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 ${\Omega }_5^{\text{Spin}}(BU(2))$ 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 $r\in {\mathbb{Z}}_{\ge 0}$, 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 $r\in {\mathbb{Z}}_{\ge 0}$, 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.

arXiv comment23 pages, 2 figures. Description of the mapping tori for U(2) vs SU(2) has been made clearer

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