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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.E. Fisher, S.-k. Ma and B.G. Nickel, Critical Exponents for Long-Range Interactions, Phys. Rev. Lett.29 (1972) 917 [INSPIRE].
M.F. Paulos, S. Rychkov, B.C. van Rees and B. Zan, Conformal Invariance in the Long-Range Ising Model, Nucl. Phys.B 902 (2016) 246 [arXiv:1509.00008] [INSPIRE].
E. Yu. Lerner and M.D. Missarov, Scalar Models of P −adic Quantum Field Theory and Hierarchical Models, Theor. Math. Phys.78 (1989) 177 [INSPIRE].
F.J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys.12 (1969) 91 [INSPIRE].
S.S. Gubser, C. Jepsen, S. Parikh and B. Trundy, O(N) and O(N) and O(N), JHEP11 (2017) 107 [arXiv:1703.04202] [INSPIRE].
D.H. Friedan, Nonlinear Models in Two + Epsilon Dimensions, Annals Phys.163 (1985) 318 [INSPIRE].
A. Huang, B. Stoica and S.-T. Yau, General relativity from p-adic strings, arXiv:1901.02013 [INSPIRE].
S.S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys.352 (2017) 1019 [arXiv:1605.01061] [INSPIRE].
M. Heydeman, M. Marcolli, I. Saberi and B. Stoica, Tensor networks, p-adic fields and algebraic curves: arithmetic and the AdS 3/CFT 2correspondence, Adv. Theor. Math. Phys.22 (2018) 93 [arXiv:1605.07639] [INSPIRE].
L. Brewin, Riemann Normal Coordinate expansions using Cadabra, Class. Quant. Grav.26 (2009) 175017 [arXiv:0903.2087] [INSPIRE].
L. Brewin, Riemann normal coordinates, http://users.monash.edu.au/ leo/research/papers/files/lcb96-01.pdf.
E. Gava and R. Jengo, A four-dimensional nonlinear σ-model, Nucl. Phys.B 140 (1978) 510 [INSPIRE].
J. Sak, Recursion relations and fixed points for ferromagnets with long-range interactions, Phys. Rev.B 8 (1973) 281.
J. Honkonen and M.Y. Nalimov, Crossover between field theories with short-range and long-range exchange or correlations, J. Phys.A 22 (1989) 751.
C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys.A 50 (2017) 354002 [arXiv:1703.05325] [INSPIRE].
B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys.B 305 (1988) 545 [INSPIRE].
B. de Wit, M. Lüscher and H. Nicolai, The Supermembrane Is Unstable, Nucl. Phys.B 320 (1989) 135 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1906.10281
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gubser, S.S., Jepsen, C.B., Ji, Z. et al. Non-local non-linear sigma models. J. High Energ. Phys. 2019, 5 (2019). https://doi.org/10.1007/JHEP09(2019)005
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2019)005