Euclidean Wilson loops and minimal area surfaces in lorentzian AdS 3

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Introduction
The Wilson loop operator is one of the most fundamental operators of a gauge theory. Its expectation value distinguishes a confining theory from one that is non-confining, is used to compute the quark/anti-quark potential, and determines the expectation value of gauge invariant operators as well as their correlation functions in various limits. Analytical methods to compute Wilson loops in the large N-limit [1,2] and for case of strong 't Hooft coupling proceeds by utilizing the AdS/CFT correspondence [3][4][5] whenever applicable. To leading order in strong coupling, the Wilson loop is computed by finding a minimal area surface in a higher dimensional space [6,7]. For the standard case of N = 4 SYM, considered in this paper, the minimal area surfaces live in AdS 5 × S 5 . This case is of particular interest because the dual string theory is described by an integrable model [8]. Consequently, the relationship between Wilson loops and minimal area surfaces has motivated much work in the area . The most studied one is the circular Wilson loop [30][31][32][33][34][35][36][37] including small perturbations around it [38][39][40]. Also, a particularly important role has been played by Wilson loops with light-like cusps [41] due to their relation with scattering amplitudes [42][43][44][45][46][47][48][49]. More recently new results for Wilson loops of more general shape have started to appear [20,[50][51][52][53], which includes solutions using Riemann theta functions. Such solutions were obtained using the methods of [54,55] and similar techniques that had been JHEP12(2015)083 previously used to find closed string solutions [11,[56][57][58][59][60][61][62][63][64]. It is also important to recall that in the large-N limit the Wilson loop in the gauge theory obeys the loop equation [65] that can also be studied within AdS/CFT [66,67].
In this paper, further insight into the properties of the Wilson loop operator is gained through study of the minimal area surfaces in AdS 5 . Such surfaces are obtained utilizing the simple but powerful Pohlmeyer [68] 1 reduction. Beginning from a Euclidean worldsheet living in AdS 3 ⊂ AdS 5 the surface is parameterized by the complex coordinate z using conformal gauge. The world-sheet metric then reads ds 2 = 4e 2α dz dz. (1.1) Here α(z,z) is a real function on a region of the complex plane that can be taken as the unit disk by a conformal transformation. Further, an important observation is that α(z,z), the conformal factor of the world-sheet metric, obeys a non-linear equation similar to the sinh-Gordon equation, where f (z) is an unknown holomorphic function. Such an equation is solvable independent of the other variables and yields that finding a minimal area surface means solving a set of linear differential equations once a solution is obtained for α(z,z). Further, the linear equations are deformable by a complex parameter λ called the spectral parameter. When |λ| = 1 a one-parameter family of minimal area surfaces is obtained which all have the same area. Such deformations are called λ deformations 2 and lead to an infinite number of conserved quantities given by the holonomy of certain associated currents around a non-trivial loop on the world-sheet. One can use the Pohlmeyer reduction in two different ways. The first one is to find new minimal area surfaces. Thus, an arbitrary function f (z) is chosen and then the solution for the conformal factor is found and used to construct a surface. The Wilson loop where the surface ends is then determined as part of the procedure. For example, an infinite parameter family of solutions were found in [20,50,52] for the case where f (z) does not vanish anywhere on the surface. These solutions are analytic and can be written in terms of Riemann theta functions. The second way to use this method, is to try to find a minimal area surface ending in any arbitrary given curve. The specified curve is used to compute the boundary conditions for f (z) and α from which those functions, and the corresponding surface, can be reconstructed. For the Euclidean case, this was discussed in [70] where it was found that the Schwarzian derivative of the contour with respect to the conformal angle 3 determines all the boundary conditions necessary to reconstruct the surface. However, finding the correct parameterization of the contour in terms of the conformal angle requires solving a non-trivial problem involving reconstructing a potential depending on the spectral parameter such that all its solutions are antiperiodic [70].

