Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \overrightarrow{X} $\end{document}(s) we define, using the integrable properties of the system, a family of curves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \overrightarrow{X} $\end{document}(λ, s) depending on a complex parameter λ known as the spectral parameter. This family has remarkable properties. As a function of λ, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \overrightarrow{X} $\end{document}(λ, s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \overrightarrow{X} $\end{document}(λ, s) has an essential singularity at the origin λ = 0. The coefficients of the expansion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \overrightarrow{X} $\end{document}(λ, s) around λ = 0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.


Introduction
The AdS/CFT correspondence [1][2][3] set on firm footing the conjectured relation between string theory and gauge theories in the planar limit. It opened the possibility of computing gauge theory quantities in the strong coupling limit by using the dual string theory. Perhaps the most important observable in gauge theories is the Wilson loop. In AdS/CFT it is related to surfaces ending on the curve associated to the Wilson loop [4][5][6]. In the large 't Hooft coupling limit, the surface that needs to be considered is the one of minimal area. Among all Wilson loops the simplest ones are those determined by a two dimensional, closed, smooth curve. Until recently, the only such example whose dual minimal area surface could be computed was the circle [7][8][9][10][11]. The minimal area surface was simply given by a half sphere. This was somewhat surprising given the known integrability properties of the system [12,13].
Here we continue the study of such solutions and show that, in particular, they also provide an infinite parameter family of examples of surfaces ending on multiple curves. The case we concentrate on is the one of concentric curves. The main technical difference with the previous case is that a certain periodicity condition is required, which implies a restriction on the position of the cuts and on the values of the spectral parameter. Since it is only one condition, it still implies an infinite parameter family of solutions.
The paper is organized as follows: in the next section we review briefly the previous results. Since the paper is essentially a continuation of the previous one we do not attempt to make this paper self-contained and should be read in conjunction with [14]. On the other hand we give new expressions for the boundary curve and the area that were derived by simplifying the ones in [14]. After that, we present the method to find minimal area surfaces ending on multiple curves and give a few examples with plots of the corresponding surfaces and boundaries. In the following section we compute the area for the case of multiple curves in complete parallel with the result for a single curve. Finally, as illustration, we give an example of a surface ending in eight different curves and give our conclusion in the last section.

Single curve case revisited
In [14] it was shown, based on [15,16] how to find an infinite parameter family of minimal area surfaces ending on a curve at the boundary. The method uses integrability properties of the equations to solve them in terms of Riemann theta functions. The technique is common to many integrable non-linear differential equations and is well described in several references, for example, in the book [16].
In this section we briefly review those results and derive a simplified expression for the boundary curve which allows the explicit computation of other important quantities. For example, a simple expression is found for the Schwarzian derivative of the boundary curve. Finally, a remarkable formula for the area of the curve is found in terms of the behavior of the boundary curve near singular values of the spectral parameter.
To be concrete, we look for minimal area surfaces in hyperbolic space (EAdS 3 ) which can be parameterized by coordinates (X, Y, Z > 0) with a metric

