Abstract
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.
Highlights
Becomes classically scale invariant, we find logarithmic divergences in one-loop diagrams which can be canceled by counterterms that can be expressed in terms of the target space laplacian of the square of the distance function, together with field redefinitions
The particular structure of counterterm we find suggests that renormalization of our theories have less to do with renormalization of the local metric as normally understood (i.e. Ricci flow) than with an augmentation of the action (1.1) to include the target space laplacian of d(φ(x), φ(y)
A conservative expectation is that once non-local terms are allowed in a field theory, they proliferate and the theory becomes non-renormalizable
Summary
One may derive (2.13) by subtracting an appropriate number of terms in the Taylor series expansion of φ(y) and finding appropriate type II subtractions to bring the result into the form (2.13). The ρ integral in the last line of (2.15) converges, provided s is positive but not an even integer, and provided the coefficients ar are coefficients in the Taylor series expansion of the Bessel function around ρ = 0. These coefficients ar are well known, and from them one can recover the expression (2.14) for the br
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