How do $theta$-terms not violate gauge invariance?

Question

In the context of QCD (and more generally, any quantum gauge theory in even dimensions), the $theta$-term is
$$
frac{theta}{8pi^2}langle F_Awedge F_Arangle = frac{theta}{32pi^2}langle F_A^{munu}, F_A^{rhosigma}rangleepsilon_{munurhosigma}
$$
and its integral over spacetime is not exactly gauge invariant—instead it transforms under a general gauge as
$$
intfrac{theta}{8pi^2}langle F_Awedge F_Arangle
overset{g}{mapsto}
intfrac{theta}{8pi^2}langle F_Awedge F_Arangle + theta n_g
tag{1}label{1}
$$
where $n_g in mathbb Z$ is the winding number of the gauge transformation $g$ (Ref. Tong’s lecture notes, §2.2). (Edit: I think I am confused—eqref{1} may be wrong.)

(Bonus question: what does eqref{1} look like with $hbar$ not set to unity?)

Ultimately, physical predictions are made with the partition function or ‘quantum-mechanical amplitude’ given by the path-integral
$$
mathscr A = int_{partialOmega}! mathcal D[psi, A] expleft(frac{i}{hbar} int_{Omega}! mathcal Lleft[psi, nabla_{!A}psi, F_Aright]right)
.$$
This is an integral over all gauge field configurations $A$. However, it appears that two physically equivalent gauge field configurations $A$ and $A^g$ separated by a ‘large’ gauge transformation $g$ would contribute differently to the partition function: $A$ contributes $exp(frac{i}{hbar}intmathcal L)$ while $A^g$ contributes $exp(frac{i}{hbar}intmathcal L)exp(itheta n_g)$, which appear to differ by a phase.

Doesn’t this make $mathscr A$ ill-defined, unless $theta in 2pimathbb{Z}$? How does the $theta$ term not spoil gauge invariance in this sense?

Note: I think I might be confusing “the integral of the $theta$-term is discrete over instanton configurations” with “the $theta$-term is gauge invariant modulo a discrete additive factor”. Are both of these accurate?

ɪdɪət strəʊlə · Answer

$newcommand{D}{mathrm{D}}newcommand{Tr}{mathrm{Tr}}$

In the usual treatment, $theta$ is a coupling constant, i.e. a fixed parameter and not a field. Therefore shifting $theta$ to $theta' color{gray}{(neq theta+2pi n,  ninmathbb{Z})}$ definitely changes the physics. Your question is equivalent to asking whether changing the parameters of a potential changes the physics or not; the answer of course is that it changes the physics. Changing $thetamapstotheta'$ is totally unrelated to the gauge transformation $Amapsto A^g$. As @MannyC correctly pointed out in a comment, the $theta$-term is exactly gauge invariant under $Amapsto A^g$.

What you probably had in mind is the observation that $thetamapstotheta+2pi$ does indeed leave the partition function (what you call the quantum-mechanical amplitude) invariant
$$Z[theta+2pi] = Z[theta],tag{1} $$
which is essentially because the second Chern class is integral
$$int c_2(F) = frac{1}{4pi^2}int Tr(F_Awedge F_A) in mathbb{Z}.$$

For a more modern treatment, that goes beyond your question, but is very cool one may wonder whether there is a gauge principle behind (1), I.e. can I consider $theta$ as a gauge field for some kind of symmetry? 
The answer is in fact yes. Using what's known as higher-form symmetries [1], one finds that the gauge field for a $p$-form symmetry is a $(p+1)$-form connection. So the case $p=0$ corresponds to the normal 1-form connections $A$ that you know of, that transform under normal gauge transformations. Putting $p=-1$ we have a $(-1)$-form symmetry whose gauge field is a $0$-form, i.e. a scalar. Then you can think of $theta$ as a gauge field and $theta+2pisimtheta$ as a gauge transformation for the gauge group $mathbb{Z}$, yielding another explanation to why (1) holds.

References

[1] D. Gaiotto, A. Kapustin, N. Seiberg, B. Willet, Generalized global symmetries, arXiv:1412.5148

How do $theta$-terms not violate gauge invariance?

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