CALT682518
Notes on fast moving strings
Andrei Mikhailov^{1}^{1}1email:
California Institute of Technology 45248,
Pasadena CA 91125
and
Institute for Theoretical and
Experimental Physics,
117259, Bol. Cheremushkinskaya, 25, Moscow, Russia
We review the recent work on the mechanics of fast moving strings in antide Sitter space times a sphere and discuss the role of conserved charges. An interesting relation between the local conserved charges of rigid solutions was found in the earlier work. We propose a generalization of this relation for arbitrary solutions, not necessarily rigid. We conjecture that an infinite combination of local conserved charges is an action variable generating periodic trajectories in the classical string phase space. It corresponds to the length of the operator on the field theory side.
1 Introduction.
The AdS/CFT correspondence is a strongweak coupling duality. Weakly coupled YangMills is mapped to the string theory on the highly curved AdS space. When AdS space is highly curved, the string worldsheet theory becomes strongly coupled. Therefore, the weakly coupled YangMills maps to the strongly coupled string worldsheet theory. Nevertheless, in some situations elements of the YM perturbation theory can be reproduced from the string theory side. One of the examples are the “spinning strings”. Spinning strings are a class of solutions of the classical string worldsheet theory. They were first considered in the context of the AdS/CFT correspondence in [1, 2, 3]. These are strings rotating in with a large angular momentum. It was noticed in [1] that the energy of these solutions has an expansion in some small parameter which is similar in form to the perturbative expansion in the field theory on the boundary. Then [4] computed the anomalous dimensions of single trace operators with the generic large Rcharge, making the actual comparison possible. In [5] more general solutions were considered, having large compact charges both in and in . For all these solutions, computations in the classical worldsheet theory lead to the series in the small parameter which on the field theory side is identified with where is the ’tHooft coupling constant and a large conserved charge. Moreover it was shown in [6] that the quantum corrections to the classical worldsheet theory are suppressed for the solutions with the large conserved charge (see also the recent discussion in [7]). This opened the possibility that the results of the calculations in the classical mechanics of spinning strings, which are valid a priori only in the large limit, can be in fact extended to weak coupling and therefore compared to the YangMills perturbation theory. It was conjectured that the YangMills perturbation theory in the corresponding sector is reproduced by the classical dynamics of the spinning strings. The following picture is emerging.
Single string states in correspond to singletrace operators in the supersymmetric YangMills theory. (We consider the large limit.) The dynamics of the singletrace operators is described in the perturbation theory by an integrable spin chain. This spin chain has a classical continuous limit [8] which describes a class of operators with the large Rcharges. In this limit the spin chain becomes a classical continuous system. We have conjectured in [9] that this classical system is equivalent to the worldsheet theory of the classical string in . The YangMills perturbative expansion corresponds to considering the worldsheet of the fast moving string as a perturbation of the nullsurface [8, 9, 10, 11, 12]. The nullsurface perturbation theory was previously considered in a closely related context in [13].
In this paper we will try to make the statement of equivalence more precise. We will argue that the string worldsheet theory has a “hidden” symmetry which is defined unambiguously by its characteristic properties which we describe. This commutes with the group of geometrical symmetries of the target space. It corresponds to the length of the spin chain on the field theory side. We conjecture that the phase space of the classical continuous spin chain is equivalent to the Hamiltonian reduction of the phase space of the classical string by the action of this . The equivalence commutes with the action of geometrical symmetries.
We should stress that the hidden symmetry which we discuss in this paper was constructed already in [12], but the explicit calculation was carried out only at the first nontrivial order of the nullsurface perturbation theory. The main new result of our paper is that we discuss this hidden symmetry from the point of view of the integrability. We conjecture the relation between the symmetry and the local conserved charges which if true gives a uniform description of this symmetry at all orders of the perturbation theory.
