Singular Calabi-Yau Manifolds and ADE Classification of CFTs

TL;DR

This paper explores the relationship between singular Calabi-Yau manifolds with ADE singularities and their dual descriptions via Landau-Ginzburg and Liouville theories, providing explicit calculations and character identities.

Contribution

It establishes a detailed correspondence between ADE singularities in Calabi-Yau manifolds and specific conformal field theories, including explicit modular invariant computations and character identities.

Findings

Computed modular invariants for D4, E6, E8 singularities.

Identified and proved character identities across minimal models.

Linked Landau-Ginzburg models with ADE classifications in Calabi-Yau contexts.

Abstract

We study superstring propagations on the Calabi-Yau manifold which develops an isolated ADE singularity. This theory has been conjectured to have a holographic dual description in terms of N=2 Landau-Ginzburg theory and Liouville theory. If the Landau-Ginzburg description precisely reflects the information of ADE singularity, the Landau-Ginzburg model of $D_4,E_6,E_8$ and Gepner model of $A_2\otimes A_2, A_2\otimes A_3, A_2\otimes A_4$ should give the same result. We compute the elements of $D_4,E_6,E_8$ modular invariants for the singular Calabi-Yau compactification in terms of the spectral flow invariant orbits of the tensor product theories with the theta function which encodes the momentum mode of the Liouville theory. Furthermore we find the interesting identity among characters in minimal models at different levels. We give the complete proof for the identity.