Differences and relations between CFTs defined on the complex plane and CFTs defined on the torus? 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?
 A: 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.
