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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?

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I guess you are aware of the article where Prof. Rehren includes the heterotic case from the beginning.... –  Marcel Nov 21 '11 at 11:19
That's a nice paper... I was more looking for actual examples of heterotic CFTs: ones that are particularly easy to describe, or that are specially relevant for other purposes. –  André Nov 21 '11 at 15:20

2 Answers 2

The first example that comes to mind is the heterotic string worldsheet theory, described in the original paper of Gross, Harvey, Martinec, & Rohm.

I don't know if there is a classification result for rational heterotic CFTs which generalizes the FRS result. However, if you want to understand the global space of CFTs, you may not want to emphasize rational CFTs anyways. Most CFTs aren't rational.

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Thanks for your answer. I'm now reading this article. If I understand, there's 2 CFTs constructed: one compactified on the $E_8\times E_8$-torus, and one compactified on the $\Gamma_{16}$-torus. Quote: "In order to achieve a consistent string theory involving only left-moving coordinates $X^I$ to cancel anomalies and to preserve the geometrical structure of string interactions, we are forced to compactify on a special torus". Do I understand that, as far as constructing CFTs is concerned, I may disregard those constraints and compactify on any torus? (or not compactify at all) –  André Nov 22 '11 at 14:27

I just found by incidence a simple example in some proceedings of Böckenhauer and Evans below. Namely for $\mathrm{Spin}(8\ell)_1$ (so $D_{4\ell}$ lattice) with $\ell=1,2,\ldots$ there exist modular invariants, which should give rise to heterotic models (by Rehrens paper).

see section 7 in

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