Anomaly interplay in $U(2)$ gauge theories

TL;DR

The paper analyzes anomaly cancellation in four-dimensional U(2) gauge theories, using cobordism and η-invariant techniques to distinguish global from perturbative anomalies. It shows that a spin-structured U(2) theory has no global anomalies because $Ω_5^{ ext{Spin}}(BU(2))=0$, while the familiar SU(2) global anomaly appears as a local mixed anomaly requiring an even number of fermions with isospin $2r+1/2$. In theories without a spin structure, the old SU(2) anomaly is still a perturbative mixed anomaly, and the so-called new SU(2) anomaly corresponds to a mixed gauge–gravity perturbative anomaly, cancelable only if there are an even number of fermions with isospin $4r+3/2$. The authors further discuss how Wess–Zumino terms can cancel perturbative anomalies to yield a low-energy theory with a residual global anomaly that may be cancelled by coupling to topological degrees of freedom, and they interpret these results in terms of cobordism invariants, showing there are no additional independent anomalies in the spin-$U(2)$ setting. The work has implications for electroweak-like gauge theories based on U(2) and clarifies the interplay between global and perturbative anomalies in extended gauge groups.

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 $Ω_5^{\text{Spin}}(BU(2))$ 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 $r\in \mathbb{Z}_{\geq 0}$, 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 $r\in \mathbb{Z}_{\geq 0}$, 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.

Figures (2)

• Figure 1: The $\mathcal{A}_1$-module structure for $\mathbb{Z}/2[w_2^\prime,w_3^\prime,w_2^{\prime\prime}]\{UV\}$, up to degree ten.
• Figure 2: The $E_2$ page of the Adams spectral sequence (\ref{['eq: main Adams ss']}), from which one can read off the bordism groups $\Omega_{d \leq 5}^{\frac{\text{Spin} \times U(2)}{\mathbb{Z}/2}}(\text{pt})$.