$\alpha$-induction for bi-unitary connections

TL;DR

This paper explores $ ext{ extalpha}$-induction in the context of bi-unitary connections, linking algebraic structures to topological order and conformal field theory, and characterizing flatness conditions.

Contribution

It extends $ ext{ extalpha}$-induction to bi-unitary connections, providing new insights into their flatness and applications in topological order and conformal field theory.

Findings

$ ext{ extalpha}$-induction yields flat bi-unitary connections with commutative Frobenius algebras

Characterization of finite-dimensional nondegenerate commuting squares

Examples related to chiral conformal field theory and Dynkin diagrams

Abstract

The tensor functor called $\alpha$-induction arises 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 giving a braided fusion category of endomorpshisms of $N$. It is also understood in terms of Ocneanu's graphical calculus. We study this $\alpha$-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares and gives 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 have a commutative Frobenius algebra, or a local Q-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.