The classical string on is an integrable system (see [14, 15, 16, 17, 18] and references there), and our corresponds to an action variable. The existence of the action variables for integrable systems with a finitedimensional phase space is a consequence of the Liouville theorem [19]. The classical string has an infinitedimensional phase space. We are not aware of the existence of a general theorem which would guarantee that the action variables can be constructed in the infinitedimensional case. But we will give two arguments for the existence of one action variable for the string in , at least in the perturbation theory around the nullsurfaces. The first argument gives an explicit procedure to construct the action variable order by order in the perturbation theory (Sections 3, 4.4 and 4.6). The second argument uses the existence of the local conserved charges [20] (known as higher Pohlmeyer charges) and the results of the evaluation of these charges on the socalled “rigid solutions” performed in [21, 22]. The arguments in Section 4 of our paper together with the results of [21, 22] suggest that the action variable is an infinite linear combination of the Pohlmeyer charges and allow in principle to find the coefficients of this linear combination.
The plan of the paper. In Section 2 we will review the classification of the nullsurfaces following mostly [11, 9] and stress that the moduli space of the nullsurfaces is a bundle over a loop space. Therefore it has a canonically defined action of . In Section 3 we will explain how to extend the action of from the nullsurfaces to the nearlydegenerate extremal surfaces using the perturbation theory. A large part of Section 3 is a review of [12]. In Section 4 we discuss the geometrical meaning of this as an action variable and argue that it is an infinite sum of the local conserved charges.
Note added in the revised version. The coefficients of the expansion of the action variable in the local conserved charges were fixed to all orders in the first paper of [29]. Here we consider only the Pohlmeyer charges for the part of the string sigmamodel. The role of the Pohlmeyer charges for was discussed in the second paper of [29]. In the special case when the motion of the string is restricted to the action variable discussed here corresponds to the action variable of the sineGordon model, see the third paper of [29].
2 Nullsurfaces.
2.1 The definition.
A twodimensional surface in a spacetime of Lorentzian signature is called a nullsurface if it has a degenerate metric and is ruled by the light rays. There is a connection between nullsurfaces and extremal surfaces. An extremal surface is a twodimensional surface with the induced metric of the signature which extremizes the area functional. Extremal surfaces are solutions of the string worldsheet equation of motion in the purely geometrical background (no field). When the string moves very fast, the metric on the worldsheet degenerates and the worldsheet becomes a nullsurface. Therefore a nullsurface can be considered as a degenerate limit of an extremal surface.
In there are two types of the light rays. The light rays of the first type project to points in . The light rays of the second type project to the timelike geodesics in and the equator of . The operators of the large Rcharge correspond to the nullsurfaces ruled by the light rays of the second type^{2}^{2}2The nullsurfaces of the first type have a boundary. They describe the shock wave propagating from the cusp of the worldline of a spectator quark in ..
2.2 The moduli space of nullsurfaces.
It is straightforward to explicitly describe all the nullsurfaces of the second type in . We have to first describe the moduli space of the nullgeodesics of the second type. An equator of is specified by a point in the coset space . Similarly, a timelike geodesic in is specified by . Given and , let and be the corresponding equator in and timelike geodesic in , respectively. To specify a light ray in we have to give also a map which pulls back the angular coordinate on to the length parameter on (see Fig.1). Such maps are parametrized by . We see that each light ray is defined by a triple . Therefore, the moduli space of lightrays of the second type in is geometrically:
(1) 
A nullsurface is a oneparameter family of light rays. Therefore it determines a contour in bundle over connection on this bundle. The definition of this connection is: the curve in the total space is considered horizontal, precisely if the corresponding collection of light rays is a degenerate surface. What is the curvature of this connection? Both and are Kahler manifolds (if we forgive that the metric on the first coset is not positivedefinite). Let us denote the Kahler forms and . The curvature of our bundle is . A curve in the base space over this film should be an integer). Moreover, it is lifted as a horizontal curve almost unambiguously, except that there is a “global” action of shifting on every light ray by the same constant. Therefore, the moduli space of nullsurfaces is the bundle over the space of contours in subject to the integrality condition which we described. can be lifted to the horizontal curve in the total space if and only if a twodimensional film ending on this curve has an integer Kahler area (integral of . The condition of the degeneracy of the metric defines a is a . But we have to also remember that an arbitrary collection of the light rays is not necessarily a nullsurface. It is a nullsurface only if the induced metric is degenerate. To understand what it means, let us choose a spacelike curve belonging to our surface. This spacelike curve is a collection of points, one point on each light ray. For the surface to be null, the tangent vector to this curve at each point of the curve should be orthogonal to the light ray to which the point belongs. (This condition does not depend on how we choose a spacelike curve.) What kind of a constraint does it impose on the contour? The space
To summarize, the moduli space of the nullsurfaces of the second type is:
(2) 
Here means the space of maps from the circle to ; for a Kahler manifold means the space of maps satisfying the integrality condition. At this point we consider the nullsurfaces without a parametrization; therefore we divide by the group of the diffeomorphisms of the circle. Turning on the fermionic degrees of freedom on the worldsheet we get the moduli space of supersymmetric nullsurfaces [9]:
(3) 
Here is the phase space of the continuous spin chain [9]. Therefore the moduli space of nullsurfaces is “almost” equivalent to the phase space of the continuous spin chain, except for the fiber and the reparametrizations . We have to explain what happens to the fiber and why the nullsurface actually comes with the parametrization. Also, we have to explain how the symplectic structure is defined on the moduli space of null surfaces. Let us start with the parametrization.