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Given a contour (X(s), Y (s), Z = 0) ≡ (X(s) = X(s) + iY (s), Z = 0) at the boundary Z = 0 we seek a surface of minimal area ending on it and parameterized as where the coordinates (z,z) parameterize the complex plane. In that plane we find a closed curve z(s) such that Z(z(s),z(s)) = 0. Such curve maps to the contour in the boundary of (EAdS 3 ) space, and the interior maps to a minimal area surface. At this time there is no generic solution to such problem of finding the minimal area surface for an arbitrary boundary. However we can give an infinite parameter family of minimal area surfaces and the corresponding contours where they end.
It is an open problem if such solutions approximate all contours (and if so, in which sense). The construction starts by choosing an auxiliary hyperelliptic Riemann surface M, known as the spectral curve and given, as a subspace of C 2 by There is a reality condition that requires the set of branch points {0, ∞, λ j=1...2g } to be symmetric under the involution T : λ ↔ −1/λ. A basis of one-cycles a i , b i is chosen such that the non-trivial intersections are a i • b j = δ ij and that under the involution behave as where T ij is a g × g symmetric matrix such that T 2 = 1. As a matter of notation, a generic point in the Riemann surface M is denoted as p i = (µ p i , λ p i ) ∈ M. The coordinate λ p i is called the projection of p i on the complex plane. The origin and infinity play an important role and are denoted as p 1 = 0, p 3 = ∞. Except for the branch points, given a point p 4 in one sheet there is another point denoted as p4 on the other sheet of the Riemann surface such that λ p 4 = λ p4 and µ p 4 = −µ p4 . The relation between p 4 and p4 is the hyperelliptic involution associated with M. Now we consider the basis of holomorphic differentials and define the matrices and the basis of normalized holomorphic differentials ω i=1···g such that  9) and the theta function 10) It also defines the Jacobi map φ : M → C g for which the standard notation is where the point p 1 = 0 is chosen as a distinguished point, p 4 is an arbitrary point and 4 1 is just notation for the same function. There is an ambiguity in choosing the path of integration. Since two different paths from p 1 to p 4 can only differ by a closed cycle, φ(p 4 ) changes by where 1 and 2 are vectors with integer components. Further, if p 4 is another branch point, the path 1 → 4 can be traced back on the lower sheet defining a generically non-trivial cycle. By choosing a certain path from p 1 = 0 to p 3 = ∞ we define the vectors ∆ 1,2 ∈ Z g through The path should be chosen such that ∆ t 1 .∆ 2 is an odd integer. With this vector we define another function (2.14) Because ∆ t 1 .∆ 2 is odd it follows thatθ(−ζ) = −θ(ζ) and in particularθ(0) = 0. Further, consider the vector ω = (ω 1 , · · · , ω g ) and expand it around p 1 = 0. Since p 1 = 0 is a branch point an appropriate coordinate can be chosen as y = −2i √ λ, namely λ = −y 2 /4. It follows that ω = (ω 1 + y 2 ω 12 + y 4 ω 14 + · · · )dy (2.15) for some constant vectors ω 1 , ω 12 , etc. that can be computed from eq. (2.7) by expanding µ(λ). Near infinity, on the other hand, an appropriate coordinate isỹ = 2 √ λ and the expansion is ω = (ω 3 +ỹ 2 ω 32 +ỹ 4 ω 34 + · · · )dỹ (2.16) The vectors ω 1 , ω 3 , ω 12 , ω 32 , . . . ∈ C g play an important role and it is convenient to introduce a notation for the gradient of a function along those directions Notice that D 1 is a first derivative. Second derivatives are denoted e.g. as D 2 1 F (ζ) = (ω 1 .∇)(ω 1 .∇)F (ζ). With all these ingredients in place, the solutions are given by where the vector ζ(z,z) ∈ C g is given by and the constants µ, ν are given by which uses the directional derivatives D 1,3 defined in eq. (2.20). Notice that eq. (2.23) implies ∂ z F (ζ(z,z)) = 2D 1 F (ζ(z,z)) and ∂zF (ζ(z,z)) = 2D 3 F (ζ(z,z)). The solution contains an arbitrary parameter p 4 = (µ p 4 , λ p 4 ) ∈ M. For eq. (2.22) to define a solution it is necessary that |λ p 4 | = 1. Therefore, each spectral curve M actually defines a one real parameter family of surfaces. It was shown in [14] that all those surfaces have the same area. Since the boundary is at Z = 0, to find the curve where the surface ends we need to find the zeros of Z or, equivalently, the zeros ofθ. Typically the zeros are given by isolated curves z(s) on the world-sheet from which one can be chosen. The solution eq. (2.22) maps such curve z(s) to a curve X(s) at the boundary Z = 0 defining the shape of the Wilson loop. The region of the world-sheet inside the curve z(s) is mapped, by eq. (2.22) to the minimal area surface ending at the curve X(s).

Boundary curve
As just described the shape of the boundary curve, or equivalently of the dual Wilson loop, follows from finding a curve z(s) on the world sheet such that Once this function ζ s is found, the shape of the curve follows from replacing ζ → ζ s in the expression for X, namely eq. (2.22) to get X(s) = X(s) + iY (s) [14]:

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However, it turns out that the result can be simplified by using eq. (A.9) derived in the appendix. With this identity, eq. (2.27) reduces to which is a much simpler expression for the boundary curve. In fact the function X(s) describes a family of curves parameterized by the point p 4 in the Riemann surface. Now we study the analytic properties of X(s, p 4 ) as a function of p 4 .

