Particle on a circle with magnetic flux$.$ I am trying to understand the model studied in 1905.09315 §2, to wit, a $0+1$ dimensional theory with target space $\mathbb S^1$ with non-trivial magnetic flux:
$$
\mathcal L=\frac12m\dot q^2-\frac{i}{2\pi}\theta\dot q\tag{2.1}
$$
where $q\colon \mathbb R\to \mathbb S^1$ an angular variable, and $\theta\in \mathbb S^1$ the magnetic flux.
The main goal of the section is to study what happens when we make $\theta$ position dependent,
$$
\mathcal L_\theta\to -\frac{i}{2\pi}\theta(t)\dot q(t)
$$
whose action
$$
-\frac{i}{2\pi}\int_\mathbb R\theta(t)\dot q(t)\mathrm dt\tag{2.3}
$$
is apparently ill-defined. To fix this, the authors introduce local patches $\mathbb S^1=\cup_i U_i$ and a lift $\tilde\theta_i\colon U_i\to\mathbb R$ such that, on $U_i\cap U_{i+1}$, we have $\theta_i=\theta_{i+1}+2\pi n_i$ for some integer $n_i$. With this, the authors set
$$
S_\theta=i\sum_{i=1}^n\left[\frac{1}{2\pi}\int_{t_{i-1}}^{t_i}\tilde\theta_i(t)\dot q(t)\mathrm dt-n_i q(t_i)\right]\tag{2.6}
$$
where $t_i$ is some point in $U_i\cap U_{i+1}$. It seems as if the authors are implicitly regarding $\theta$ as a section of a circle bundle rather than as a function. (In the paper they always write the action in exponentiated form; this is probably more correct, because it is only defined mod $2\pi$, but I only write the argument here to simplify the notation).
Furthermore, the authors claim that if $\theta$ is not constant (and has non-zero winding number), then the shift symmetry $q\to q+\chi$ is broken, and this is regarded as an anomaly, from where one can deduce that the ground state cannot be trivial for all $\theta\in\mathbb S^1$.
Questions:


*

*Why is $(2.3)$ ill-defined, and how does $(2.6)$ fix this problem?

*Why does the breaking of $q\to q+\chi$ imply that the ground state must become degenerate at some point in parameter space?

 A: *

*The Lagrangian density is supposed to be a $\mathbb{R}$-valued function. As long as it only includes derivatives of functions on the circle, it is straightforwardly $\mathbb{R}$-valued because the derivatives are tangent space-valued and the tangent space of a 1d manifold is $\mathbb{R}$.
However, $\theta$ is $S^1$-valued, so on the face of it, the term $\theta(t)\dot{q}(t)$ is not an $\mathbb{R}$-valued object. It's actually not an object at all because it is not clear what the operation between $\theta$ - a point on $S^1$ - and $\dot{q}$ - a vector in the tangent space of $S^1$ - is supposed to be. Hence (2.3) is ill-defined without further elaboration.
The expression in local patches fixes this because the local $\bar{\theta}_i$ are $\mathbb{R}$-valued. This expression "should" be thought of as $\exp\left(\int_{S^1} \theta \mathrm{d}q\right)$ and is the equivalent of Wilson line integrals $\exp\left(\int_\gamma A\right)$ for generic gauge fields, where if the Wilson line lies in more than one patch, you get a very similar expression when you write it in local coordinates. 
$\theta$ here really is a section of the bundle $\mathbb{R} \to \mathbb{R}/2\pi\mathbb{Z} = S^1$, where you should think of the l.h.s. $\mathbb{R}$ as a helix with the same radius as the circle.