2.3 Parametrized nullsurfaces.
The phase space of the classical string has a boundary which consists of strings “moving with the speed of light”. A string moving very fast can be approximated by a nullsurface. But one nullsurface can approximate many different fast moving strings. The nullsurface as we defined it so far “remembers” only the direction of the velocity at each point of the approximated string, but it misses the information about the ratios of the relativistic factors at different points of the string. Although in the nullsurface limit, the ratio for two different points on the worldsheet remains finite. Therefore, if we want to think of the moduli space of the nullsurfaces as the boundary of the phase space, we have to equip the nullsurfaces with an additional structure. This additional structure is the parametrization.
A nullsurface is a oneparameter family of the light rays. The parameterization is a particular choice of the parameter. In other words, it is a monotonic function from the family of light rays forming the nullsurface to the circle, defined modulo . One can also think of it as a density on the set of light rays forming the nullsurface. This density is roughly speaking proportional to the density of energy on the worldsheet of the fastmoving string, in the limit when it becomes the nullsurface. We will now give the definition of .
Consider the family of string worldsheets converging to the nullsurface . We will introduce a parametrization of in the following way. Consider a Killing vector field on , corresponding to some rotation of the sphere:
(4) 
Here parametrizes the : .
When is large, is close to , the string moves very fast and the conserved charge corresponding to is very large. We can approximate this charge by an integral over a spacial contour on the nullsurface of times some density :
(5) 
Here is the part of the nullsurface; we choose the coordinate on the nullsurface to be the affine parameter on the light ray normalized by the condition . Eq. (5) with the condition is the definition of , and also the precise definition of the large parameter , modulo . We choose as the parametrization.
We can now say that the moduli space of parametrized nullsurfaces is the boundary of the phase space of a classical string. We say that a family of extremal surfaces has a parametrized nullsurface as a limit when if and only if

has as a limit when , as a continuous family of smooth twodimensional surfaces in a smooth twodimensional manifold, and

the density of approaches Eq. (5) in the limit .
This definition of the parametrization does not depend on which particular geometrical symmetry we use. An alternative way to define the same parametrization is to use a special choice of the worldsheet coordinates on . Let us choose the worldsheet coordinates so that
where is the projection of the string worldsheet to and is the projection to . Then we define . In the nullsurface limit defines the parametrization of the nullsurface.
2.4 The symplectic structure.
The moduli space of parametrized nullsurfaces as a manifold depends only on the conformal structure of the target space. But we can introduce additional structures on this moduli space which use the metric on .
An important additional structure is the closed 2form which originates from the symplectic form of the classical string. Strictly speaking a differential form in the bulk of the manifold does not automatically determine a differential form on the boundary. Indeed, suppose that we have a differential form, for example a 2form in the bulk. We can try to define the “boundary value” of on the boundary in the following way. Given two vector fields on the boundary, we find two vector fields in the bulk such that and . Then we define . But the problem is that this definition will depend on the choice of and . Intuitively, if is some other choice of a pair of vector fields inducing on the boundary, and the “vertical component” of is not small enough near the boundary, then .
