non-local field theory (of Jaffe type) vs strings
As it is well known expectation values of fields are distributions. One usually works with tempered distributions or even with Jaffe fields (https://journals.aps.org/pr/abstract/10.1103/PhysRev.158.1454), all defining local field theories because the space of test functions contains those of compact support. In a sense, rather than writing $\phi(f)$ for a test function $f$ that smeares the operators, one can loosely write $\phi(x)$ because the function $f$ can be made arbitrarily localized in a compact region around a point $x$. That's why we talk about local fields in the first place. Moreover, one can talk about vanishing commutators at spacelike distances (microlocality) precisely because one is able to sharply decide whether two test functions are causally connected, once they have no overlap. 651
This again explains why these fields are considered local.
On the other hand, one could in principle work with non-local fields (non-localizable in the terminology of <PERSON>) where the space of test functions doesn't include those of compact support (this is the case for correlators in momentum space that grow faster than a linear expontential). There is an inherent non-local aspect in these type of fields and perhaps calling them fields is misleading (and some theorems do not longer apply, e.g. the <PERSON> bound, renormalization theorems, only macrolocality holds, not sure about the OPE,...). Moreover, they need infinitely many counterterms in perturbation theory, which is why people started to study those in the sixties, in order to attach the pion-theory and even gravity with a field theory approach.
After this introductory background, here is my question: is there a relation between these non-local field theories and string theory? Have people explored a connection between the type of fields generated by string theory and the non-local ones of <PERSON>? I am asking this because the fact that one can't localize arbitrarily these fields is reminiscent, to me at least, of the problems for gravitons in quantum gravity (e.g. black-hole production that prevents one from exploring arbitrarily short-distance scales) and the notion of local observables. 651