Non-local non-linear sigma models

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.