  The study of rational conformal field theories in the moduli space is of
particular interest since these theories correspond to points in moduli space
where the algebraic and arithmetic structure are usually richer, while also
being points where non--trivial physics occurs (such as in the study of
attractor black holes and BPS states at rational points). This has led to
various attempts to characterize and classify such rational points. In this
paper, a conjectured characterization by Gukov--Vafa of rational conformal
field theories whose target space is a Ricci flat K\"ahler manifold is analyzed
carefully for the case of toroidal compactifications. We refine the conjectured
statement as well as making an effort to verify it, using $T^4$
compactification as a test case. Seven common properties in terms of Hodge
theory (including complex multiplication) have been identified for $T^4$-target
rational conformal field theories. By imposing three properties out of the
seven, however, there remain $\mathcal N = (1,1)$ SCFTs that are not rational.
Open questions, implications and future lines of work are discussed.

High Energy Physics - Theory;Algebraic Geometry;Number Theory