Rational conformal field theory with matrix level and strings on a torus

Study of the matrix-level affine algebra $U_{m,K}$ is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of $U_{m,K}$ modular-invariant partition functions. Here we connect the algebra $U_{2,K}$ to strings on 2-tori describable by rational conformal field theories. As Gukov and Vafa proved, rationality selects the complex-multiplication tori. We point out that the rational conformal field theories describing strings on complex-multiplication tori have characters and partition functions identical to those of the matrix-level algebra $U_{m,K}$. This connection makes obvious that the rational theories are dense in the moduli space of strings on $T^m$, and may prove useful in other ways.