Sigma model renormalization group flow, 'central charge' action, and Perelman's entropy
Abstract
Zamolodchikov's ctheorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be a gradient one to all loop orders. However, the monotonicity of the flow of the targetspace metric is not obvious since the metric on the space of metricdilaton couplings is indefinite. To leading (oneloop) order when the RG flow is simply the Ricci flow the monotonicity was proved by Perelman [G. Perelman, math.dg/0211159.] by constructing an 'entropy' functional which is essentially the metricdilaton action extremized with respect to the dilaton with a condition that the targetspace volume is fixed. We discuss how to generalize the Perelman's construction to all loop orders (i.e. all orders in {alpha}{sup '}). The resulting entropy is equal to minus the central charge at the fixed points, in agreement with the general claim of the ctheorem.
 Authors:
 Blackett Laboratory, Imperial College, London SW7 2AZ (United Kingdom)
 Publication Date:
 OSTI Identifier:
 21020170
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevD.75.064024; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ACTION INTEGRAL; COUPLING; ENTROPY; METRICS; QUANTUM FIELD THEORY; RENORMALIZATION; SIGMA MODEL; SPACE; STRING MODELS
Citation Formats
Tseytlin, A. A. Sigma model renormalization group flow, 'central charge' action, and Perelman's entropy. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.064024.
Tseytlin, A. A. Sigma model renormalization group flow, 'central charge' action, and Perelman's entropy. United States. doi:10.1103/PHYSREVD.75.064024.
Tseytlin, A. A. Thu .
"Sigma model renormalization group flow, 'central charge' action, and Perelman's entropy". United States.
doi:10.1103/PHYSREVD.75.064024.
@article{osti_21020170,
title = {Sigma model renormalization group flow, 'central charge' action, and Perelman's entropy},
author = {Tseytlin, A. A.},
abstractNote = {Zamolodchikov's ctheorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be a gradient one to all loop orders. However, the monotonicity of the flow of the targetspace metric is not obvious since the metric on the space of metricdilaton couplings is indefinite. To leading (oneloop) order when the RG flow is simply the Ricci flow the monotonicity was proved by Perelman [G. Perelman, math.dg/0211159.] by constructing an 'entropy' functional which is essentially the metricdilaton action extremized with respect to the dilaton with a condition that the targetspace volume is fixed. We discuss how to generalize the Perelman's construction to all loop orders (i.e. all orders in {alpha}{sup '}). The resulting entropy is equal to minus the central charge at the fixed points, in agreement with the general claim of the ctheorem.},
doi = {10.1103/PHYSREVD.75.064024},
journal = {Physical Review. D, Particles Fields},
number = 6,
volume = 75,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}

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