What are the Dirac operators on $S^1$? This is crossposted at stack exchange as https://math.stackexchange.com/questions/248391/dirac-operators-on-s1.
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.
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 $f(-x) = -f(x).$ With this extra condition, there cannot be harmonic spinors on the spinor bundle associated to the connected spin structure. Where have I gone wrong?
 A: This one is tricky and extremely confusing. The bundle of spinors  is a complex line bundle so it is trivializable.   You detect the   spin structure only if you look at  the Dirac operator.     For one  spin structure the Dirac operator has  a kernel, for the other, it does not. 
For a more detailed  discussion on the pathological case of spin structures on $S^1$ I  recommend you to have a look at the discussion on spin structures on page 150-151 of these notes. There I discuss Milnor's point of view on  this subject, but even Milnor in  his nice little paper  on this subject (see the precise  reference in the above notes) is not very careful about this issue. 
Addendum.    Over   $S^1$ we have two real line bundles, $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ the trivial line bundle $\underline{\bR}$ and the nontrivial line bundle $\tilde{\underline{\bR}}$ with  nontrivial first Stieffel-Whitney class $w_1$. Note that $\tilde{\underline{\bR}}$ can be identified with the tautological (real) line bundle over the projective line $\bR\mathbb{P}^1$.  These line bundles are equipped with natural metrics and compatible connections $\nabla$ and respectively $\tilde{\nabla}$. Something miraculous happens. Although these two real lines bundles are not isomorphic, the  complex line bundles $\underline{\bR}\otimes\bC$ and $\tilde{\underline{\bR}}\otimes\bC$ are isomorphic  as  complex line bundles over $S^1$ (duh!).   The connections $\nabla$ and $\tilde{\nabla}$ induce connections on the complexifications  $\underline{\bR}\otimes\bC$ and $\tilde{\underline{\bR}}\otimes\bC$ and we obtain two Dirac operators $\newcommand{\ii}{\boldsymbol{i}}$  $D=-\ii\nabla_\theta$ and $-\ii\tilde{\nabla}_\theta$ on the same  line bundle.   However, the isomorphism $\Phi$ that maps $\underline{\bR}\otimes\bC$ to  $\tilde{\underline{\bR}}\otimes\bC$ does not map the complexification of $\nabla$ to the complexification of $\tilde{\nabla}$ so that the operators $D$  and $\tilde{D}$ are not conjugate to each  other.   The operator $D$ is $-\ii\frac{d}{d\theta}$,  and
$$ \Phi^{-1}\tilde{D}\Phi =-\ii\frac{d}{d\theta}+\frac{1}{2}= D+\frac{1}{2}. $$
