A 2d rational conformal field theory (RCFT) gives rise to a modular tensor category (MTC) equipped with a Frobenius algebra object (see, for example, <http://arxiv.org/abs/hep-th/0204148>).


Is there an example of inequivalent RCFTs that give rise to the same underlying MTC and Frobenius algebra?



Distinct 2D RCFTs with the same underlying MTC
The moonshine module VOA has trivial representation theory, i.e. the MTC is $Vec$ (and so the Frobenius algebra inside will be trivial). So this should give an example (i.e. take a trivial RCFT). 


