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Heterotic (0,2) Gepner Models and Related Geometries

On the sad occasion of contributing to the memorial volume ``Fundamental Interactions'' for my teacher Wolfgang Kummer I decided to recollect and extend some unpublished notes from the mid 90s when I started to build up a string theory group in Vienna under Wolfgang as head of the particle physics group. His extremely supportive attitude was best expressed by his saying that one should let all flowers flourish. I hope that these notes will be useful in particular in view of the current renewed interest in heterotic model building. The content of this contribution is based on the bridge between exact CFT and geometric techniques that is provided by the orbifold interpretation of simple current modular invariants. After reformulating the Gepner construction in this language I describe the generalization to heterotic (0,2) models and its application to the Geometry/CFT equivalence between Gepner-type and Distler-Kachru models that was proposed by Blumenhagen, Schimmrigk and Wisskirchen. We analyze a series of solutions to the anomaly equations, discuss the issue of mirror symmetry, and use the extended Poincaré polynomial to extend the construction to Landau-Ginzburg models beyond the realm of rational CFTs. In the appendix we discuss Gepner points in torus orbifolds, which provide further relations to free bosons and free fermions, as well as - simple currents in N=2 SCFTs and minimal models.