Singular Calabi–Yau manifolds and ADE classification of CFTs Original Research Article

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 D4,E6,E8 and Gepner model of A2⊗A2,A2⊗A3,A2⊗A4 should give the same result. We compute the elements of D4,E6,E8 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.