I know the definition of the ring of differential operators, by using the smooth coverings. But I do not know how to calculate the global section. 

Can anyone help explain that why the global section of twisted differential operators on $Bun_{G}$ are constant except twisted by $\omega^{1/2}$? 


Here $Bun_{G}$ is the moduli stack of $G$-bundles on a smooth algebraic curve $X$ over $\mathbb{C}$, $G$ is semisimple. $\omega$ is the canonical line bundle. I do not know whether this is true for other algebraically closed field