Analytic properties of X(s, p 4 ), analogy with the monodromy matrix
As discussed in the previous section, the function can be thought of as a family of curves parameterized by a point p 4 on the Riemann surface M. When |λ p 4 | = 1 the curve is such that the minimal area surface can be found from eqs. (2.21) and (2.22). In this section we study the analytic properties of X(s, p 4 ) as a function of p 4 and show that it is well defined on the Riemann surface but not on the complex plane parameterized by λ p 4 where it has cuts. In that manner, X(s, p 4 ) defines the spectral curve in the same way as the monodromy matrix does in many integrable systems. Furthermore, it turns out that X(s, p 4 ) has essential singularities at p 4 = 0, ∞, the precise behavior around the singularities determines the area of the corresponding surface. Finally, the Schwarzian derivative {X, s} is computed, and shown to have a remarkable simple dependence on λ p 4 .
To understand the properties of X(s, p 4 ) as a function of p 4 first it has to be proven that it is a well defined function on the Riemann surface. The reason it might not be is that there is an ambiguity in choosing the path of integration defining 4 1 . Choosing another path changes the integral by a period of the theta function. Using the quasi periodicity properties ofθ i.e. eq. (A.2), it follows that where we used that ∆ t 1 2 − t 1 ∆ 2 is an integer since 1,2 and ∆ 1,2 are integer vectors. Therefore the ratio of theta functions is not a well defined function on the Riemann surface. However, the exponential factor also depends on p 4 through the constants µ, ν defined in eq. (2.24): To study their periodicity properties recall the definition

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and introduce a variable ζ to compute the derivatives Therefore the exponential factor transforms as The factor precisely cancels the factor in eq. (2.31) coming from the ratio of theta functions showing that X(s, p 4 ) is the correct combination that is well defined on the Riemann surface.
Having proven that X(s, p 4 ) is well defined on the Riemann surface, it is interesting to study if it takes different values on the two sheets of the Riemann surface. This is indeed so. Changing sheets on the Riemann surface is equivalent to change 4 →4 or equivalently Since θ is even andθ is odd, the functions µ and ν change sign under such change. The ratio of theta functions is obviously inverted resulting in: Therefore the function X(s, λ p 4 ) defined on the complex plane has cuts. Thus, in some sense, it is analogous to the monodromy matrix. Namely, the cuts in X(s, λ p 4 ) can be removed by extending the function to X(s, p 4 ) defined on a Riemann surface, the spectral curve of the problem. Let us now study the analytical properties of X(s, p 4 ). Riemann's theorem [54] ensures that theta functions defined on a Riemann surface have g zeros and no poles. The ratio that appears in the definition of X(s, λ p 4 ) therefore can have at most g zeros and g poles. One however is at p 4 = p 1 and therefore cancels. In fact for the others it is clear that if p 4 is a zero then p4 is a pole. So X(s, p 4 ) has (g − 1) zeros and (g − 1) poles conjugate from each other on the Riemann surface (that means on opposite sheets). On the other hand the exponential function is quite interesting. The function has potential singularities on the g zeros of θ( 4 1 ). However, the derivative also vanishes on those zeros except at p 4 = p 1 . Since µ has a pole at p 4 = p 1 = 0 then X(s, p 4 ) has an essential singularity. The same happens at p 4 = p 3 in view of the function ν.

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To summarize: X(s, p 4 ) as a function of p 4 is a well defined function on the Riemann surface with (g − 1) zeros and (g − 1) poles and essential singularities at p 4 = p 1 = 0 and Given the symmetry λ ↔ −1/λ it suffices to study the function near λ = 0. Since λ = 0 is a branch point a more appropriate coordinate y can be chosen as Using the expansion from eq. (2.15) ω(y) = (ω 1 + ω 12 y 2 + · · · )dy (2. 43) it follows that Finally the expansion and eq. (A.37) in the appendix, result in This expansion is remarkable in two respects. First, the residue of the pole 1 y contains z(s), namely the parameterization of the conformal world-sheet coordinate in terms of the boundary parameter. Second, as we will see later the constant term determines the area of the minimal area surface with this boundary. Actually the area of the whole family of minimal area surfaces parameterized by p 4 such that |λ p 4 | = 1. It is also instructive to compute the expansion of the Schwarzian derivative {X, p 4 } based on the expansion of ln X. If, as shown, X has an expansion where the two leading terms come from the last term in eq. (2.50). It follows that