Given this difficulty, how do we define the symplectic form on the space of nullsurfaces given the symplectic form on the string phase space? When we lift the vector field on the boundary to the vector field in the bulk, let us require that goes to zero when . We define by Eq. (5); it is only an approximate definition at , but this is good enough for the purpose of our definition:
(6) 
where and are such that . One can see that has a kernel, which is precisely the tangent space to the fiber in the numerator of Eq. (3). The moduli space has a symmetry rotating this fiber; we will discuss this symmetry in the next section; we will call it . Therefore is the symplectic form on the moduli space of nullsurfaces modulo .
Eq. (3) implies that the moduli space of parametrized nullsurfaces modulo is the space of parametrized contours in the Grassmanian:
(7) 
One can see that is equal to the integral of the symplectic form on the superGrassmanian pointwise on the contour, with the measure . The symplectic area of the film filling the contour is the generating function of the shift of the origin of the circle. Therefore the integrality condition guarantees that the symplectic form does not depend on the choice of the origin on ; the symplectic form is horizontal and invariant with respect to the shifts of the origin of .
Our definition of the symplectic form on the space of nullsurfaces used the target space metric (just the conformal structure would not be enough) and also the fact that the target space is a product of two manifolds.
3 Nearlydegenerate extremal surfaces and the role of the engineering dimension.
Our discussion in this and the next section will be limited to the classical bosonic string.
3.1 Definition of .
The moduli space (3) of nullsurfaces is a bundle. The symmetry shifting in the fiber plays an important role in the formalism. We will call it . On fig. 3 we have shown schematically how acts on the nullsurfaces.
We conjecture that corresponds to the length of the spin chain. Generally speaking, the length of the spin chain is not conserved in the YangMills perturbation theory [26], but it is probably conserved in the continuous limit (this should be related to the discussion of the “closed sectors” in [27]). It should be conserved modulo the corrections vanishing in the continuous limit. We therefore conjecture that there is a continuation of from the space of nullsurfaces to the phase space of the classical string, at least to the region of the phase space corresponding to fast moving strings. We conjecture that this continuation is uniquely defined by the following properties:

The action of preserves the symplectic structure.

The action of does not change the projection of the worldsheet to . Moreover, it preserves the projection to of the nulldirections on the worldsheet.

We require that the orbits of are closed (otherwise, we would not have called it ).

The restriction of to the nullsurfaces acts as we described (see fig. 3).
The second property reflects the fact that corresponds to the length of the operator rather than its engineering dimension.
Let denote the Hamiltonian of . Let denote the phase space of the classical string, and denote the Hamiltonian reduction of the phase space on the level set of . The basic conjecture is:
There is a onetoone map from the phase space of the spin chain of the length to the reduced phase space of the classical string preserving the symplectic structure and commuting with the action of .
The reduction by was discussed in [23] but only in a sector [24] in which acts as some element of . The perturbation theory in this sector was discussed in [25] (see also Section 2 of [11]).
3.2 Action of on nearlydegenerate extremal surfaces.
In this subsection we will explain how to continue the action of from the boundary of the phase space. Most of this section is a partial review^{3}^{3}3Section 3 of [12] has more than just a construction of . The next step is considering the action of the Killing vector field where is the global time in on the invariants of and bringing the result to the form suitable for the comparison with the field theory computation. Here we are discussing only the first step. of Section 3 of [12].
3.2.1 Particle on a sphere.
Consider the phase space of a particle moving on , and restrict to the domain where the velocity of the particle is nonzero. This domain is naturally a bundle over the moduli space of equators of ; let denote the projection map in this bundle. A point of the phase space, corresponding to the position and the velocity , projects by to the equator going through and tangent to . See the discussion in [11].
The symplectic form on the phase space is expressed in terms of the symplectic form on the base and the connection form :
(8) 
where , ( is the momentum of the particle) and is the symplectic form on the moduli space of equators. The moduli space of equators is a Kahler manifold, the symplectic form is the Kahler form.