*Coupling the theory to a background gauge field $A$ at constant $\theta$, we see that varying it smoothly from $0$ to $2\pi$ incurs an anomalous term for the shift symmetry $\chi$ in the partition function (the authors' eq. (2.11)) $A$:
$$ Z[\theta + 2\pi,A] = Z[\theta,A]\exp\left(-{\mathrm{i}\int A \mathrm{d}t}\right)$$
and the anomalous term is discrete ("flux quantization") and can be interpreted as a discrete jump in the Chern-Simons level of the background field. 
Note that the anomalous term is exactly of the form of the anomaly for the global shift symmetry at varying $\theta$ if one identifies $A = \dot{\theta}\chi$. It is not the breaking of the global symmetry that implies the degenerate ground state, but the breaking also implies this discrete jump anomaly in a background where that global symmetry is gauged.
This means that there must be a non-unique ground state somewhere in $\theta\in[0,2\pi)$ by the following reasoning (I'm merely rephrasing the authors' argument on page 23 a bit more explicitly):
As we vary $\theta$ smoothly, the energy of the ground state we start with also varies smoothly. We also know that the theory is truly $2\pi$ periodic when we turn the background $A$ off, i.e. reaches the same spectrum again. Since in principle one can scale the energy levels as far apart as one likes, there is a version of this theory where the partition function is strongly dominated by the ground state, but anomaly hasn't shrunk at all, so it looks as if the "ground state" has changed. 
This happens where there is level crossing for $\theta \in [0,2\pi)$, since then the ground state becomes degenerate with a higher-level state at some point, after which the higher-level state becomes the ground state. We are not "allowed" to "switch" to this new ground state at the degeneracy point because we are varying $\theta$ smoothly, but this would not yield a smooth $E(\theta)$ (if you take the lower half of a crossing of two lines, you get a triangle shape, not a smooth line), so the smooth variation in $\theta$ cannot detect this.
A: The flux parameter $\theta$ belongs to $U(1)$ because the only way that it can be measured is by means of an Aharonov-Bohm scattering experiment whose cross section depends on $\sin(\frac{\theta}{2})$ which implies that $\theta$ can be measured only modulo $2 \pi$, or $e^{i\theta} \in U(1)$.
Thus, to be precise, the condition $\theta \in \mathbb{R}/2\pi \mathbb{Z}$ indicates that $\theta$ belongs to the Lie algebra of the electromagnetic group $U(1)$ (not a section in a $U(1)$ bundle).
It should be worthwhile to mention that there is a non-Abelian generalization where fluxes like $\theta$ are generalized to holonomies belonging to a non-Abelian gauge group $G$:
$$G \ni h = \mathrm{P} e^{i\int A}$$
With $A$ being a flat connection($F_A=0$). For example, Non-Abelian Higgs models in $2+1$ dimensions have these kind of non-Abelian vortex solutions.
Before answering your second question, let me refer you to Balachandran and de Queiroz, where they evaluate explicitly in section 2, the action of the parity $P$ and the time reversal $T$ operator on this system. They show that $P$ and $T$ are anomalous except at $\theta=0$ and $\theta = \pi$, but their product $PT$ is not. (According to the $CPT$ theorem the product can be taken as the charge conjugation $C$).
That being the case, the problem that Cordova, Freed, Lam and Seiberg want to address is why at the point $\theta=0$ the ground state is unique and at $\theta = \pi$ it is not.  While the $\theta=0$ can be explained by Kramer's theorem for Bosons: There is a unique vacuum and the theory is gapped, the case $\theta=\pi$ needs an explanation.
Before, proceeding, let me mention that their  attribution of the ground state degeneracy to an anomaly is based on the relation of the anomaly to the index theorem: whenever there is an anomaly (manifested by the non-invariance of the partition function by a $1-$ cocycle), there is a degenerate vacuum representation of the symmetry.
Their explanation is based on the notion of mixed (generalized) gauge symmetry where they argue that the charge conjugation together with U(1) symmetry, together, join into a mixed gauge symmetry in which the $U(1)$ symmetry is generated by a $-1$ form (negative one form) gauge field and the time reversal by means a discrete gauge field. This is the symmetry which is anomalous at $\theta=\pi$ and results in the ground state degeneracy.
