I was studying this paper where the authors construct some field theory solutions by wrapping M5-branes on holomorphic curves on Calabi-Yau. I have some questions about their construction.
What they do is to consider the local geometry (2.1) $$\mathbb{C}^2\rightarrow X \rightarrow C_g$$ where $X$ is a Calabi-Yau three-fold. They proceed by stating that to preserve supersymmetry the determinant line bundle of $X$ should be the canonical bundle $K_{C_g}$ of the curve.
Here is my first question: why and how this geometric constrain is needed to preserve supersymmetry? Is it some sort of topological twist?
Then they proceed in stating that this condition imposes that $X$ has to be of the form $X=K_{C_g}\otimes V$ where $V$ is an arbitrary $\mathrm{SU}(2)$-bundle over the curve (are they talking about principal bundle?). Now, depending on the specific form of this last bundle, we get different solutions and here is the big catch:
When the structure group of $V$ reduces from $\mathrm{SU}(2)$ to $\mathrm{U}(1)$, then $X$ is decomposable and the local geometry becomes $$\mathbb{C}^2\rightarrow \mathcal{L}_1\oplus\mathcal{L}_2\rightarrow C_g$$ with the condition $$\mathcal{L}_1 \otimes \mathcal{L}_2=K_{C_g}$$
Could someone shed some light on this step? First I don't understand the dimensionality, the direct sum of the two line bundles has complex dimension $2$ while the Calabi-Yau had complex dimension $3$. I would also like some details on the concept of "decomposable" and how this can be understood physically.
