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?