Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics

Abstract

An 't Hooft anomaly is the obstruction for gauging symmetries, and it constrains possible low-energy behaviors of quantum field theories by excluding trivial infrared theories. Global inconsistency condition is recently proposed as a milder condition but is expected to play an almost same role by comparing high symmetry points in the theory space. In order to clarify the consequence coming from this new condition, we discuss several quantum mechanical models with topological angles and explicitly compute their energy spectra. It turns out that the global inconsistency can be saturated not only by the ground-state degeneracy at either of high symmetry points but also by the level crossing (phase transition) separating those high symmetry points.

Modified instanton sum in QCD and higher-groups

“…Thus, the shift θ YM → θ YM + 2π/p (⇔θ →θ + 2π) gives the extra phase of the partition function by 24) and the overall Z N p phase is characterized by the background gauge fields. When θ YM is quantized in an odd-integer multiple of π/p (half of its domain 2π/p), this relation can be used to obtain the mixed 't Hooft anomaly or the global inconsistency between CP symmetry and Z [1] N× Z [3] p , generalizing [21][22][23][24]68] to cases involving 4-group symmetry. Recently, this type of relation (for 1-form symmetry) itself is understood as the generalized 't Hooft anomaly with the θ-angle periodicity [69], and it naturally leads to the existence of multi-branch structure as will be discussed in the following subsections.…”

Fractional θ angle, ’t Hooft anomaly, and quantum instantons in charge-q multi-flavor Schwinger model

“…However, since the coefficient of the local counterterm is quantized, we cannot take the simultaneous UV regularization so that the gauged partition functions at θ = 0 and θ = π are both C invariant. This is called global inconsistency condition [27,44,123]. The matching condition for global inconsistency [123] is…”

“…This is called global inconsistency condition [27,44,123]. The matching condition for global inconsistency [123] is…”

θ =π in S U N / ℤ N $$ \mathrm{S}\mathrm{U}(N)/{\mathbb{Z}}_N $$ gauge theories

“…However, the trivial theory cannot produce the mixed CP-Z N anomaly, indicating that CP should be spontaneously broken in the SU(N ) gauge theory, since the 1-form Z N symmetry is preserved in the confining phase. This anomaly matching argument has been applied to other theories in [12][13][14][15] for the discussion on their phase structure.…”

“…See also ref. [15] for detailed discussion on the case with odd N . Our discussion confirms that the same conclusion, spontaneous CP violation at θ = π, can be derived independent of N under the assumptions that the mass gap persists for all values of θ and a phase transition happens only once between θ = 0 and θ = 2π.…”