Conformal invariant ratios and Schwarzian derivative
In the case of the monodromy matrix one can consider its trace which is a well defined function on the complex plane since the cuts are removed. In the present case we can consider functions of X(s, p 4 ) which are invariant under inversion. Those are well defined in the complex plane. For example the functions are invariants depending on four, two and one point of the curve respectively. In fact they are invariant under the full Möbius group. The last one introduces the Schwarzian derivative which is conformally invariant. On the left hand side we emphasized that these are functions of λ p 4 namely the projection of p 4 on the complex plane. As functions of λ p 4 they have no cuts. The invariant Y [4] can be simply written by using eq. (2.29). On the other hand Y [2] requires evaluating ∂ s X. From eqs. (2.29), (2.26) it follows that Using the definition of µ ,ν i.e. eq. (2.24) and eqs. (A.23), (A.29) for the derivatives of the theta function we immediately find that Finally, the Schwarzian derivative is harder to compute since it involves up to third order derivatives. However, it turns out that the result is remarkably simple. The second derivative can be computed in a similar way as the first one resulting in In fact now, the Schwarzian derivative can be evaluated as The term in square brackets can be simplified by first using eq. (A.39) to replace D 2 3 lnθ(ζ s − is obtained. The last term can be rewritten using eq. (A.41) A completely equivalent expression can be found in terms of derivatives D 1 : The most important point is that the dependence in λ is very explicit in eq. (2.61). We believe this to be one of the main results of this paper since {X, s} gives a conformal invariant characterization of the Wilson loop.

Area
In [14] the following formula for the finite part of the area of the minimal area surface was derived: where (σ, τ ) are world-sheet coordinates such that z = σ + iτ and is the element of arc-length on the boundary curve z(s) = σ(s) + iτ (s). In this section we simplify the result and relate it to the expansion of X(s, p 4 ) near the singularity at p 4 = 0.
In fact, the second term in eq. (2.66) can be rewritten as Using that ∂ zθ = 2D 3θ , ∂zθ = 2D 1θ and that the contour of integration is such that θ(ζ s ) = 0, the formula for the area can be rewritten as where the sign was chosen according to the conventions in figure 1.
We can further use that and eq. (A.35) to find where s i,f are the initial and final values of s. Since the curve is periodic, for an integer n known as the turning number of the curve. It is the number of times the unit normal to the curve goes around the unit circle when the point goes once around the curve. With the assumptions of figure 1 it is n = 1. The final result for the area is therefore which is simpler than the one found in the previous paper [14]. Equivalently we can write Figure 1. The world-sheet is parameterized by a complex coordinate z. The functionθ(ζ) depends on z through the vector ζ = 2ω(p 1 )z + 2ω(p 3 )z where ω p1,3 are complex vectors. Our convention is that the shaded region whereθ(ζ) > 0 maps to the minimal area surface. The contourθ(ζ s ) = 0 maps to the boundary of the surface, i.e. the Wilson loop. The contour is parameterized counterclockwise by s. The gradient ∇θ(ζ s ) is normal to the boundary and point towards the interior. Finally, the region whereθ(ζ s ) < 0 also maps to a minimal area surface and the corresponding can be easily changed to cover that case. Slightly more involved is the case whenθ(ζ) is purely imaginary but nothing fundamentally different changes. (2.74) Consider now the expansion in eq. (2.48). It immediately follows that where we recall that y = −2i λ p 4 is the appropriate coordinate around the branch point p 4 = 0. Also the integral of a total derivative was discarded. Quite remarkably, the function z(s) is also contained in the expansion, in fact

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We can then write the slightly more cumbersome formula dy ∂ s ln X(s, y ) ds (2.77) The integrals on spectral parameters y, y are complex contour integrals (given by the residues) around the points y = 0, y = 0. The integral along the real parameter s is an integral around the boundary curve.

