Alpha-induction is a tensor functor producing a new fusion category from a unitary modular tensor category and a Q-system in it. Last year, we proved equivalence of locality (commutativity) of the Q-system and flatness of the alpha-induced bi-unitary connection. We further showed correspondence between the local part of the Q-system in the sense of Böckenhauer-Evans and the flat parts of the alpha-induced bi-unitary connection in this year.
We also proved equivalence of modular invariance of the coupling matrix appearing in a rational 2-dimensional conformal field theory and triviality of its representation theory. We further clarified its physical meaning in connection to violation of the Haag duality and computations of the Rényi entropy.