Abstract: In this talk we examine the physical significance of modular invariance in unitary d=2 CFTs. In this context, invariance under T-transformations is essential for locality. In contrast, invariance under the S-transformations represents a form of completeness of the theory, which can be precisely understood as Haag duality for arbitrary multi-interval regions. We apply these ideas to Minimal models. These are traditionally classified using modular-invariant partition functions. However, a more refined classification exists for non-complete models, where the failure of modular invariance—or equivalently, Haag duality—can be quantified by a finite Jones index which related to the quantum dimensions of the model. We use this index to impose constraints on the renormalization group flows between minimal models. These constraints establish selection rules that not only unify several known results but also reveal new insights. Based on arXiv 2408.04011 and  2412.16587