Now it is easy to construct the action of . One takes
(9) 
This is a vertical vector field, it does not act on the base. The coordinate is essentially the angle along the equator on which the particle is moving. More explicitly:
(10) 
It is easy to see that the trajectories of the vector field on the phase space of a particle on are periodic with the period . One has to remember that this vector field is defined only on the open subset of the phase space, where the velocity of the particle is nonzero. But we consider fast moving strings, and the region of the phase space where the velocity is nearly zero is not important for us.
3.2.2 String on a sphere.
In some sense, a string is a continuous collection of particles. Therefore, it is natural to apply a similar construction to the string. Treating the string as a continuous collection of particles requires the choice of the coordinates on the worldsheet. We will therefore introduce the conformal gauge:
(11) 
In this gauge the symplectic form is:
(12) 
In the Hamiltonian formalism, we introduce — the component of the momentum, and — the component of the momentum. Now we will interpret the string as a collection of particles parametrized by . We are tempted to interpret the vector field (9),(10) acting pointwise in as the required symmetry. The generator of this symmetry would be . But this would be wrong. This field preserves the symplectic structure, does have periodic trajectories and acts correctly on the null surfaces. But unfortunately it does not preserve the gauge (11). It only commutes with the second constraint, . But it does not commute with the first one, . Indeed, it commutes with but not with . Therefore we should modify this vector field so that it still has periodic trajectories, but also commutes with the constraint. There is a systematic procedure to do this, order by order in , developed in [12].
Let us summarize this procedure, or perhaps a variation of it. To make sure that the modified vector field is Hamiltonian (preserves the symplectic structure) we construct it as a conjugation of with some canonical transformation, which we denote :
(13) 
or schematically . Since is a canonical transformation, is automatically a Hamiltonian vector field. Since is singlevalued, generates periodic trajectories. It remains to construct such that commutes with the constraint . But to require that commutes with is the same as to require that commutes with — the pullback of by . Therefore we have to find such a canonical transformation that the pullback of with is annihilated by the vector field . In other words, we have to find a canonical transformation which removes from ; after this canonical transformation becomes where all the for are in involution with and is of the order . This was done in Section 3 of [12]. The canonical transformation can be expanded in ; the corresponding generating function is expanded in the odd powers of . The authors of [12] gave the explicit expression for to the first order in , but they also give a straightforward algorithm for constructing the higher orders. (We will reconsider the higher orders from a slightly different point of view in Section 4.4, perhaps making this algorithm more precise.)
At the first order we need to find such that the canonically transformed constraint, which is a function of :
has zero Poisson bracket with up to the terms of the order , for every . And should be of the order . In other words, we should have:
(14) 
One can see that
(15) 
works. Notice also that this is reparametrization invariant (where transforms as a density of weight one). Therefore it commutes also with the second constraint . Therefore, to the first order in the canonical transformation we are looking for is generated by this . Then the generator of the is, up to the terms of the order :
(16)  
One can see immediately that the trajectories of this charge are closed up to the terms of the order subleading to . Indeed, we have explained in Section 3.2.1 why the leading term gives periodic trajectories. And the second term (which as we have seen is needed to make the charge commuting with the Virasoro constraints) averages to zero on the periodic trajectories of the first term. Therefore (see for example Section 3 of [11]) the trajectories of do not drift at this order.
We will discuss the higher orders in Section 4.4.
4 Length of the operator and local conserved charges.
We have seen that the nullsurface perturbation theory has a “hidden” symmetry . The existence of symmetries acting on the phase space is typical for integrable systems, at least for those which have a finitedimensional phase space. Corresponding conserved charges are called action variables [19]. Classical string in is an integrable system. Therefore, we should not be surprised to find such an action variable^{4}^{4}4Strictly speaking, the integrability is not necessary for the construction of the action variable in perturbation theory. A typical example is a particle on a sphere in an arbitrary (polynomial) potential. When the particle moves very fast, it does not feel the potential. All the trajectories are periodic in the limit of an infinite velocity. Therefore on the boundary of the phase space, when the velocity is infinite, we have an action variable — the absolute value of the momentum. It is well known that the perturbation theory in allows us to extend this action variable from the boundary inside the phase space, but only in the perturbation theory. For an arbitrary potential, the perturbation series must diverge, because in fact there is no additional conserved quantity besides the energy. Therefore the will be actually broken by effects which are not visible in the perturbation theory, unless if the potential is such that the system is integrable. We want to thank V. Kaloshin and A. Starinets for discussions of this subject..
