Centralizers of discrete Temperley-Lieb-Jones subfactors

Abstract

Discrete, unimodular inclusions of factors $(N\subseteq M,E)$ with $N$ of type ${\mathrm{II}}_1$ have a natural notion of standard invariant, generalizing the finite index case. When the unitary tensor category of $N$-$N$ bimodules generated by ${}_N{L}^2(M,\tau \circ E{)}_N$ is equivalent to the Temperley-Lieb-Jones category $\text{TLJ}(\delta )$, the associated discrete standard invariants are classified in terms of fair and balanced $\delta$-graphs. Many examples of these subfactors naturally arise in the context of the Guionnet-Jones-Shlyakhtenko (GJS) construction for graphs. In this paper, we compute the discrete standard invariant of the centralizer subfactor $N\subseteq M^{\varphi }$ for the canonical state $\varphi =\tau \circ E$, which is again a discrete subfactor of $\text{TLJ}(\delta )$-type. We show that the associated fair and balanced $\delta$-graph behaves analogously to a universal covering space of the original fair and balanced $\delta$-graph. As an application, we obtain an obstruction to the realization of discrete tracial TLJ-type standard invariants by subfactors of a ${\mathrm{II}}_1$ factor $M$ in terms of the fundamental group of M.