Centralizers of discrete Temperley-Lieb-Jones subfactors

Discrete, unimodular inclusions of factors $(N\subseteq M, E)$ with $N$ of type $\rm{II}_{1}$ 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 $N\subseteq M^{\phi}$ for the canonical state $\phi=\tau\circ E$, 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 $\rm{II}_{1}$ factor $M$ in terms of the fundamental group of M.