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Conformal symmetry of the worldsheet theory arises from diffeomorphism and Weyl invariance after gauge fixing of the worldsheet metric. (Is that true? Why can the metric be gauge fixed? Why can one not fix "more", s.t. also the conformal invariance vanishes?)

I think my confusion arises from the fact, that in the lecture we studied the corresponding conformal field theory on the complex plane. I understand, that there is a conformal map, for example, from the cylinder to the complex plane. I do not see how one would obtain such a map if the topology of the worldsheet is more complicated. But this does not mean, that the worldsheet theory is not a conformal field theory, right?

Thank you very much for your help.

physicus