Rotating Wilson loops and open strings in
The AdS/CFT correspondence relates super Yang-Mills on to type IIB string theory on . In this context, a quark/anti-quark pair moving on an following prescribed trajectories is dual to an open string ending on the boundary of . In this paper we study the corresponding classical string solutions. The Pohlmeyer reduction reduces the equations of motion to a generalized sinh-Gordon equation. This equation includes, as particular cases, the Liouville equation as well as the sinh/cosh-Gordon equations. We study generic solutions of the Liouville equation and finite gap solutions of the sinh/cosh-Gordon ones. The latter ones are written in terms of Riemann theta functions. The corresponding string solutions are reconstructed giving new solutions and a broad understanding of open strings moving in . As a further example, the simple case of a rigidly rotating string is shown to exhibit these three behaviors depending on a simple relation between its angular velocity and the angular separation between its endpoints.
The AdS/CFT correspondence establishes a duality between certain gauge theories and string theory . The most well-known, and the one we consider here, is between super Yang-Mills on and type IIB string theory on . This duality has motivated the study of strings moving in AdS backgrounds. Two kinds of strings play an important role. One are closed strings moving inside , which are dual to single trace operators in the gauge theory. They have played a most important role in unraveling the AdS/CFT correspondence. For a review see . The other are open strings ending in the boundary, which are dual to Wilson loops in the gauge theory . The case we consider here is time-like Wilson loops that can be understood physically as an (infinitely) heavy quark/anti-quark pair moving along prescribed trajectories. In this paper, we study the case where the motion is on a circle which corresponds to an open string moving in . To solve the equations of motion for the string, we first use the Pohlmeyer reduction  to obtain a generalized sinh-Gordon equation for the world-sheet metric. This equation can be reduced, under certain conditions, to the Liouville, the sinh-Gordon or cosh-Gordon equations. In the case of the Liouville equation, we discuss its most generic solution explicitly which turns out to be equivalent to the solutions obtained by Mikhailov in . In the case of the cosh- and sinh-Gordon, we discuss finite gap solution that can be written in terms of theta functions. This gives a rather generic picture of the type of solutions that appear. The solutions in terms of theta functions are closely related to the solutions for closed strings described by Jevicki and Jin  (see also [7, 8]) and, more in particular, by Dorey and Vicedo in , Sakai and Satoh in . Moreover, for the case of Wilson loops, theta functions were recently used to find surfaces dual to an infinite parameter family of closed Wilson loops . It should be noted that the study of strings moving in AdS space predates the AdS/CFT correspondence for example in the work of de Vega and Sanchez  (see also ) where the relation with the sinh/cosh-Gordon equation is discussed. The present paper applies those techniques to the case of open strings.
In a broader context, the study of Wilson loops has provided deep insight into the AdS/CFT correspondence, for example in the case of the circular Wilson loop [14, 15, 16, 17], the light-like cusp as applied to computation of anomalous dimensions of twist two operators , or more recently to scattering amplitudes [19, 20, 21]. The Pohlmeyer reduction as used here is well-known  but further applications of the method are still being developed,  is a recent example. It is part of the use of integrability techniques to understand the AdS/CFT correspondence . Finally, the possibility of solving the equations of motion using algebro-geometric methods is also known [24, 25, 26, 27] in the mathematical literature related to minimal areas surfaces and the theory of solitons.
2 A simple example
In this section, the simple case of a rigidly rotating string from reference  is revisited; namely, the situation depicted in fig.1. The conformal factor in the world-sheet metric is seen to obey a cosh-Gordon, Liouville or sinh-Gordon equation depending on the relation between the angular velocity and the separation between the end points. Defining , the equation is cosh-Gordon if , Liouville if , or sinh-Gordon if . Interestingly, the Liouville case where is particularly simple and the shape of the string can be written in terms of trigonometric functions.
Let us proceed now to describe the solution. The manifold can be defined as a subspace of given by the constraint
written in terms of the coordinates (). The action for a Lorentzian world-sheet in conformal gauge is given by
where the Lagrange multiplier enforces the embedding constraint. The equations of motion are
and, due to the gauge choice, the solution must additionally satisfy the conformal constraints,
The same equations can be written using complex coordinates
or global coordinates defined through
in which case, the metric takes the well-known form:
In the coordinates the rotating string depicted in fig.1 is found by using the following ansatz:
Here, and are conformal coordinates on the world-sheet. Notice that these are open strings and, therefore, there is no periodicity condition111For closed strings this ansatz can be problematic because depends explicitly on through the function , which is generically not periodic as follows from the equations of motion.. The ansatz is justified by writing the equations of motion, which are all solved if
and obeys a differential equation better written in terms of ; namely,
Above, are constants such that and
The solution is actually determined by only two constants: the angular velocity and the minimal value of denoted as and given by .
