Wilson loops and minimal area surfaces in hyperbolic space
The AdS/CFT correspondence relates Wilson loops in N = 4 SYM theory to minimal area surfaces in AdS space. If the loop is a plane curve the minimal surface lives in hyperbolic space H _{3} (or equivalently Euclidean AdS _{3} space). We argue that finding the area of such extremal surface can be easily done if we solve the following problem: given two real periodic functions V _{0,1}(s), V _{0,1}(s +2π) = V _{0,1}(s) a third periodic function V _{2}(s) is to be found such that all solutions to the equation $-{\mathrm{\partial}}_{S}^{2}\varphi +[{V}_{0}+\frac{1}{2}(\lambda +\frac{1}{\lambda}){V}_{1}+\frac{i}{2}(\lambda -\frac{1}{\lambda}){V}_{2}]\varphi =0$ are anti-periodic in s ϵ [0, 2π] for any value of λ. This problem is equivalent to the statement that the monodromy matrix is trivial. It can be restated as that of finding a one complex parameter family of curves X(λ, s) where X(λ = 1, s) is the given shape of the Wilson loop and such that the Schwarzian derivative {X(λ, s), s} is meromorphic in λ with only two simple poles. We present a formula for the area in terms of the functions V _{0,1,2} and discuss solutions to these equivalent problems in terms of theta functions. Finally, we also consider the near circular Wilson loop clarifying its integrability properties and rederiving its area using the methods described in this paper.
- Publication Date:
- Grant/Contract Number:
- SC0007884; PHY-0952630
- Type:
- Accepted Manuscript
- Journal Name:
- Journal of High Energy Physics (Online)
- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2014; Journal Issue: 11; Journal ID: ISSN 1029-8479
- Publisher:
- Springer Berlin
- Research Org:
- Purdue Univ., West Lafayette, IN (United States)
- Sponsoring Org:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; gauge-gravity correspondence; AdS-CFT correspondence
- OSTI Identifier:
- 1454534
Kruczenski, Martin. Wilson loops and minimal area surfaces in hyperbolic space. United States: N. p.,
Web. doi:10.1007/JHEP11(2014)065.
Kruczenski, Martin. Wilson loops and minimal area surfaces in hyperbolic space. United States. doi:10.1007/JHEP11(2014)065.
Kruczenski, Martin. 2014.
"Wilson loops and minimal area surfaces in hyperbolic space". United States.
doi:10.1007/JHEP11(2014)065. https://www.osti.gov/servlets/purl/1454534.
@article{osti_1454534,
title = {Wilson loops and minimal area surfaces in hyperbolic space},
author = {Kruczenski, Martin},
abstractNote = {The AdS/CFT correspondence relates Wilson loops in N = 4 SYM theory to minimal area surfaces in AdS space. If the loop is a plane curve the minimal surface lives in hyperbolic space H3 (or equivalently Euclidean AdS 3 space). We argue that finding the area of such extremal surface can be easily done if we solve the following problem: given two real periodic functions V 0,1(s), V 0,1(s +2π) = V 0,1(s) a third periodic function V 2(s) is to be found such that all solutions to the equation − ∂ S 2 ϕ + [ V 0 + 1 2 ( λ + 1 λ ) V 1 + i 2 ( λ − 1 λ ) V 2 ] ϕ = 0 are anti-periodic in s ϵ [0, 2π] for any value of λ. This problem is equivalent to the statement that the monodromy matrix is trivial. It can be restated as that of finding a one complex parameter family of curves X(λ, s) where X(λ = 1, s) is the given shape of the Wilson loop and such that the Schwarzian derivative {X(λ, s), s} is meromorphic in λ with only two simple poles. We present a formula for the area in terms of the functions V 0,1,2 and discuss solutions to these equivalent problems in terms of theta functions. Finally, we also consider the near circular Wilson loop clarifying its integrability properties and rederiving its area using the methods described in this paper.},
doi = {10.1007/JHEP11(2014)065},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2014,
place = {United States},
year = {2014},
month = {11}
}