  Tree tensor network descriptions of critical quantum spin chains are
empirically known to reproduce correlation functions matching CFT predictions
in the continuum limit. It is natural to seek a more complete correspondence,
additionally incorporating dynamics. On the CFT side, this is determined by a
representation of the diffeomorphism group of the circle. In a remarkable
series of papers, Jones outlined a research program where the Thompson group T
takes the role of the latter in the discrete setting, and representations of T
are constructed from certain elements of a subfactor planar algebra. He also
showed that for a particular example of such a construction, this approach only
yields - in the continuum limit - a representation which is highly
discontinuous and hence unphysical. Here we show that the same issue arises
generically when considering tree tensor networks: the set of coarse-graining
maps yielding discontinuous representations has full measure in the set of all
isometries. This extends Jones' no-go example to typical elements of the
so-called tensor planar algebra. We also identify an easily verified necessary
condition for a continuous limit to exist. This singles out a particular class
of tree tensor networks. Our considerations apply to recent approaches for
introducing dynamics in holographic codes.
