I am trying to understand the Dirac operators associated to the 2 spinor bundles on $S^1.$ I have been getting very confused about why one bundle has nontrivial harmonic spinors and the other doesn't.(Harmonic spinors are solutions $s$ to the equation $Ds = 0$ where $D$ is the Dirac operator and $s$ is a section.)
Here is my argument (which must be wrong somewhere). We have 2 spin structures given by the connected 2-fold covering and the disconnected 2-fold covering. Since the tangent bundle $TS^1$ is trivial, we can choose the trivial connection on it given by $f \rightarrow df.$ When considered as a connection on the principal bundle of frames (also isomorphic to $S^1$), i.e. as a Lie algebra valued one form on $S^1,$ it must be the zero form.
(As a quick aside, the Lie algebra of $SO(1)$ is just the $0$ Lie algebra, so it seems like there is only one connection on the tangent bundle of $S^1$ since we could only ever have the $0$-form as the connection form on our frame bundle. But this is not true, we can add a 1-form to any connection and get another connection. How can this be?)
(EDIT: Answer provided by Eric: because we have implicitly reduced the structure group of the frame bundle to $SO(1),$ an $so(1)$ valued one form corresponds to a connection compatible with the given metric, and there is only one of these since the torsion of any connection on $S^1$ is zero.)
Ok, so now given either spin structure, the connection must lift to the $0$ connection. Furthermore, any complex line bundle over the circle is trivial, so both cases look exactly the same, and the Dirac operator appears to be $f \rightarrow i\frac{df}{dx}.$
However, I am told that in the case of the connected double cover we should have an additional condition on our $f,$ namely that it should satisfy $f(-x) = -f(x).$ Where have I gone wrong?
(2nd EDIT) I think I know where my confusion stems from. Given a spin structure $P$ on a manifold $M^n$ we can identify sections of the spinor bundle with smooth maps $f: P \rightarrow \mathbf{C}^n$ such that $f(pg) = g \cdot f(p)$ as follows. Take any discontinuous section $s: M \rightarrow P.$ This gives a section $t:M \rightarrow P \times_{Spin(n)} \mathbf{C}^n$ of the spinor bundle by the formula $t(m) = [s(m), f \circ s(m)],$ and by the compatibility condition on $f$ this is independent of the choice of section $s.$
It is via this identification that I understand how sections of the spinor bundle for the connected double cover must be functions $f: S^1 \rightarrow \mathbf{C}$ such that $f(-x) = f(x).$ And, as luck would have it, in this case the Spin structure on $S^1$ is itself, as a space, $S^1,$ so the description of the Dirac operator translates easily via this identification. But in the case of $S^2,$ for example, sections can be identified with maps from $S^3$ to $\mathbf{C}^2$ but the local description of the Dirac operator I see in books (like Lawson/Michelson's Spin Geometry) tells me how to differentiate maps $U \subset S^2 \rightarrow \mathbf{C}^2$ in a trivializing nbhd $U.$ How do I translate the local description of the Dirac operator so that it tells me how to differentiate the map $S^3 \rightarrow \mathbf{C}^2$?