Let $\mathbb{S} \rightarrow S^{1}$ be the spinor bundle associated to the connected double cover $\text{Spin}(S^{1}) \rightarrow S^{1}$. Let $\mathbb{D} \rightarrow D^{2}$ be the spinor bundle associated to the bundle $\text{Spin}(D^{2})$. We may view $\text{Spin}(S^{1})$ as sitting in $\text{Spin}(D^{2})$, (as discussed in for example this question). This means that there is a an induced map $\mathbb{S} \rightarrow \mathbb{D}$, and thus if we have a section $f: S^{1} \rightarrow \mathbb{S}$ we may ask if there is a section $F: D^{2} \rightarrow \mathbb{D}$ with the property that $F|_{S^{1}} = f$.
Are there simple conditions on a section $f: S^{1} \rightarrow \mathbb{S}$ that guarantee that $f$ extends to a holomorphic section $F: D^{2} \rightarrow \mathbb{D}$.
The space of sections, $\Gamma(S^{1}, \mathbb{S})$, may be identified with the space of anti-periodic functions on the circle. We may write down an explicit basis \begin{equation} \eta_{n}(t) = e^{i(n+1/2)t}, \quad (t \in [0,2\pi], n \in \mathbb{Z}). \end{equation} A guess would be that functions that are linear combinations of $\eta_{n}$ with $n \geqslant 0$ extend to holomorphic sections $\Gamma(D^{2}, \mathbb{D})$, but I'm not sure how to continue from there.
As an aside, the context that I'm trying to understand this in are these lecture notes on conformal nets by André Henriques.
