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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!

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up vote 4 down vote accepted

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.

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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
    
The analogy with drums is a bit misleading. The statement does hold under some further assumptions and it's definitely not obvious that at least some class of CFTs wouldn't in fact be completely determined by the spectrum. If you think it is, please update your answer with the reason :) –  Marek Mar 6 '11 at 10:02
    
@Marek, it is only an analogy, some aspects of any analogy will not really be analogous, otherwise you'd have an isomorphism (or duality, I guess). To your question, I am not aware of a set of conditions under which two CFTs are guaranteed to be identical. –  user566 Mar 6 '11 at 18:35
    
@Moshe: there are still better and worse analogies. I just wanted to know which one is yours :) –  Marek Mar 6 '11 at 23:56
    
@Marek, mine is the excellent kind of course, distingushed by it's insight and clarity. Or at least that's how it resonates with me... –  user566 Mar 7 '11 at 0:52
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