Abstract
SU(N ) gauge theory is time reversal invariant at θ = 0 and θ = π. We show that at θ = π there is a discrete ’t Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at θ = π the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at θ = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at θ = π, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at θ = π is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O(2)-symmetric fixed points. It may also be that the four-dimensional theory around θ = π is gapless, e.g. a Coulomb phase could match the underlying anomalies.
Highlights
One of the central tools for analyzing strongly coupled systems is ’t Hooft’s anomaly matching [1]
It is sometimes impossible to do that in spite of the fact that G is a true symmetry of the theory. ’t Hooft argued that the obstruction to coupling G to classical background gauge fields is preserved under the Renormalization Group flow
It is believed that SU(N ) gauge theory without matter confines for all values of θ and the 1-form ZN symmetry is always unbroken at zero temperature
Summary
One of the central tools for analyzing strongly coupled systems is ’t Hooft’s anomaly matching [1]. The obstruction is usually referred to as an anomaly This idea has had powerful applications, especially when the associated obstruction and symmetry G are continuous and the theory has a weak coupling limit. This is due to the fact that the anomaly could be computed explicitly (for a review see [2]). There is no convincing argument that the theory is gapped at zero temperature around θ = π, or that it is confined at finite temperature around θ = π (in the sense that space-like loops have an area law) We investigate these different scenarios in light of the constraints from anomaly matching. In this case a natural proposal involves a four-dimensional Coulomb phase. (There could be an interacting 4d CFT for special values of θ.)
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