  We review the physical meaning of modular invariance for unitary conformal
quantum field theories in d=2. For QFT models, while T invariance is necessary
for locality, S invariance is not mandatory. S invariance is a form of
completeness of the theory that has a precise meaning as Haag duality for
arbitrary multi-interval regions. We present a mathematical proof as well as
derive this result from a physical standpoint using Renyi entropies and the
replica trick. For rational CFT's, the failure of modular invariance or Haag
duality can be measured by an index, related to the quantum dimensions of the
model. We show how to compute this index from the modular transformation
matrices. The index also appears in a limit of the Renyi mutual informations.
Cases of infinite index are briefly discussed. Part of the argument can be
extended to higher dimensions, where the lack of completeness can also be
diagnosed using the CFT data through the thermal partition function and
measured by an index.