Genus g = 1 case
In the case of genus one the theta functions reduce to Jacobi theta functions and the solutions can be written in terms of elliptic functions or, equivalently, elliptic integrals.
In the previous work [14] this case was not described since it had already been studied in [25] and [26]. More recently, these solutions were analyzed including the evaluation of the first quantum correction [38]. In this subsection however, revisiting this case is useful as a starting point for the next section. For genus one there is only two cuts, namely four branch points. They are taken to be p 1 = 0, p 3 = ∞, λ 1 = a ∈ R, λ 2 = − 1 a in view of the λ ↔ −1/λ symmetry. The spectral curve is described by the equation an is illustrated in figure 2 where it is also shown a choice of cycles a 1 , b 1 . The only holomorphic differential is which needs to be properly normalized. Computing where the standard notation for elliptic functions was used (see [55] ) The formulas from [14] reduce to

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We also have ∆ 1 = 1, ∆ 2 = 1. Taking the involution T = −1 we find as it should be for the reality condition to be satisfied. Let us introduce also the half-period a that definesθ and which is a zero of θ: Now it is possible to define the variable ζ where z = σ+iτ . Now we can relate the Riemann theta functions to Jacobi Theta functions (we use the convention in [55]) by: and The world-sheet metric (or Lagrange multiplier Λ) is determined by In the last equation α was written in terms of Jacobi elliptic functions, perhaps the easiest expression to check the validity of the cosh-Gordon equation for such function. The last step is to choose the spectral parameter λ such that |λ| = 1.
In terms of λ we define The solution can then be written in terms of Jacobi theta functions as where v is an additional parameter which corresponds to 4 1 in eq. (2.22). The formulas for ψ 1,2 andψ 1,2 refer to the functions defined in [14] and are included for completeness. They are not used in this paper. Finally, the solution is written as As before the boundary curves are found by finding the zeros ofθ = −θ 1 . Since the variable ζ + depends only on τ ( or ζ − on σ), the world-sheet region is a strip between two zeros of θ 1 . Generically the edges of the strip map to two infinite boundary curves with the shape of a spiral. However if we take λ = 1 for a > 1 and λ = −1 for a < 1 we have the following interesting result • a > 1, λ = 1, the Wilson loop is given by two concentric circles of radii r 1,2 , satisfying ln The ratio, by definition, is smaller than one, but, analyzing the previous result, it shows that it has a minimum value larger than zero. For each value of the ratio r 2 r 1 in that range there are two possible values of v. However they do not give the same solution, namely, for each allowed ratio r 2 r 1 there are two solutions, one world-sheet is close to the boundary and the other extends towards the interior.
The following limits are of interest • a → ∞ gives a circle.
• a → 1 − gives two parallel lines. • a → 0 gives a straight line.

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For the case a = 1 corresponding to two parallel line the spectral curve coincides with the one given in [39]: suggesting that both approaches are equivalent.

Surfaces ending on two contours
In the previous section we discussed, following [26] a solution ending on two concentric circles and corresponding to two cuts. It seems evident that adding two small cuts to the spectral curve will just slightly deform the solution. This is correct except that generically the periodicity is broken and therefore the curves become infinite spirals instead of two closed curves. To be precise, the theta functions are periodic but there is also the exponential factor in eq. (2.28) which is periodic when the exponent is purely imaginary, a condition that needs to be satisfied. Both periods have to match. It is clear that for particular positions of the cuts the periodicity would be restored but it requires some numerical effort to find them. In this section we present several examples. The number of JHEP05(2014)037 branch cuts is four, the situation is depicted in figure 3 where also the choice of cycles is made evident. Due to the involution λ → −1/λ , the branch point c = −1/b. The points b andc are the complex conjugates of b and c, respectively. As we shrink the bb and cc branch cuts we should get results close to the concentric circles for a > 1. Unlike the g = 1 case, even if we take λ = 1, the solutions are not automatically periodic; we must pick a suitable value of a > 1 as we move b to get periodic concentric curves.
The ratio of Riemann theta functions in eq. (2.28) is periodic if the argument shifts by an integer vector, also, the exponential function is periodic under a 2π shift in the imaginary direction. Both periods have to match. This can be achieved in practice by moving the branch points around. We found values for a and b such that we can match 2,3,4 and 5 periods of the theta function into a single period of the exponential. In each case we obtained concentric Wilson loops. The curves deviate from a circle in a way such that the periodicity is manifest as can be seen in figures 4 and 5. The corresponding positions of the branch points is shown in table 1 below. It is also possible to fit one period of the theta function into e.g. two periods of the exponential in (2.28). It turns out that we get a boundary made of self-intersecting concentric curves.  each individual curve resembles a circle which is described twice as shown in figure 6. It is also made clear the claim that as we shrink the bb branch cut the surface approaches the one corresponding to the concentric circular Wilson loops.