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At this moment it is not clear how to solve such a problem but in a recent important paper by Dekel [53] it was shown that such problem is solvable by studying perturbations around the circle. Although such a perturbative approach had been considered before [38], in [53] new methods extend the expansion to much higher orders than before providing a useful tool for solving the problem.
Another, related approach is to extend the results associated with light-like cusps [49] by considering the limit where the number of cusps goes to infinity in such a way that a smooth curve is reproduced. This approach is used to great effect in a recent paper by J. Toledo [71] where he managed to obtain a Y-system type of equation for the cross ratios associated with a given curve. The Y-system uses as an input a curve in the world-sheet describing the world-sheet boundary in the world-sheet coordinates where f (z) = 1. In the language of the Pohlmeyer reduction this is equivalent to giving f (z) in the coordinates where the world-sheet is the unit disk. Instead of using the more difficult approach of solving for α and then computing the area, Toledo showed that, from the solution to the Y-system of equations, the shape of the Wilson loop and the area of the associated surface follow. As mentioned before, this approach was derived in a roundabout way and a direct derivation that connects it with the methods discussed here and in [70] would make the discussion more complete.
It should be noticed that in the case of Wilson loops with light-like cusps the worldsheet is Euclidean while the target space has Lorentzian signature. This particular combination has neither been analyzed with the methods of [70], nor exact solutions have been constructed as in [20,50,52]. For this reason, this paper is devoted to studying Euclidean world-sheets in Lorentzian AdS 3 . Our main result is to extend the results of [70] to this case and the construction of new exact solutions using theta functions. This requires implementing the Pohlmeyer reduction for this new case and results in a construction analogous to [70]. The Schwarzian derivative of the contour with respect to the conformal angle provides boundary conditions for the functions f (z) and α(z,z). The conformal angle is found in principle by requiring that all conserved charges vanish. When computing the area we find a new, simpler and more direct way to derive the formula for the area in terms of the Schwarzian derivative of the contour. It avoids taking limits of the spectral parameter and using the WKB approximation. The formula for the area is valid when f (z) has no zeros in the unit disk, a condition that also applies to the formula given in [70] although it was not made explicit there. After that, we construct an infinite parameter family of solutions in terms of Riemann theta functions. Particular examples are used to check the previous results in this paper. The same examples can be used to check the Y-system method of [71], although we leave that for future work. Finally we derive some useful identities for the theta functions that simplify some calculations with respect to previous work. This paper is organized as follows: in the next section we derive the Pohlmeyer reduction relevant for this case. In section 3 we make contact with [70]. In particular we find a simpler derivation for the area formula. In section 4 we present new solutions correspond to the case where f (z) has no zeros in the unit disk and used them to test the results of the previous sections. Finally, in the last section, we give our conclusions. In an appendix we collect several useful formulas for theta functions and perform the computation of the Schwarzian derivative of the contour in terms of those functions.

JHEP12(2015)083 2 Integrability and Pohlmeyer reduction
Surfaces of minimal area are found by implementing the well-known Pohlmeyer Reduction [68] which is based on the integrability of the string Sigma Model. The utility of the method is due to its simplification of the problem; namely, it reduces solving the nonlinear string equations of motion (including the conformal constraints) to solving a single Sinh-Gordon equation plus a set of linear differential equations.
This work builds upon previous results found in [20] by again considering general open string solutions in Lorentzian AdS 3 but now for the case of a world-sheet with Euclidean signature. The Lorentzian AdS 3 manifold is defined as a subspace of R 2,2 subject to a constraint on the coordinates X µ (µ = −1, 0, 1, 2), For later convenience, the relationships between the embedding coordinates and global coordinates (t, φ, ρ)) and Poincare coordinates are now defined through the expressions (2.2) and (2.3) respectively.
Further, the world-sheet is parameterized by the conformal coordinates (σ, τ ) or equivalently by the complex combinations z = σ + iτ andz = σ − iτ which are more useful for this work. For this choice, the world-sheet metric has the form Working in conformal gauge, the action for the string Sigma Model is given by where the Lagrange multiplier Λ enforces the embedding constraint. Consequently, following from the action and the gauge choice, the equations (2.6), (2.7), and (2.8) determine a surface of minimal area describing the string.
Proceeding, the equations (2.6)-(2.8) are reduced to a single Sinh-Gordon equation. The procedure utilized here begins by forming a 2 × 2 real matrix X using particular combinations of the embedding coordinates,

