$\alpha$-induction for bi-unitary connections

Abstract

The tensor functor called $\alpha$-induction produces a new unitary fusion category from a Frobenius algebra object, or a $Q$-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of $N$ to $M$ arising from a subfactor $N\subset M$ of finite index and finite depth, which gives a braided fusion category of endomorphisms of $N$. It is also understood in terms of Ocneanu’s graphical calculus. We study this $\alpha$-induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain $4$-tensors appearing in recent studies of $2$-dimensional topological order. We show that the resulting $\alpha$-induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local $Q$-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.