Abstract
We explore the low energy dynamics of the four siblings of Lorentz symmetry enriched SU(2) Yang-Mills theory with a theta term at $\theta=\pi$ in $(3+1)$d. Due to a mixed anomaly between time reversal symmetry and the center symmetry, the low energy dynamics must be nontrivial. We focus on two possible scenarios: 1) time reversal symmetry is spontaneously broken by the two confining vacua, and 2) the low energy theory is described by a U(1) Maxwell gauge theory (e.g. U(1) spin liquid in condensed matter) which is deconfined and gapless while preserving time reversal symmetry. In the first scenario, we first identify the global symmetry on the time reversal domain wall, where time reversal symmetry in the bulk induces a $\mathbb{Z}_2$ unitary symmetry on the domain wall. We discuss how the Lorentz symmetry and the unitary $\mathbb{Z}_2$ symmetry enrich the domain wall theory. In the second scenario, we relate the symmetry enrichments of the SU(2) Yang-Mills to that of the U(1) Maxwell gauge theory. This further opens up the possibility that SU(2) QCD with large and odd flavors of fermions could be a direct second order phase transition between two phases of U(1) gauge theories as well as between a U(1) gauge theory and a trivial vacuum (e.g. a trivial paramagnet), where the gauge group is enhanced to be non-Abelian at and only at the transition. We characterize these transitions, and name them as Gauge Enhanced Quantum Critical Points.
Highlights
The SU(N ) Yang-Mills theory is a non-Abelian gauge theory with a gauge group SU(N ) described by the action S=− 1 Tr(F ∧ F ) + θ Tr(F ∧ F ), (1) 4g2 M48π 2 M4 which admits a topological term parameterized by a variable θ
Various evidences including ’t Hooft anomalies [2,8], deformation of supersymmetric Yang-Mills [9,10,11], and holographic calculation in the large N limit [12,13] provide various constraints on the low-energy dynamics, which we summarize as the standard lore of YangMills
We review the dynamics of SU(N ) Yang-Mills as a function of θ ∈ [0, 2π )
Summary
The SU(N ) Yang-Mills theory is a non-Abelian gauge theory with a gauge group SU(N ) described by the action. Various evidences including ’t Hooft anomalies [2,8], deformation of supersymmetric Yang-Mills [9,10,11], and holographic calculation in the large N limit [12,13] provide various constraints on the low-energy dynamics, which we summarize as the standard lore of YangMills. It is widely believed that at the low energy, the theory confines and the center symmetry is unbroken. States [1,2,18] Such spontaneous broken of time reversal has been shown for large N Yang-Mills, where as one tunes from θ < π to θ > π a first-order phase transition has been observed [12,13]. It is desirable to study all possible scenarios of SU(2) Yang-Mills in detail
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