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A result of choosing these combinations is that the embedding constraint requires that det X = 1 namely X ∈ SL(2, R). Further, any such matrix can be written as the product of any other two SL(2, R) group elements A a (a = 1, 2) . Convenient for the current work, this product is defined by the expression A useful consequence of this choice, used later, is the introduction of a redundancy in the description of X which implies an invariance under a world-sheet gauge transformation, In addition, these two group elements are used to define two one-forms, which satisfy the relationships (2.13) and (2.14) where no summation on a is implied.
which is simplified by the fact that the currents J a are traceless, (2.13),

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In terms of the currents, the system of equations to be solved are the equations of motion and conformal constraints (2.26)-(2.28) as well as the defining equations for the currents (2.24) and (2.25).
Inspection of these equations reveals a more convenient description by defining two new currents.
Summarizing, the system of equations to solve are the following.
While seemingly more complicated, everything is now in place to complete the reduction and solve the problem. A flat current a is defined as a linear combination of the currents A and B which is also traceless.
The importance of the current a is the realization that a one parameter family of nontrivial solutions exists given by α 2 + β 2 = 1 and γ = 1. This family is parameterized in terms of the spectral parameter λ for which α + iβ = iλ and α − iβ = 1 iλ . Using these facts, the flat current is written as follows.

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An additional restriction must be imposed since A and B are real whereas λ is generically complex which means the flat current a also satisfies the following reality condition.
Note that the original currents J a can be recovered using the newly defined current a: To determine a, first expand the current A in terms of the Pauli matrices, σ a=1,2,3 , and generically complex coefficients n i using the notationn i = n * i .
In this way, the conditions det A z = 0 and det Az = 0 are reinterpreted as a condition that the coefficients are the components of a light-like vector defined by the metric diag(−, +, −): For the coefficients written generically as n i = n i,R + in i,I , the above requirement produces two conditions on the real and imaginary parts: Since the real and imaginary parts of the coefficient vector have the same signature and are orthogonal they must be proportional to each other and are both either space-like or light-like. Now the gauge symmetry discussed earlier, (2.11), is re-expressed in terms of A as which amounts to an SL(2, R) = SO(2, 1) rotation of the vectors n R and n I . Assuming that n 2 R = 0, such a transformation always allows these vectors to be put into the following forms.
In the above expressions, α(z,z) is a real function. Thus. the flat current is and the flatness condition of the current determines the components of B:

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Here, f (z) is an arbitrary holomorphic function. In addition, α satisfies At this point, conventions chosen for this work make it convenient to rotate the flat connection with the SU(2) matrix to put it in a simpler form: The new flat current satisfies the reality conditioñ Sinceã is flat, we can solve the linear problem We can choose Ψ(λ; z,z) to satisfy the reality condition where the factor of i is chosen for convenience. With that choice, however, and since J 1 = a(1), J 2 = a(−1), we can take A 1 = Ψ(1)R, A 2 = Ψ(−1)R since A 1,2 turn out to be real. Thus, the solution to the non-linear problem reads Therefore, the strategy is to solve the equation for α, replace it in the flat current, solve the linear problem, and reconstruct the solution X. Actually, this procedure gives a oneparameter family of real solutions that can be written as The reason is that eqs.
These surfaces end in different boundary contours but they all have the same regularized area that, for any value of λ, is given by [42][43][44][45][46][47][48]: where the integral is over the domain D of the solution.

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3 Schwarzian derivative and the condition of vanishing charges In [70] a method of approaching the problem using the condition of vanishing charges was described. In particular the area was computed in terms of the Schwarzian derivative of the contour. Those results were derived for Euclidean AdS 3 , in this section we rewrite them for Lorentzian AdS 3 to get some further insight into the surfaces. Later we are going to provide concrete solutions in term of theta functions. Following [70], in this section we take the world-sheet to be the unit disk in the complex plane z. The boundary of the disk maps to the contour in the boundary of AdS 3 and the interior of the disk maps to the surface of minimal area that we seek. Near the boundary the induced metric diverges implying that α → ∞. Introducing a coordinate we find that eq. (2.54) implies the behavior From here we can compute the leading behavior of the flat current as we approach the boundary. It is best written in terms of where, from (2.58) and (2.56), ψ 1,2 satisfy the equations and the same forψ 1 ,ψ 2 It follows that

