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I was talking about K3 surfaces with some physicists, and one of them told me that the ${\cal N}=4$ superconformal field theories with central charge 6 are expected to be relatively scarce. In particular, one should expect a lot of a priori different theories (e.g., those coming from sigma models whose targets are different hyperkähler surfaces, or the Gepner model) to be isomorphic. I have not found similar statements in the mathematical literature, but it sounds like a statement that, if suitably tweaked, could conceivably make sense to mathematicians.

Question: Where can I find such a claim (and perhaps additional justification)?

Also, I am curious to know if there are underlying physical principles behind such a claim, or if it was conjectured due to a scarcity of characters (i.e., the space of suitable modular/Jacobi forms is small), or perhaps some combination.

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For a survey on what was known in 1999 on the subject, there is the review A Hiker's Guide to K3 - Aspects of N=(4,4) Superconformal Field Theory with central charge c=6 by Werner Nahm and Katrin Wendland. I have not been following this subject, so I am not sure whether the current picture is substantially different.

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Some of the papers citing that review might also be relevant.

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  • $\begingroup$ Thank you. The Nahm-Wendland paper suggests that the moduli space of such theories is somewhat more subtle than I had been led to believe. $\endgroup$
    – Scott Carnahan
    Commented Sep 15, 2011 at 12:09

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