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✁ ✂ ✁ ✄ Figure 5. Wilson loops and corresponding minimal area surfaces for two concentric curves. Now, four and five periods of the theta functions are fit into one period of the exponential as is evident from the shape of the curves.

More than two boundary curves
As the size of the cuts is varied, closed curves appear in the region between the two periodic curves giving rise to surfaces with more boundaries. An example is shown in figure 7. The curves depicted are zeros ofθ in the world-sheet and correspondingly of the function Z(z,z). That means that they map into multiple curves at the boundary. The region surrounded by one of the closed curves maps into a minimal area surface ending in the corresponding single curve. On the other hand the complement of those regions maps to a minimal area surface ending in multiple curves, those closed curves we described plus two infinite curves (of spiral shape) given by the upper and lower periodic curves. Again, for λ = 1 and particular positions of the cuts, the period of those curves match those of the exponential and the result is a surface ending in multiple closed curves. A particular example is illustrated in figures 8, 9 and 10. Admittedly, the surface is not simple to describe or plot since it ends in eight different and self-intersecting curves. The boundary curves are shown JHEP05(2014)037 Figure 6. In this case, two periods of the exponential are fit in two periods of the theta function giving rise to concentric self-intersecting curves. In the first example the position of the cuts is given by a = 2.412712, b = 2 + 0.05I and in the second a = 2.414205, b = 2 + 0.003I. The second one is shown as illustration of the fact that the surface can be made to approximate the (double cover) of the concentric circles.
in figure 9, notice however that different scales are used. As an intermediate representation, contours of equal Z are depicted in figure 8. Finally, the surface is plotted in figure 10.

Area
Consider again the surfaces ending on two concentric curves, namely those in figure 4, 5 and compute their area. The formula in eq. (2.75) applies to this case if we take the turning number of the curve to be n = 0. Remember that this is the turning number of the boundary curve in the world-sheet. In figure 7 it is clear that, as one moves along the upper and lower curves, the tangent to the curves does not wrap around the unit circle. The area of the world-sheet can also be explicitly written term of a parameterization τ 1,2 (σ) of the JHEP05(2014)037 upper and lower curves giving Recall also that the total area is given by where L is the length of the Wilson loop and → 0 is the regulator defined by cutting the surface at Z = . The analytical results thus obtained are in agreement with a direct numerical evaluation of the Area as shown in table 2.  JHEP05(2014)037

Conclusions
In this paper we have simplified and extended the results in [14]. Given a boundary curve X(s), a one complex parameter family of curves X(s, λ) can be given by solving the linear problem associated with the flat connection. Studying the analytical properties of X(s, λ) it turns put that the function is naturally defined on a hyperelliptic Riemann surface M, the spectral curve. We then denote it as X(s, p 4 ) with p 4 ∈ M. In that way it plays a similar role to the monodromy matrix for standard integrable systems. The first part of the paper is devoted to the study of such function. In particular we find a simple expression for the Schwarzian derivative {X, s}. It is also shown that X(s, p 4 ) has an essential singularity at p 4 → 0 and the behavior near the singularity determines the area of the surface. The second part of the paper is devoted to the study of surfaces ending in multiple contours. In particular examples for surfaces ending in two or more contours are given, generalizing previous results of Drukker and Fiol [26]. Again, a relatively simple expression is given for the area of such curves.

A Theta functions identities
In this appendix we derive several identities for the Riemann theta functions that are needed in the main text. Some of the calculations are done in detail to illustrate the procedure. We use a basic knowledge of the theory of theta functions as can be found, for example, in the classic book on Riemann surfaces by Farkas and Kra [54]. In fact everything follows from the quasi periodicity property and the trisecant identity. Specialized books on theta functions such as [51][52][53] contain many of these results, we just briefly rederive them in this particular context and using the notation of this paper.

