Leading infrared logarithms for the~$\sigma$-model with fields on an~arbitrary Riemann manifold
Teoretičeskaâ i matematičeskaâ fizika, Tome 169 (2011) no. 1, pp. 158-166
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We derive a nonlinear recurrence equation for the infrared leading logarithms (LLs) in the four-dimensional $\sigma$-model with fields on an arbitrary Riemann manifold. The derived equation allows computing the LLs to an essentially unlimited loop order in terms of the geometric characteristics of the Riemann manifold. We reduce solving the $SU(\infty)$ principal chiral field in an arbitrary number of dimensions in the LL approximation to solving a very simple recurrence equation. This result prepares a way to solve the model in an arbitrary number of dimensions as $N\to\infty$.
Keywords:
renormalization group, sigma model, large $N$.
@article{TMF_2011_169_1_a14,
author = {M. V. Polyakov and A. A. Vladimirov},
title = {Leading infrared logarithms for the~$\sigma$-model with fields on an~arbitrary {Riemann} manifold},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {158--166},
publisher = {mathdoc},
volume = {169},
number = {1},
year = {2011},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a14/}
}
TY - JOUR AU - M. V. Polyakov AU - A. A. Vladimirov TI - Leading infrared logarithms for the~$\sigma$-model with fields on an~arbitrary Riemann manifold JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2011 SP - 158 EP - 166 VL - 169 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a14/ LA - ru ID - TMF_2011_169_1_a14 ER -
%0 Journal Article %A M. V. Polyakov %A A. A. Vladimirov %T Leading infrared logarithms for the~$\sigma$-model with fields on an~arbitrary Riemann manifold %J Teoretičeskaâ i matematičeskaâ fizika %D 2011 %P 158-166 %V 169 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a14/ %G ru %F TMF_2011_169_1_a14
M. V. Polyakov; A. A. Vladimirov. Leading infrared logarithms for the~$\sigma$-model with fields on an~arbitrary Riemann manifold. Teoretičeskaâ i matematičeskaâ fizika, Tome 169 (2011) no. 1, pp. 158-166. http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a14/