I'm trying to get a global idea of the world of conformal field theories.
Many authors restrict attention to CFTs where the algebras of left and right movers agree. I'd like to increase my intuition for the cases where that fails (i.e. heterotic CFTs).
What are the simplest models of heterotic CFTs?
There exist beautiful classification results (due to Fuchs-Runkel-Schweigert) in the non-heterotic case that say that rational CFTs with a prescribed chiral algebras are classified by Morita equivalence classes of Frobenius algebras (a.k.a. Q-systems) in the corresponding modular category.
Is anything similar available in the heterotic case?