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In the case of λ = 1 we can combine this with the reality condition ψ 2 = −iψ * 1 to obtain In the case of λ = −1 we obtain where we used φ 1,2 to denote solutions for λ = −1. The surface is then described by The normalization of the solutions should be such that det X = 1. However, when computing the solution in Poincare coordinates the normalization cancels in x ± = X ± T = ± tan t ± 2 = ± tan t±φ 2 : Near the boundary, equations (3.7)-(3.10) imply that The functions ψ 1 andψ 1 are two linearly independent solutions of the linear problem defined in the boundary along θ. It can be obtained from by eliminating ψ 2 . Defining very similar to the Euclidean case. If β 2 (θ) and f (θ) are known, we need to find two linearly independent solution of the equation for λ = 1 to determine x + as their ratio and the same for x − with λ = −1. Using the result for the Schwarzian derivative of the ratio of two solutions

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That means that, if we knew the boundary contour x ± (θ) in the conformal parameterization then we could compute β 2 (θ) and also f (z) by using a dispersion relation. As in [70], one way to find such conformal parameterization is to write eq. (3.20) after an arbitrary reparameterization θ(s): and also, more explicitly, Thus, knowing the boundary curve x ± (s) in an arbitrary parameterization allows the computation of V 0,1 (s) but leaves V 2 (s) undetermined. Similarly as in [70] the real function V 2 (s) can be computed by requiring that all solutions of the Schrödinger equation (3.25) are anti-periodic in the variable s. Once V 2 (s) is determined, it is possible to compute the area and the conformal reparameterization θ(s). For later use it is convenient to recall the relation to the boundary variables in global coordinates (t, φ):

Computation of the area
To compute the regularized area we used formula (2.63). It can be simplified by observing that the sinh-Gordon equation (2.54) implies Locally, we can rewrite this equation as

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If f has no zeros inside the unit disk then this equation defines a conserved current on the world-sheet. At this point it is useful to recall that, under a holomorphic coordinate transformation z → w(z) the sinh-Gordon equation is invariant provided we change in particular implying namely χ = √ f dz is a holomorphic 1-form and then is a function (0-form) on the disk such that χ = dW . On the other hand transforms as a Schwarzian derivative. Since the difference between two Schwarzian derivatives transforms homogeneously, we can rewrite eq. (3.35) as the conservation of the current where we used the function W (z) defined in eq. (3.39) to write a current that transforms appropriately under a coordinate transformation. Otherwise the extra term − 1 √f {W ,z} does not play any role since it is anti-holomorphic. Finally, we follow [49] and write the area as (dσ ∧ dτ = i 2 dz ∧ dz) The integral is over the unit disk whose boundary is parameterized as z = e iθ . Integrating by parts we find

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At the boundary α diverges and then, from eq. (3.42) j z vanishes whereas, from eq. (3.2) Using eq. (3.24) together with the simple result it follows that This result is invariant under reparameterizations of the boundary and therefore we can choose an arbitrary parameter s instead ofz: Finally inside the disk we can take any other conformal parameterizations. In the next section we use W (z) as a coordinate and just denote it as z. In that case the function f (z) = 1 and the boundary of the world-sheet is given by a curve z(s) that has to be found as part of the solution.

Solutions in terms of theta functions
In this section we discuss exact analytical solutions to the minimal area surface problem that can be written in terms of Riemann Theta functions. It follows along the lines of similar solutions constructed in [20,50,52]. We are going to consider the case where the analytic function f (z) appearing in eq. (2.54) has no zeros inside the unit circle and therefore can be set to f (z) = 1 by an appropriate conformal transformation of the unit circle into a new domain in the complex plane that has to be found as part of the solution.
The equation for α reduces to the sinh-Gordon equation that has known solutions in terms of Riemann Theta functions associated to hyperelliptic Riemann surfaces. We are going to define such a surface by an equation in C 2

