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arXiv 2001.07731hep-th
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$U(2)$规范理论中的Anomaly相互作用

Anomaly interplay in $U(2)$ gauge theories

Joe Davighi, Nakarin Lohitsiri

备注：
23 pages, 2 figures. Description of the mapping tori for U(2) vs SU(2) has been made clearer
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摘要：

本文讨论了四维$U(2)$规范理论中的反常抵消。对于一个定义了自旋结构的$U(2)$规范理论，边群$Ω_5^{\text{Spin}}(BU(2))$的消失意味着不可能有全局反常，这与$SU(2)$规范理论的相关情况相反。我们明确表明，熟悉的$SU(2)$全球异常被替换为本地异常时，$SU(2)$嵌入在$U(2)$。对于$r\in \mathbb{Z}_{\geq 0}$，必须有偶数个具有同位旋$2r+1/2$的费米子，才能消除这种局域反常。如果定义的$U(2)$理论没有自旋结构的选择，而是使用自旋- $U(2)$结构，当所有费米子（玻色子）都有半整数（整数）同位旋和奇（偶）$U(1)$电荷时，这是可能的，那么情况就更加微妙了。我们发现，最近发现的“新的$SU(2)$全球异常”也是等价的，虽然只是在配分函数的水平上，在$U(2)$理论，这是一个混合规范异常与规范重力异常的组合微扰异常。对于$r\in \mathbb{Z}_{\geq 0}$，如果有偶数个具有同位旋$4r+3/2$的费米子，则这种微扰反常消失，恢复了消除新的$SU(2)$反常的条件。或者，这种微扰异常可以被Wess-Zumino项抵消，留下一个具有全局异常的低能理论，它本身可以通过耦合到拓扑自由度来抵消。

英文摘要：

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.

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