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What are the differences and relations between CFTs defined on the complex plane and CFTs defined on the torus? Are they supposed to be the same CFTs?

I think they should have the same spectra of operators and OPE coefficients. But what about the correlation functions? If I consider the CFTs on the cylinder, then the correlation functions on the cylinder can be obtained from the correlations from the complex plane by a conformal transformation, but what about correlation functions on the torus?

I think another question is how does the topology of the spacetime affect the theory?

Am I missing something?

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    $\begingroup$ The second question - asking how the topology of a space-time affects a CFT - is quite broad, and I'd stick to the first question you've raised which is more focused. $\endgroup$
    – JamalS
    Dec 16, 2016 at 22:08
  • $\begingroup$ @JamalS I will appreciate it if you can provide some references for the second question, thank you! $\endgroup$
    – Nahc
    Dec 16, 2016 at 22:10

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A CFT on any surface must have an associative OPE. This constrains both the spectrum and the OPE coefficients. This condition is enough for the CFT to be consistent on the plane, but not on the torus. On the torus you have the extra condition of modular invariance of the one-point function (which implies modular invariance of the partition function).

So any CFT that exists on the torus also exists on the plane, but there are CFTs that exist on the plane and not on the torus. A trivial example is a CFT whose only primary field is the identity field, which exists on the plane for any value of the central charge $c$, but is modular invariant only if $c=0$.

The plane and the torus are not related by conformal transformations. Actually, not all toruses are conformally related to one another. (For two toruses to be conformally related, they need to have the same value of a complex parameter called the modulus.) So in principle there might be CFTs that exist on some but not all toruses, although I do not know any example.

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  • $\begingroup$ Great answer! I'm curious how this discussion generalizes to higher genus surfaces. Things must be much more complicated, aren't they? $\endgroup$
    – Student
    May 17, 2020 at 14:49
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    $\begingroup$ There are no consistency conditions beyond modular invariance of the torus one-point function, so if a CFT is consistent on the torus, then it is consistent on all Riemann surfaces. Correlation functions become technically more complicated in higher genus, though. $\endgroup$
    – Sylvain Ribault
    May 18, 2020 at 7:08

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