The local conserved charges in involution for the classical string in are explicitly known. Therefore, instead of constructing in the perturbation theory, we can try to build it as some linear combination of the already known conserved charges. In this section, we will argue that the coefficients of this linear combination are actually fixed by the calculation of [21, 22].
4.1 Local conserved charges.
Consider a string in the target space which is a product of two manifolds and . We assume that the metric on has the Lorentzian signature, and the metric on has the Euclidean signature. We will need and , but let us first consider the general . The string worldsheet will be denoted . The classical trajectory of the string is an embedding . We are going to use the fact that the target space is a direct product. A point of is obviously a pair where is a point of and is a point of . Therefore for each point we have , where and . Consider the 1forms and on the string worldvolume, taking values in and in . In other words, is a differential of .
The metric on has the Lorentzian signature, and we consider the string worldsheets which have the induced metric with the Lorentzian signature. Pick two vector fields and on , which are both lightlike but have a nonzero scalar product:
These vector fields have a simple geometrical meaning. Since the worldsheet is twodimensional, at each point we have two different lightlike directions. The vector points along one lightlike direction, and points along another. Pick a spacial contour on , and a 1form on such that and . Consider the following functional:
(17) 
We will prove that this functional does not depend on a particular choice of , , and . This is therefore a correctly defined functional on the phase space of the string. Indeed, the only ambiguity in the choice of is where is some function on the worldsheet. But this function cancels in (17). The ambiguity in the choice of and is also in rescaling which does not change (17). It remains to prove that (17) does not depend on the choice of the integration contour . To prove that (17) is independent of , let us choose coordinates on the worldsheet in such a way that the induced metric is . Then is proportional to and is proportional to . In these coordinates
(18) 
The variation of under the variation of the contour is measured by the differential of the form:
But on the equations of motion . Therefore the integral does not depend on the choice of the contour.
Let us explain why on the equations of motion we have . Let be the second quadratic form of the surface, (here is the normal bundle to in ). The second quadratic form is defined in the following way: suppose that the particle moves on with the velocity , then the acceleration of the particle is modulo a vector parallel to . For the surface to be extremal, the trace of should be zero. The trace of is the contraction of with the induced metric on ; it is a section of . The trace of is proportional to , therefore we should have:
But notice that therefore . Another conserved charge is:
(19) 
Are there charges containing higher derivatives of ? Let us consider the following expression:
(20) 
Even though it is not true that is zero. The covariant derivatives and do not commute, therefore . In fact, for any function we have
(21) 
where is the Riemann tensor of . Now we have to start using that is a sphere. For , the Riemann tensor is constructed from the metric tensor, and
(22) 
Now consider the following differential form:
(23) 
Using (22) we can show that , therefore is a local conservation law. We use the formula which is special for . We will denote this charge . There is also a charge which is obtained from (23) by replacing with and or with or .
These charges are just the first examples of an infinite family of charges, which are all in involution. This infinite family was constructed in [20].
A particularly important linear combination is
(24) 
The construction of this charge requires only that the target space is a direct product of two manifolds.
