  A characterization of rational superconformal field theories (SCFTs) on 1+1
dimensions with Ricci-flat Kahler targets was proposed by S. Gukov and C. Vafa
in terms of the Hodge structure of the target space. The article
[arXiv:2205.10299] refined this idea and extracted a set of necessary
conditions for a $T^4$-target N=(1,1) SCFT to be rational; only a partial
effort was made, however, to study whether it also constitutes a sufficient
condition. It turns out that the set of conditions in [arXiv:2205.10299] is not
sufficient, and that it becomes a set of necessary and sufficient conditions by
adding one more condition in the case of $T^4$. The Strominger--Yau--Zaslow
fibration in the mirror correspondence plays an essential role there. At the
end, we also propose a refined version of Gukov--Vafa's idea for general
Ricci-flat Kahler target spaces.
