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$\newcommand{\d}{\mathrm{d}}\newcommand{\U}{\mathrm{U}}\newcommand{\b}[1]{\overline{#1}}\newcommand{\C}{\mathbb{C}}\newcommand{\ex}[1]{\mathrm{e}^{#1}}\newcommand{\i}{\mathrm{i}}$ Consider a free theory of a complex $\U(1)$ gauge-field, $a\in\Omega^1(X;\C)$, $$S[a] :=\frac12\int_X\b{\d a}\wedge\star\d a.$$ The gauge invariance of this theory, $\U(1)_\C$, is a complexified version of $\U(1)$ where a gauge transformation is $\delta_\lambda a = \d \lambda$, with a complex parameter $\lambda\in\Omega^0(X;\C)$.

This theory has, moreover, a real $\U(1)$ zero-form global symmetry acting on the gauge field as $$a\mapsto \ex{\i\varphi} a, \qquad \varphi\in\mathbb{R}.$$ One might be tempted to background-gauge it by allowing $\varphi$ for some curvature and compensating by coupling the theory minimally to a real flat $\U(1)$-connection $B$, as $$S[a,B]:=\frac12\int_X \b{\left(\d a - B\wedge a\right)}\wedge\star \left(\d a - B\wedge a\right),$$ where $B$ transforms under $\U(1)$ as $B\mapsto B+\i\,\d\varphi$, and looking at the resulting partition function as a functional of $B$ $$ Z[B]:= \int\frac{\mathrm{D}a}{\text{vol}(\Omega^0(X;\C))}\ex{-S[a,B]}.$$

However one can quickly realise that $Z[B]$ doesn't make sense, as the coupling to $B$ broke gauge-invariance $\U(1)_\C$. Indeed, $S[a,B]$ contains a term that behaves like a mass term for $a$, thus violating gauge invariance. For example, if I took the connection $B$ to be a constant (which I can, since $B$ is just a background field), it would be exactly a mass term for $a$. Moreover, I can't see a way to say make $B$ transform under $\U(1)_\C$ in a way that cancels the unwanted terms. Nor can I imagine adding extra terms in $B$ and $a$ that could cancel it.

In some ways, this looks a lot like a mixed anomaly. The theory simply doesn't allow me to gauge $\U(1)$. But in other ways, it seems a bit more complicated. Usually, with mixed anomalies, the background-gauged theory makes perfect sense; it is just the gauge transformations of the background-gauged symmetry that don't make sense. After all, the symmetry is still global. Here, not even the background-gauged partition function makes sense. You might say, oh but that's because $\U(1)_\C$ is already gauged. That's probably right, but, if we momentarily think of $S[a]$ as just the kinetic terms for the gauging of a global $\U(1)_\C$ symmetry, in the background-gauged $\U(1)_\C\times\U(1)$ version of the theory, the anomaly is not there at all, as it only appears through the kinetic terms of $\U(1)_\C$ and in the background-gauged version $a$ would be flat. So the theory wouldn't warn me that I can't gauge both symmetries until it's too late.

So my question is, is this actually a mixed anomaly? Has it been studied in the literature? (it seems too simple to not have been studied) If so, where does it fall into, classification-wise? Or have I just missed something and everything is fine and the theory makes sense?

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Okay, I'm an idiot. Gauge invariance is still there and life still makes sense. The solution is that once $B$ is introduced, $a$ no longer transforms as $\delta_\lambda a = \mathrm{d}\lambda$, but rather $\delta_\lambda a = (\mathrm{d}-B)\lambda$ (and $\delta_\lambda B = 0$). Since $B$ is a flat one-form, the operator $\mathrm{d}_B:=\mathrm{d}-B\wedge$ is still nilpotent so $\delta_\lambda\mathrm{d}_B\; a = \mathrm{d}_B\,\delta_\lambda a = 0$.

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