Wilson loops and minimal area surfaces in hyperbolic space

Journal Article · Fri Nov 14 00:00:00 EST 2014 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/JHEP11(2014)065· OSTI ID:1454534

Kruczenski, Martin ^[1]

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

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 $-{\mathrm{\partial }}_S^2\varphi +[V_0+\frac{1}{2}(\lambda +\frac{1}{\lambda })V_1+\frac{i}{2}(\lambda -\frac{1}{\lambda })V_2]\varphi =0$ 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.

• Research Organization: Purdue Univ., West Lafayette, IN (United States)
• Sponsoring Organization: National Science Foundation (NSF) (United States); USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
• Grant/Contract Number: SC0007884
• OSTI ID: 1454534
• Journal Information: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 11 Vol. 2014; ISSN 1029-8479
• Publisher: Springer Berlin
• Country of Publication: United States
• Language: English

Related Subjects

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AdS-CFT correspondence
gauge-gravity correspondence