# 2D CFTs and permutation orbifolds

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.

BTW the most radical and trivial example are supersymmetric and other special CFTs whose $Z$ cancels, $Z(\tau,\bar\tau)=0$. There are many of them, so the value of $Z$ can't be uniquely determining the CFT. Well, it would be more interesting to quote examples with the same nonzero $Z$. ;-) –  Luboš Motl Mar 6 '11 at 8:35