4.2 Local conserved charges are invariant under .
Consider a local conserved charge acting trivially on the part of the worldsheet. In the conformal gauge, this means that is constructed as a contour integral of some combination of and . Let us decompose in the inverse powers of :
(25) 
where is a nonnegative integer, the “order” of the charge; is of the order , is of the order etc. We have to require that is in involution with the Virasoro constraints. In particular, it should be in involution with for an arbitrary . (Here we used that is trivial in AdSpart.) Let us now apply the canonical transformation which we described in Section 3.2.2. After this canonical transformation becomes where all the for are in involution with and is of the order . And becomes , where is the canonically transformed . We should have:
(26) 
for an arbitrary . At the leading order in this implies that is in involution with . At the next order, it follows that for all values of the expression is in involution with . This implies that:
(27) 
Since the vector field generated by is periodic, this equation implies that is in involution with . An analogous argument at higher orders shows that all the commute with . Therefore is in involution with the expression which is the generator of . The conserved charges of [20] do have an expansion of the form (25) therefore they should commute with . This reinforces our conjecture that should be a combination of the local conserved charges.
4.3 A geometrical meaning of .
We can try to make more transparent the geometrical meaning of by drawing an analogy with the Liouville theorem for finitedimensional integrable systems. A mechanical system with dimensional phase space is integrable if there are functions in involution with each other, and the Hamiltonian is a function of . Then, there are action variables , each of them being some combination of :
such that each generates (has periodic orbits). In this paper we are dealing with an infinitedimensional system, a classical string in . We can take the first Pohlmeyer charge as a Hamiltonian^{5}^{5}5It is a natural Hamiltonian on the phase space of a classical string in any case when the target space is a direct product of two manifolds.. This Hamiltonian is presumably integrable, because there is an infinite family of higher charges commuting with it. On the other hand, it does not have any special periodicity properties (we do not see any reason why it would). This means that the closure of the orbit of is an invariant torus. Our commutes with (This fact is seen immediately, because can be rewritten as and by definition does not act on the AdSpart of the worldsheet.) Therefore should be a shift of one of the angles parametrizing the invariant torus of . The angles parametrizing the invariant torus are in correspondence with its onedimensional cycles. Which cycle corresponds to ? Every invariant torus can be connected by a oneparameter family of invariant tori to a torus on the boundary of the phase space (or the one very close to the boundary). This means that every 1cycle is connected to some 1cycle on a torus on the boundary — the space of nullsurfaces. We should take that 1cycle which is connected to the orbit of on the nullsurfaces, described in Section 3.1. The corresponding action variable is — the generator of . These arguments show the uniqueness of .
The first Pohlmeyer charge has a special property: it actually generates on the boundary. Therefore the difference between and should be a combination of charges vanishing at the boundary. We expect that this is an infinite and linear combination. Indeed, the construction of [12] tells us that the charge we are looking for is local at each order in . A nonlinear combination of the charges would be nonlocal (a product of integrals).
4.4 A different point of view on the perturbation theory; higher orders.
In Section 3 we constructed as where is generated by and is the canonical transformation such that commutes with . This canonical transformation is constructed in the perturbation theory, order by order in .
A disadvantage of this procedure is that at each order we have to require that our commutes with for any . Since there are infinitely many values of we have to impose infinitely many conditions on at each order. At the first order, we have seen in Section 3.2.2 that these conditions are not really independent; one generating function takes care of all of them — see Eq. (14). At the higher orders, this is not immediately obvious. Therefore, we would like to propose a slightly different way of constructing . Let us forget for a moment about the Virasoro constraint; instead of the phase space of the string consider the space of harmonic maps . Instead of requiring that commutes with , let us require that commutes with . We will see that the requirement that commutes with already fixes in the perturbation theory, and the resulting will automatically commute with the Virasoro constraints.
As in Section 3, we look for the generator of as a pullback by a canonical transformation of . In other words, let us look for such a canonical transformation that commutes with (the pullback of by ). We can construct such a canonical transformation order by order in the perturbation theory. Let us denote . We have:
(28) 
Under the rescaling : , , , . The symplectic structure is of the degree 1: , therefore the Poisson brackets are of the degree : . We can construct order by order in this grading. We have:
(29) 
Suppose that we have already found such that commute with . At the order , we want to modify by the canonical transformation with the generating function of the order so that commutes with . Since is periodic, we can decompose
(30) 
where . Then we should take
(31) 
Repeating this procedure at higher orders, we end up with the function such that .
The reparametrization invariance is manifestly preserved at each order, therefore the resulting charge will commute with for any . Also, the fact that