel further behavioral intricacies of non-trivial 3D quantum systems. Continued exploration could thus bridge QCD phenomena with broader horizons in theoretical physics, impacting the comprehension of low-dimensional quantum dynamics extensively.