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where g is the (arbitrary) genus and (µ, λ) parameterize C 2 . For the solution to be real the branch points have to be symmetric under the involution T : λ → 1/λ, see also eq. (2.42). We should then choose a basis of cycles {a i , b i } such that the involution maps: This choice defines the g × g matrices as well as a basis of holomorphic differentials such that where Π =CC −1 is the period matrix of the Riemann surface. The next step is to choose two branch points p 1,3 ≡ (λ 1,3 , µ = 0) that map into each other under the involution T . In addition we require that the path connecting them is an even half-period: , with ∆ t 1 ∆ 2 and even integer. This half period define a Theta function with characteristics that we call Using the properties under the involution T : λ → 1/λ it is easy to prove that where φ is defined through These results imply that, if ζ * = ±T ζ, then θ(ζ),θ(ζ) ∈ R. As we approach the branch points p 1,3 , the vector of holomorphic differentials ω(λ) diverges as 1/µ(λ); for that reason it is convenient to define a new vector and two particular values:

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where λ 1,3 are the projections of the points p 1,3 . If we further define the constant then we obtain that the following reality condition is satisfied: Under all these conditions, from eq. (A.9) in the appendix, it follows that a real solution to the sinh-Gordon equation can be written as where C α is a constant equal to ±1, chosen so that e α is positive in the region of interest. Such region of interest is taken to be a connected domain in the complex plane bounded by a curve whereθ vanishes, namely α diverges. It should be noted that the condition that θ vanishes is only one real equation sinceθ is real, a general theta function with arbitrary characteristics would be complex and the condition that it vanishes would only be satisfied at isolated points in the world-sheet. The next step is to solve the linear problem for Ψ, namely eq. (2.58). To this end we choose an arbitrary point p 4 on the Riemann surface, for example on the upper sheet, and write the solutions as    Recall that real solutions require |λ| = 1 (see eq. (2.62)) which restricts the possible points p 4 that can be chosen, in fact, as discussed in the appendix, we have to choose |λ 4 | = 1.

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It is easy to see that |λ| = 1 implies that, if (ψ 1 , ψ 2 ) is a solution to eqs. (3.7), (3.8) then so is (ψ * 2 , ψ * 1 ). However, one can check that for the solutions in eqs. (4.17), (4.18) such solution is the same as the original (up to an overall constant). Instead, another, linearly independent solution, to equations (3.7)-(3.10) is obtained by choosing the corresponding point on the lower sheet of the Riemann surface that we denote as p4. Since p 1 is a branch point we have where the constant A was defined in eq. (4.21). Using these functions we can write a

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solution Ψ to eq. (2.58): This is not the whole story since the actual matrices Ψ also have to satisfy the reality conditions (2.59). Fortunately this problem is easily solved by first defining the linear combinations that satisfy the same equations due to the symmetry (2.57) of the flat current but in addition satisfy the reality condition that satisfy the reality condition (2.59) as required and can be checked using the definition ofR in eq. (2.55). Finally we can write the solution to the non-linear problem as The intermediate matrices Ψ F (λ) are useful since we can equally well write the solution in the form This gives the shape of the surface analytically. In the next section we give particular examples to get an idea of the shape of these solutions.

Computation of the area
The regularized area can be computed by using the formula (2.63)

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where we set f (z) = 1 since we are considering that case. The domain D is the region of the complex plane bounded by the curve whereθ vanishes. Furthermore, from eqs. (A.7) and (4.14) we find Thus, the regularized area is equal to where A W S is the world-sheet area, namely the area of the domain D of the complex plane that maps to the minimal surface. The last integral can be done using Gauss' theorem in the form where in the last equality we used that (∂F dz +∂F dz) = dF = 0. The final result for the Area is then with ∂ ln θ(ζ) = C ± D 1 ln θ(ζ) evaluated along the boundary. This gives a practical way to evaluate the area for the solutions discussed in this section. We can now verify eq. (3.52). Indeed starting from (3.52) and using eqs. (A.39), (A.41) we obtain where we renamed W → z for simplicity since we use W as the world-sheet coordinate. Furthermore, since ∂ s D 3 ln θ(ζ s ) = C ± ∂ sz D 2 3 ln θ(ζ s ) + C ± ∂ s z D 13 ln θ(ζ s ) (4.50) and also from eq. (A.18) we find Finally, since the world-sheet area A W S is given by and we can integrate by parts we find in perfect agreement with eq. (4.54).

