2D CFTs and permutation orbifolds

Question

Suppose we have 2 systems with the same partition function, does this mean the 2 systems are the same?

For example, in 2D CFTs, would the equality of two partition functions imply that the underlying theories are the same (in the CFT sense, I mean same central charge, same OPE, etc).

Suppose we take the $\text{N}^{th}$ symmetric product of a mother CFT with a partition function $Z(\tau,\bar{\tau})$ and then I orbifold by the permutation group $S_N$ or any cyclic subgroup $\mathbb{Z}_N$ to get the partition function of the permutation orbifold $\mathbf{Z}$. Now suppose we find another system with the same partition function $\mathbf{Z}$, does this mean that this system should be equivalent to the permutation orbifold?

Please feel free to edit or correct my question.

Thank you all in advance!

Answer 1

The partition function tells you about spectrum of dimensions and degeneracies. It doesn't tell you about OPEs (and higher point functions), so it doesn't determine the theory uniquely. The problem is somewhat reminiscent of the question "can you hear the shape of a drum?", for which the answer is negative: you can have drums of different shapes that produce the same harmonics. That would be analogous to two different CFTs that have the same spectrum and hence partition function, but differ in more detailed properties.