I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me such an example exists, but noone could point out a concrete example or reference to me.




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In the framework of Fuchs, Runkel, Schweigert etc. full CFTs are classified by Morita equivalence classes of Frobenius algebras in a modular tensor category $\mathcal C$ (the representation category of the chiral CFT), so my question more concretely:



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> I am looking for two special symmetric haploid Frobenius algebra objects $A,A'$ in a modular tensor category $\mathcal C$, which are not Morita equivalent but whose full centres $Z(A),Z(A')$ are equivalent as objects (they cannot be equivalent as algebra objects due to a result of Kong and Runkel), i.e. have the same modular invariant partition function $Z(A)\_{ij}=Z(A')\_{ij}$.
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I am also fine with an example in any other formulation of CFT any reference in physics literature, non-degenerate braided subfactors or similar...



Example for non equivalent rational full CFTs with same modular invariant (partition function)
I apologize since my answer will involve shameless self-promotion. You can find one example
of this kind on page 36 of [my slides](http://pages.uoregon.edu/vostrik/talks/beijing.pdf). In this example one Frobenius algebra is commutative
and another is not Morita equivalent to a commutative algebra.