The world-sheet metric is
The equations of motion then imply an equation for :
Up to a simple rescaling, this is the cosh-Gordon if , Liouville if and sinh-Gordon equation if . The reason being that all other factors are positive since and .
A more physical picture is obtained by computing the angular separation between the end points of the string (at fixed time ). Introducing the constants , we find
A simple analysis of the integrand reveals that decreases with increasing and . When , the separation simplifies to
the equation for reduces to Liouville if . After noting that and , the situation can be summarized by fig.2.
The particular case leading to the Liouville equation (i.e. ) is quite simple since two roots coincide and the shape of the string can be determined in terms of trigonometric functions:
It is also convenient to write the solution in the coordinates defined in eq.(6):
Here we used world-sheet coordinates:
Finally, in the original coordinates , the conformal factor is
Recalling that , it is easy to check that obeys the Liouville equation:
Here, is the one defined in eq.(21).
As discussed later in this paper, the rigidly rotating string for is a particular case of the solutions found by Mikhailov in .
3 The general case
In the previous section, a rigidly rotating string demonstrated that the conformal factor can obey the cosh-Gordon, Liouville, or sinh-Gordon equations. Now, we consider the general, non-rigid, case. It is convenient to use embedding coordinates,
and arrange them in a real matrix
This matrix obeys , i.e. . Consider the decomposition
with , . There is now a redundancy in the description and, as a result, a gauge symmetry
leaves invariant. Define now two one-forms:
It follows, with no summation on implied, that
To be precise, the conventions used for differential forms in coordinates are:
The equations of motion (3) become
Expanding the derivatives in terms of the currents yields:
Since the currents are two by two traceless matrices the traceless part can be extracted by using commutators; namely,
Here, the right-hand side is the traceless part of the left-hand side. Thus, we get a simple equation
Also, the equations of motion can be written in terms of the currents:
These equations need to be supplemented with the constraints ; equivalently,
At this point, it is convenient to define two new currents,
in terms of which the equations of motion can be rewritten:
A flat current can be found as a linear combination:
which in addition satisfies
There is a one parameter family of non-trivial solutions given by , . It can be conveniently parameterized in terms of the spectral parameter as , . Thus:
Since and are real but is generically complex, the flat current satisfies the reality condition
It is also useful to note that , . Returning to the current , and expanding it in terms of the Pauli matrices :
the condition implies that , are light-like vectors, i.e.
and the same for . The gauge symmetry , in terms of , gives which is an rotation of the vectors , . Assuming that , they can always be put in the form:
where is a real function. In this way, the current is
where , . The equation for determines that
where and are arbitrary functions and, in addition, satisfies
By an appropriate change of coordinates and a redefinition of , one can set and to a constant except at those points where they vanish. In the rest of the paper we consider only the case where is constant everywhere and can therefore be set equal to either, , or . The case where vanishes at a finite set of points will not be considered since we do not know how to write the general solution in those cases.
The flat connection can be written as
which is valid in general even if only the case of constant is considered here. Since is flat, we can solve the linear problem
Moreover, since , , we have , ; namely,
Therefore, the strategy is to solve the equation for , replace it in the flat current, solve the linear problem, and reconstruct the solution . In fact, a family of solutions satisfying the equations of motion and the constraints can be introduced:
Furthermore, the reality condition is satisfied if .
4 Liouville case ()
When , equation (61) reduces to the Liouville equation:
The general solution to this equation is
In this case, to reconstruct the solution , we can side-step the procedure described in the previous section and directly solve the linear problem
Using separation of variables (and taking as coordinates ) or by any other technique, the general solution is found to be
where and are arbitrary functions. Although this satisfies the equations of motion, it also needs to satisfy the constraints
Although particular solutions exist, it appears that generic solutions cannot be found in the case where both and are different from zero. In this paper, only the case where is considered and the solution is
The constraints then reduce to
These are relatively easy to study by noting that different choices of the functions , are related by reparameterizations of the world-sheet and, therefore, equivalent. In this way, we can choose for example
without loosing generality. Now, the solution can be written as
The string ends at the boundary on two curves determined by . The curves are given by
Notice that which is a parameterization of the boundary of space. Furthermore, these solutions are valid for any and not only . In fact they are already known since they were found by Mikhailov in . In particular, they contain the solution corresponding to the rotating string described in the first section. To show that, start by parameterizing the boundary curves as
Next, observe that, according to eq.(78), the two boundary curves are the same up to a shift by in the parameter and a change in sign of , which is equivalent to a shift , . Since the shift in parameter does not change the shape of the curve, one curve is obtained from the other by rotating by and shifting also by . Therefore, the shape of one boundary curve can be chosen arbitrarily, which determines the functions and also the shape of the other boundary curve.
