Anomaly interplay in U(2) gauge theories

Abstract We discuss anomaly cancellation in U(2) gauge theories in four dimensions. For a U(2) gauge theory defined with a spin structure, the vanishing of the bordism group $$ {\Omega}_5^{\mathrm{Spin}} $$ Ω 5 Spin (BU(2)) implies that there can be no global anomalies, in contrast to the related case of an SU(2) gauge theory. We show explicitly that the familiar SU(2) global anomaly is replaced by a local anomaly when SU(2) is embedded in U(2). There must be an even number of fermions with isospin 2r + 1/2, for r ∈ ℤ≥0, for this local anomaly to cancel. The case of a U(2) theory defined without a choice of spin structure but rather using a spin-U(2) structure, which is possible when all fermions (bosons) have half-integer (integer) isospin and odd (even) U(1) charge, is more subtle. We find that the recently-discovered ‘new SU(2) global anomaly’ is also equivalent, though only at the level of the partition function, to a perturbative anomaly in the U(2) theory, which is this time a combination of a mixed gauge anomaly with a gauge-gravity anomaly. This perturbative anomaly vanishes if there is an even number of fermions with isospin 4r + 3/2, for r ∈ ℤ≥0, recovering the condition for cancelling the new SU(2) anomaly. Alternatively, this perturbative anomaly can be cancelled by a Wess-Zumino term, leaving a low-energy theory with a global anomaly, which can itself be cancelled by coupling to topological degrees of freedom.