Discrete theta angles, symmetries and anomalies

Abstract

Gauge theories in various dimensions often admit discrete theta angles, that arise from gauging a global symmetry with an additional symmetry protected topological (SPT) phase. We discuss how the global symmetry and ’t Hooft anomaly depends on the discrete theta angles by coupling the gauge theory to a topological quantum field theory (TQFT). We observe that gauging an Abelian subgroup symmetry, that participates in symmetry extension, with an additional SPT phase leads to a new theory with an emergent Abelian symmetry that also participates in a symmetry extension. The symmetry extension of the gauge theory is controlled by the discrete theta angle which comes from the SPT phase. We find that discrete theta angles can lead to two-group symmetry in 4d4d QCD with SU(N),SU(N)/\mathbb{Z}_kSU(N),SU(N)/ℤk or SO(N)SO(N) gauge groups as well as various 3d3d and 2d2d gauge theories.

Revisiting Wess-Zumino-Witten terms

“…There is, in addition, a mixed anomaly between the one-form symmetry and the spacetime symmetry, as pointed out recently in [25]. We will now review this mixed anomaly.…”

“…This operation is known as the transgression in algebraic topology, and Γ is said to transgress to α. 25 Summarizing, we find that, under the fibration (4.6),…”

“…3, we review the anomaly of four dimensional SU and SO QCD. In particular, we review the fact that SO(2n c ) QCD has 1-form symmetry in addition to other ordinary 0-form symmetries, and that there is a mixed 't Hooft anomaly between them, as recently pointed out in [25]. In Sec.…”

“…In other words, the gauge theory on the 4d boundary has a mixed 't Hooft anomaly between the 2 1-form symmetry and the spacetime rotation symmetry. The existence of this mixed anomaly was first pointed out in [25], where [SO(N c )×SU(N f )] / 2 was used instead. In that paper it was also pointed out that this mixed anomaly between the magnetic 2 1-form symmetry and the rest of the symmetry in the SO(2n c ) gauge theory is transformed into a 2-group structure between the electric 2 1-form symmetry and the rest of the symmetry in the S pin(2n c ) gauge theory.…”

“…Then what we need to do is understand the mixed anomaly of the 2 1-form symmetry with other symmetries in the ultraviolet, and describe how it is represented in the low-energy sigma model. The first task was very recently performed in [25] which we briefly review. 5 Our remaining task is then to demonstrate how it is realized in the infrared.…”