Consider a Calabi-Yau threefold $X$ with an elliptic curve $E$ living inside it. Furthermore, assume $E$ is super-rigid, i.e., for any *stable* map from a nonsingular curve $\mu:C\to E$ we have $H^0(C,\mu^*(N_{E|X}))=0$. In particular it's rigid, i.e., $H^0(E,N_{E|X})=0$. I'm trying to understand the geometric implications of super-rigidity.

Then (see 1.1 here) rigidity implies that there is a short exact sequence: $$ 0\to L\to N\to L^\vee\to 0, $$ where $L$ is a non-trivial degree 0 line bundle (I think I can show this).

On the other hand, in the paper linked above it's claimed that super-rigidity is equivalent to $L$ being a nontorsion element of $Pic^0 E$, which I don't know how to prove. Any help with it is greatly appreciated.