# Minimal area surfaces in AdS _{n+1} and Wilson loops

## Abstract

The AdS/CFT correspondence relates the expectation value of Wilson loops in N = 4 SYM to the area of minimal surfaces in AdS _{5}. In this paper we consider minimal area surfaces in generic Euclidean AdS _{n+1} using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS _{3}. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the λ-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the λ-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all λ deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.

- Authors:

- Purdue Univ., West Lafayette, IN (United States). Dept. of Physics and Astronomy

- Publication Date:

- Research Org.:
- Purdue Univ., West Lafayette, IN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1505222

- Grant/Contract Number:
- SC0007884

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AdS-CFT Correspondence; Wilson; 't Hooft and Polyakov loops

### Citation Formats

```
He, Yifei, Huang, Changyu, and Kruczenski, Martin.
```*Minimal area surfaces in AdSn+1 and Wilson loops*. United States: N. p., 2018.
Web. doi:10.1007/jhep02(2018)027.

```
He, Yifei, Huang, Changyu, & Kruczenski, Martin.
```*Minimal area surfaces in AdSn+1 and Wilson loops*. United States. doi:10.1007/jhep02(2018)027.

```
He, Yifei, Huang, Changyu, and Kruczenski, Martin. Mon .
"Minimal area surfaces in AdSn+1 and Wilson loops". United States. doi:10.1007/jhep02(2018)027. https://www.osti.gov/servlets/purl/1505222.
```

```
@article{osti_1505222,
```

title = {Minimal area surfaces in AdSn+1 and Wilson loops},

author = {He, Yifei and Huang, Changyu and Kruczenski, Martin},

abstractNote = {The AdS/CFT correspondence relates the expectation value of Wilson loops in N = 4 SYM to the area of minimal surfaces in AdS5. In this paper we consider minimal area surfaces in generic Euclidean AdSn+1 using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS3. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the λ-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the λ-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all λ deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.},

doi = {10.1007/jhep02(2018)027},

journal = {Journal of High Energy Physics (Online)},

issn = {1029-8479},

number = 2,

volume = 2018,

place = {United States},

year = {2018},

month = {2}

}