Modular Invariance in Superstring on Calabi-Yau n-fold with A-D-E Singularity

We study the type II superstring theory on the background $\br^{d-1,1}\times X_n$, where $X_n$ is a Calabi-Yau $n$-fold ($2n+d=10$) with an isolated singularity, by making use of the holographically dual description proposed by Giveon-Kutasov-Pelc (hep-th/9907178). We compute the toroidal partition functions for each of the cases $d=6,4,2$, and obtain manifestly modular invariant solutions classified by the standard $A-D-E$ series corresponding to the type of singularities on $X_n$. Partition functions of these modular invariants all vanish due to theta function identities and are consistent with the presence of space-time supersymmetry.