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

TL;DR

This paper develops a non-critical string description for Type II strings on Calabi–Yau n-folds with isolated singularities, employing a holographic throat built from ${\cal N}=2$ Liouville theory and ${\cal N}=2$ minimal models. It constructs toroidal partition functions across $d=6,4,2$ that are modular-invariant and organized by the ADE classification, with partition functions vanishing due to theta-function identities, signaling space-time supersymmetry. The results reveal parafermion-like conformal blocks for $d=4$ and $d=2$, and demonstrate that the Liouville sector induces a gap that excludes the graviton, aligning with a decoupled, non-gravitational theory. The work provides a consistency check for this holographic non-critical framework and points to generalizations to broader CY singularities and Gepner-type constructions.

Abstract

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.