Particle on a circle with magnetic flux$.$

  1. 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.

  2. 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.