For the rigidly rotating string, we have that one curve is determined by
Consequently, the other is
This implies that
which is precisely the result in eq.(16). Physically, this means that a ray of light emitted from the quark reaches the anti-quark exactly after a time . If it reaches earlier or later we then have the cosh or sinh Gordon equation. In Poincare coordinates, the two end points of the string describe arbitrary trajectories that are asymptotically null. In this case, it should be noted that both trajectories are arbitrary since they come from different portions of the trajectory in global The only condition is that one asymptotes to a null direction that is equal to the incoming direction of the other one. See fig.3.
5 Sinh/cosh Gordon case ()
The other case to consider is when the world-sheet conformal factor obeys the sinh or cosh-Gordon equation, i.e. eq.(61) with . The solution can be written in terms of theta functions associated with a hyperelliptic Riemann surface. The reality condition implies that the Riemann surface can be thought as two planes connected by a set of cuts symmetric under the interchange . In the examples considered here, the cuts are on the real axis although other situations are possible.
To be concrete, the hyperelliptic Riemann surface of genus is defined by the equation
where parameterize . A basis of 1-cycles is chosen together with a basis of normalized holomorphic differentials . Points on the Riemann surface are denoted as whereas their projection on the complex plane is denoted as . Obviously, each corresponds to two points on the Riemann surface (upper and lower sheets) except for the branch points . Two of the branch points are going to be singled out and denoted as and . Take a path from to on the upper sheet and close it by tracing the same path backwards on the lower sheet. When written in the basis , the closed path defines two integer vectors and such that
These vectors, together with the periodicity matrix of the Riemann surface, define a theta function with characteristics:
This function and the usual theta function without characteristics determine a function
which solves the sinh/cosh-Gordon equation. The constant is such that and should be chosen such that the right hand side is real and positive. This is always possible since the reality conditions ensure that is real and is either real or purely imaginary. Moreover, as discussed later, this is the only reality condition needed. As explained below, once is real, a real solution for can always be obtained. More details on the notation and derivations can be found in the appendix including the definition of the vectors . Finally, the constants are such that and should be chosen such that
With these definitions, it is just a matter of algebra to check that satisfies the equation
as shown in the appendix. Consequently, we have to identify
In what follows, it is convenient to choose
Summarizing, if we choose , such that is even, we have the sinh-Gordon equation. If we choose them such that is odd we obtain the cosh-Gordon one.
It should be noted that there is a different solution
which is interesting but simply related by to the previous one. For that reason it is not considered further.
The next step is to write the flat current and find the matrix that satisfies
The matrix is a two by two matrix that we write as
where and are two linearly independent solutions of the equation
where is now a two dimensional row vector. Using expression (62) for , the equations can be rewritten:
Here and are as in eq.(90). It is important to note that once is real and the spectral parameter is taken real, obey real equations and, therefore, can always be taken to be real. For example, given a solution, its real and imaginary part are real and also solve the equations.
Now we can proceed to solve the equations. One way to do it is to convert them into an equation for the ratio which can then be solved using the identities (152). Afterwards, one can solve for individually, again using eqns.(152). The result is
where the constants , and need to be determined. By comparing with the theta function identities found in the appendix, eq.(152), one finds that the spectral parameter is given by
That the ratio of theta functions simplifies as in the last equation is explained in the appendix. Here is a constant independent of but dependent on the other parameters of the solution; namely, the Riemann surface and the points , . The other constants are such that
The constants turn out to be
As discussed in the appendix, there are three possible cases:
, are real. The equation is sinh-Gordon and we can take and such that . Finally .
, are real. The equation is cosh-Gordon and we can take , and such that . Finally .
, real, purely imaginary. The equation is cosh-Gordon and we can take , and such that . Finally .
As discussed above, obey real equations; therefore, if the previous solutions are complex, in fact, two solutions are obtained and are given by its real and imaginary part. If those are independent we are done. Otherwise, we have to find another linearly independent solution which can be easily done:
The notation denotes the point which is different from but has the same projection . That is, one is in the lower sheet and the other in the upper sheet. Notice that in eq.(101), the value of depends only on meaning that it does not matter if is in the upper or lower sheet; therefore, and satisfy the same differential equation. Furthermore, define the constants by replacing .
In this way, we have found . However, we know that the solution is given by so we still need to find . Since satisfies the same equation, the solution is similar, we only need to find a point such that
Given the formula