Boundary curve
The boundary curve associated with these minimal area surfaces can be derived by using eqs. (4.42), and (4.33)-(4.36) and similarly forx − . This can be greatly simplified by studying the behavior of the functions near the boundary as in eq. (3.11). However, since we are using here a worldsheet parameterization such that f (z) = 1, the world-sheet is bounded by a curve z(s) which generically is not a circle. For that reason we revisit the derivation. Consider a point z 0 at the world-sheet boundary and expand the coordinate z as where s represents fluctuations along the boundary and ξ towards the inside of the worldsheet (ξ = 0 is the boundary). Instead of eq. (3.11) we now find Since ψ 1,2 obey the same equations as ψ 1,2 but with λ ↔ −λ it follows that φ 1,2 behave in the same way with We can now simplify (4.55) to e i(t+φ) = φ 21 +φ * 11 φ 21 + φ * 11 bdry. (4.60) As mentioned before, in this case (φ 11 , φ 21 ) and (φ * 21 , φ * 11 ) are linearly dependent solutions implying that φ * 11 /φ 21 is constant on the world-sheet. In fact, using that where the matrix T is defined in eq. (4.3), we obtain

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Now the constant can be computed explicitly as where A was defined in eq. (4.21). Finally we obtain Similarly and It is important to note thatx ± and x ± are related by an SL(2, C) transformation (as follows from eqs. (3.33), (4.66)) implying that Namelyx ± (s) is a conformally equivalent (but generally complex) description of the Wilson loop.

Examples
To illustrate the solutions we describe two Wilson loops associated with genus g = 2 auxiliary surfaces. These examples make clear the shape of the solutions we are discussing and also provide the reader with concrete numbers that s/he can reproduce and use as a basis for further work. For the same reason the results are rounded to just a few significant figures. 4

Example 1
In example 1, we choose a surface with branch points −2, − 1 2 , 0, 1 3 , 3, ∞. This surface has the required invariance under λ ↔ 1/λ. In addition it also has the symmetry λ ↔ 1/λ that plays no role in the construction but simplifies the calculations. 5 We choose a basis 4 These calculations can be easily done using Maple or Mathematica. Now we choose two branch points p 1,3 such that the half-period C 13 = 1 2 (∆ 2 + Π∆ 1 ) connecting them is even. We select p 1 = 1 3 , p 3 = 3 and thus which, from eq. (4.7), defineθ. Furthermore, the vectors ω 1,3 in eq. (4.13) follow from eq. (4.12) as Finally the regularized area can be found to be

Example 2
In this case we choose the branch points at −1 − i, − 1 2 (1 + i), 0,    In this case the regularized area is given by:

Acknowledgments
We are very grateful to P. Vieira for comments and discussions as well as to J. Toledo for collaboration during the initial stages of this paper.

A Theta function identities
In this work we use the notation in [72], the calculations are similar to those in [20,50]. However, there some small differences, the main one being thatθ(ζ) is defined by an JHEP12(2015)083 even period and therefore it does not vanish at ζ = 0. For that reason we introduced an additional odd half-period a such that θ(a) = 0. This modifies the formulas enough that it is worth rewriting them. On the other hand the procedure is exactly the same as in [20,50], namely all identities follow from the quasi-periodicity of the theta function and the fundamental trisecant identity [73][74][75], so we do not give detailed derivations. The trisecant identity is where a is a non-singular zero of the theta function. Now we can take the limit p i → p j and obtain the first derivative identity

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Choosing various combination of points p j,k,l the following first derivative identities are obtained They can be combined with the trisecant identity (A.1) to obtain, for example Second derivatives can be obtained similarly, for example, from the first equation in (A.4) we obtain, by taking derivative with respect to p 4 : They can be combined into that becomes the sinh-Gordon equation in the main text. The reason is that one takes where C ± is a constant defined in eq. (4.14).

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Other useful identity can be obtained from (A.3) by taking p j = p 3 , p l = p 4 and expanding for p k → p 3 . The first non-trivial order gives Eq. (A.9) together with the identities in eq. (A.4) is all that is needed to check the equations of motion. However we are also interested in computing the Schwarzian derivative of the boundary contour. This is a more involved calculation for which we derive several identities in the next subsection.

