  These notes combine results from two papers by the present authors viz., Part
I (arXiv:2205.10299) and Part II (arXiv:2212.13028) into one streamlined
version for better readability, along with a review on theory of complex
multiplication for non-singular complex projective varieties and complex tori
that is aimed at string theorists. We think that it is worth posting this
edition as a separate entry in arXiv for those reasons, although this edition
contains no essential progress beyond Part I and Part II.
  S. Gukov and C. Vafa proposed a characterization of rational N=(1,1)
superconformal field theories (SCFTs) on 1+1 dimensions with Ricci-flat Kahler
target spaces in terms of the Hodge structure of the target space, extending an
earlier observation by G. Moore. We refined this idea and obtained a
conjectural statement on necessary and sufficient conditions for such SCFTs to
be rational, which we indeed prove to be true in the case the target space is
T^4. In the refined statement, the algebraicity of the geometric data of the
target space turns out to be essential, and the Strominger--Yau--Zaslow
fibration in the mirror correspondence also plays a vital role.
