  We study non-local non-linear sigma models in arbitrary dimension, focusing
on the scale invariant limit in which the scalar fields naturally have scaling
dimension zero, so that the free propagator is logarithmic. The classical
action is a bi-local integral of the square of the arc length between points on
the target manifold. One-loop divergences can be canceled by introducing an
additional bi-local term in the action, proportional to the target space
laplacian of the square of the arc length. The metric renormalization that one
encounters in the two-derivative non-linear sigma model is absent in the
non-local case. In our analysis, the target space manifold is assumed to be
smooth and Archimedean; however, the base space may be either Archimedean or
ultrametric. We comment on the relation to higher derivative non-linear sigma
models and speculate on a possible application to the dynamics of M2-branes.
