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arXiv 1406.4945hep-th
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双曲空间中的威尔逊圈与极小面积曲面

Wilson loops and minimal area surfaces in hyperbolic space

Martin Kruczenski

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摘要：

AdS/CFT对应将$N$ =4 SYM理论中的威尔逊圈与AdS空间中的极小面积曲面联系起来。如果回路是平面曲线，则极小曲面存在于双曲空间$H_3$（或等价的欧几里德AdS $_3$空间）中。我们认为，找到这样的极值曲面的面积可以很容易地做，如果我们解决以下问题：给定两个真正的周期函数$V_{0,1}(s)$，$V_{0,1}(s+2π)=V_{0,1}(s)$，第三个周期函数$V_2(s)$是要找到的，使所有的解决方案，以方程$- ϕ"(s) + \big[V_0+{1\over 2} (λ+{1 \over λ}) V_1 + {i\over 2} (λ-{1 \over λ}) V_2\big] ϕ(s)=0$是反周期的$s\in[0,2π]$的任何值的$λ$。这个问题等价于单值矩阵是平凡的。它可以被重述为找到一个复杂的参数族的曲线$X(λ,s)$，其中$X(λ=1,s)$是威尔逊环的给定形状，使得施瓦茨导数$\{X(λ,s),s\}$是亚纯的$λ$只有两个简单的极点。我们提出了一个公式中的功能$V_{0,1,2}$的面积和讨论的解决方案，这些等价的问题，θ函数。最后，我们还考虑了近圆Wilson回路，阐明了它的可积性，并使用本文所述的方法重新推导了它的面积。

英文摘要：

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 $- ϕ"(s) + \big[V_0+{1\over 2} (λ+{1 \over λ}) V_1 + {i\over 2} (λ-{1 \over λ}) V_2\big] ϕ(s)=0$ are anti-periodic in $s\in[0,2π]$ 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.

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