Modular invariance in superstring on Calabi-Yau n -fold with A-D-E singularity

We study the type II superstring theory on the background Rd-1,1xXn , where Xn is a Calabi-Yau n -fold ( 2n+d=10 ) with an isolated singularity, by making use of the holographically dual description proposed by Giveon et al., 1999. 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 Xn . Partition functions of these modular invariants all vanish due to theta function identities and are consistent with the presence of space-time supersymmetry