A.2 Trisecant identity
By studying the analytic properties of ratios such as those in eq. (A.2) Fay derived the following fundamental identity known as trisecant identity [54]: valid for an arbitrary vector ζ ∈ C and four arbitrary points p i,j,k,l on the Riemann surface. The coefficient γ ijkl is called a generalized crossed ratio and defined as where, a is a non-singular zero of the theta function i.e. θ(a) = 0 but |∇θ| = 0. Under such conditions it is an important property of the cross ratio that it does not depend on the zero a chosen to defined it. In this paper we definedθ, see eq. (2.14), through the odd period The trisecant identity plays a central role in using the theta function to solve non-linear differential equations in integrable systems. For us, it is the starting point for deriving all the identities below. It is interesting to note that in string theory it has an interpretation as a bosonization identity [56].

A.3 Generalized cross ratio
As it was just stated, γ ijkl in eq. (A.5) is independent of a as long as a is a zero of the theta function (θ(a) = 0). In the case at hand, the world-sheet boundary curve z(s) defines a set of zeros ofθ, namely ζ s , for any s is a zero ofθ (see eqs. (2.25), (2.26)). However, from the definition ofθ i.e. eq. (2.14) it is clear thatθ(ζ s ) = 0 ⇐⇒ θ(ζ s + 3 1 ) = 0, it follows that we can take a = ζ s + Up to a constant independent of ζ s , the cross ratio can be written as which is therefore independent of ζ s as long as it is a zero ofθ. However, it is also true thatθ(0) = 0. Replacing ζ s → 0 and using that θ is symmetric whereasθ is antisymmetric it immediately follows thatγ = −1. Therefore the simple but very useful identity is derived.

A.4 First derivatives
Identities for the derivatives of the theta functions can be obtained from the trisecant identity by deriving with respect to the position of the points p i p j , p k , p l . Consider deriving the trisecant identity with respect of the position of p i and afterwards taking the limit p j → p i . The result reads valid for any vector ζ ∈ C g and any three pints p i , p k , p l in M. In our case, such points are usually chosen among p 1 = 0, p 3 = ∞, and p 4 , p4 arbitrary and in opposite sheets of M. As particular cases it is useful to derive valid for any vector ζ. The functionθ is defined in eq. (2.14) and is odd, a fact used in writing eq. (A.11). In the solutions eq. (2.22), the vector ζ is a function of the world-sheet coordinates (z,z): ζ(z,z) = 2ω 1z + 2ω 3 z (A. 12) and therefore ∂ z F (ζ(z,z)) = 2D 1 F (ζ), ∂zF (ζ(z,z)) = 2D 3 F (ζ) (A. 13) In that way, the identities (A.11) can be used as a table of derivatives of the theta functions and allow the verification of the equations of motion.

A.5 Second derivatives
Second derivatives can be computed in the same way as first ones. In this subsection we just derive a special case that is useful for our purposes. Consider eq. (A.10) and take the points p i = p 3 , p l = p 4 and use the definition ofθ, i.e. eq. (2.14). After reordering the terms, the result is The pint p k is now taken to be very close to p 3 so that the expansion in eq. (2.16) can be used, namely Similarly, taking p i = p 1 and expanding fro p k → p 1 it follows that A.6 First derivatives at ζ = ζ s A special role in the construction is played by the world-sheet boundary curve z(s) defined such thatθ (ζ s ) = 0 (A.18)

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where ζ s = ζ(z(s),z(s)) = 2ω 1z (s) + 2ω 3 z(s) (A. 19) It therefore becomes necessary to evaluate various derivatives at such values of ζ = ζ s . Consider the first and third equations in (A.11) and evaluate them at ζ = ζ s . The result is Furthermore, using thatθ is antisymmetric and θ symmetric it follows that    Another possibility to cancel the pole atθ(ζ) = 0 is by subtracting the second equation in (A.11) from itself evaluated at p4: where again we used thatθ is odd:θ(− 4 1 ) = −θ( 4 1 ). From the trisecant identity, taking the points p i = p 4 , p j = p 1 , p k = p 3 , p l = p4 and using that          Expanding both sides inỹ two useful identities are derived: where D 3 denotes the derivative along the direction ω 32 and we also took into account thatθ is an odd function and therefore its even derivatives vanish at 0. Similarly, starting from eq. (A.21) one finds

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Finally, taking two derivatives in direction ω 3 i.e. applying D 2 3 to eq. (A.24) and evaluating at ζ = ζ s follows that  The result is not exactly the same, they only agree if which is another useful identity.

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