Spinor Bundle on $S^{2}$

Question

What is the spinor bundle on $S^{2}$, I mean how does it look like. Is $S$ a spinor bundle on $S^{2}$ :

If $K$ be the canonical bundle, then $S = K^{1/2} \otimes \Lambda^{0,1}(S^{2})$.

Answer 1

it is isomorphic to the tautological bundle over $\mathbb{CP}^1=S^2$, i.e. to $K^{-1/2}$. The corresponding principal $U(1)$ bundle is the Hopf fibration $S^3\to S^2$.