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We consider type II and heterotic string compactifications on an isolatedsingularity in the noncompact Gepner model approach. The conifold-type ADEnoncompact Calabi-Yau threefolds, as well as the ALE twofolds, are modeled by atensor product of the SL(2,R)/U(1) Kazama-Suzuki model and an N=2 minimalmodel. Based on the string partition functions on these internal Calabi-Yauspreviously obtained by Eguchi and Sugawara, we construct new modular invariant,space-time supersymmetric partition functions for both type II and heteroticstring theories, where the GSO projection is performed before the continuousand discrete state contributions are separated. We investigate in detail themassless spectra of the localized modes. In particular, we propose aninteresting three generation model, in which each flavor is in the 27+1representation of E6 and localized on a four-dimensional space-time residing atthe tip of the cigar.

Localized Modes in Type II and Heterotic Singular Calabi-Yau Conformal Field Theories