# Sigma model renormalization group flow, 'central charge' action, and Perelman's entropy

## Abstract

Zamolodchikov's c-theorem 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 target-space metric is not obvious since the metric on the space of metric-dilaton couplings is indefinite. To leading (one-loop) 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 metric-dilaton action extremized with respect to the dilaton with a condition that the target-space 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 c-theorem.

- 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 c-theorem 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 target-space metric is not obvious since the metric on the space of metric-dilaton couplings is indefinite. To leading (one-loop) 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 metric-dilaton action extremized with respect to the dilaton with a condition that the target-space 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 c-theorem.},

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}

}