A.1 Identities at the world-sheet boundary
The previous identities are valid for any vector ζ ∈ C g . Since the points at the boundary of the world-sheet are zeros ofθ, in this section we derive identities valid when ζ = ζ s is an arbitrary zero ofθ, i.e.θ(ζ s ) = 0. From (A.4) we immediately get from where we find where, in the second one we also took the limit p 4 → p 1 . Multiplying the second and fourth equations in (A.4) byθ(ζ) and taking ζ = ζ s it follows that   6 One could also take derivative with respect to p1 but then one has to be careful with a hidden dependence on p1 through the definition ofθ.

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Also, multiplying the second equation in (A.4) byθ(ζ) taking derivative D 3 with respect to ζ and setting ζ = ζ s it follows that where (A.13) was used to simplify the result. Taking derivative D 3 with respect to ζ in the third equation in (A.4), taking ζ = ζ s , and using (A.20) we obtain Taking derivative D 3 with respect to ζ in the first equation in (A.4), taking ζ = ζ s , and using (A. 19) we obtain where we replaced with respect to ζ and setting ζ = ζ s it follows that which is the last identity we need. It is equivalent to say that the left hand side is independent of the zero ifθ that we take. In particular if we take ζ s = a + 3 1 we obtain the right-hand side.

A.2 Identities at particular points
One last type of identity is needed in order to fix the spectral parameter λ to any desired value. Indeed, according to eq.

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In practice we fix first λ and then choose p 4 accordingly, namely we need to invert the function λ(p 4 ). The main observation is that the right hand side of the equation, as a function of p 4 , has the following properties: it is a well-defined function on the Riemann surface, namely independent of the path used to define the integral 4 1 . Second it takes the same values on both sheets of the Riemann surface, namely it has no cuts and therefore it is a well-defined function of λ 4 , the projection of p 4 onto the complex plane. Finally, as a function of λ 4 it has a zero at λ = λ 1 and a pole at λ 4 = λ 3 (where λ 1,3 are the branch points taken to be p 1,3 ). It has no other zeros or poles. This last property is perhaps the only that requires an explanation since, as function of p 4 the theta functions in the numerator and denominator have g − 1 additional zeros. The fact is that all those zeros coincide and therefore cancel between numerator and denominator. This can be checked [72][73][74][75] using Riemann's theorem to write a = κ + q 1 where κ is the Riemann constant and q 1...g−1 are g − 1 points on the Riemann surface that turn out also to be the zeros of the numerator and denominator. Taking into account all these properties, we can write for some constant A 0 . This constant can be evaluated by considering the limits λ 4 → λ 1 and λ 4 → λ 3 . We obtain where the products are over all branch points except p 1 , p 3 , 0 and ∞. Since the two expressions for A 0 have to agree we find that Finally we get, for the spectral parameter λ λ = ±i|λ 3 |e iπ∆ t which allows us to easily choose λ 4 to obtain any λ we desire. In fact it is easily seen that |λ| = 1 if and only if |λ 4 | = 1, thus for real solutions we just take λ 4 on the unit circle.

A.3 Schwarzian derivative
The formulas summarized in the previous subsections can be used to derive a particularly simple expression for the Schwarzian derivative of the contour similar to the one found in [50]. We will be using that, from eq. (A.11) it follows that ∂ s F (ζ s ) = ∂ s z ∂ z F (ζ s ) + ∂ sz ∂zF (ζ s ) (A.32) = C ± ∂ s z D 1 F (ζ s ) + C ± ∂ sz D 3 F (ζ s ) (A.33)

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In particular sinceθ(ζ s ) = 0, we obtain ∂ s z D 1θ (ζ s ) + ∂ sz D 3θ (ζ s ) = 0 (A. 34) which determines the direction tangent to the world-sheet contour (z(s),z(s)). Now, starting fromx + = −e −2µ 4 z−2ν 4zθ (ζ s −  as used in the main text. Notice also that x − has the same expression as x + except that the point p 4 in the Riemann surface is replaced by p 5 . In the Schwarzian derivative the only effect is to replace λ → −λ. Thus Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.