CALT-68-2545

ITEP-TH-30/05

An action variable of the sine-Gordon model

Andrei Mikhailov

California Institute of Technology 452-48,
Pasadena CA 91125

and

Institute for Theoretical and
Experimental Physics,

117259, Bol. Cheremushkinskaya, 25, Moscow, Russia

It was conjectured that the classical bosonic string in AdS times a sphere has a special action variable which corresponds to the length of the operator on the field theory side. We discuss the analogous action variable in the sine-Gordon model. We explain the relation between this action variable and the Bäcklund transformations and show that the corresponding hidden symmetry acts on breathers by shifting their phase. It can be considered a nonlinear analogue of splitting the solution of the free field equations into the positive- and negative-frequency part.

## 1 Introduction.

Studies of classical strings in was an important part of the recent work on the AdS/CFT correspondence. It was observed that the energies of the fast moving classical strings reproduce the anomalous dimension of the field theory operators with the large R-charge, at least in the first and probably the second order of the perturbation theory [1, 2, 3, 4, 5, 6, 7, 8].

Classical superstring in is an integrable system. An important tool in the study of this system is the super-Yangian symmetry discussed in [9, 10]. The nonabelian dressing symmetries were also found in the classical Yang-Mills theory, see the recent discussion in [11] and the references therein. It was conjectured in [12, 13, 14] that the super-Yangian symmetry is also a symmetry of the Yang-Mills perturbation theory. It was shown that the one-loop anomalous dimension is proportional to the first Casimir operator in the Yangian representation. It is natural to conjecture that the higher loop contributions to the anomalous dimension correspond to the higher Casimirs in the Yangian representation, in the following sense. We conjecture that there are infinitely many operators acting on the spin chain Hilbert space, commuting:

and depending on the ’tHooft coupling constant as power series:

and such that the anomalous dimension has an expansion:

(1) |

It should be true that involves nearest neighbors in the chain. Notice that explicitly depends on . Therefore the expansion of the anomalous dimension is:

where

Notice that for .

We have argued in [15] that the
relation analogous to (1)
holds for the classical string in ,
the conserved charge corresponding to the
-th improved Pohlmeyer charge.
We suggested to identify
the anomalous dimension with the deck transformation acting
on the phase space of the classical string in AdS times a sphere.
The deck transformation can be defined as the
action of the center of the conformal group. It is a geometric symmetry
of the classical string; it comes from
a geometric symmetry of the AdS space.
But it can be
expressed in terms of the hidden symmetries. String theory
in AdS times a sphere has an infinite family of local conserved
charges, the Pohlmeyer charges [16].
These Pohlmeyer charges can be thought of as the classical
limit of the Yangian Casimirs. We have argued in [15]
that the deck transformation is in fact generated by an infinite
linear combination of the Pohlmeyer charges; the coefficients
of this linear combination were fixed in [17].
We used in our arguments the existence of a special
action variable in the theory of the classical string on
which was discussed in [18] following
[19, 20, 21].
The special property
of this particular action variable is that in each order
of the null-surface perturbation theory
[22, 23]
it is given by a local
expression^{1}^{1}1The results of
[24, 25, 26] imply that this
property of the action variable does not hold for the
superstring. Indeed, the construction of the action variable
in Section 4 of [18] used the fact
that the classical bosonic sigma-model splits into the part
and the part. But the fermions “glue together” the
AdS and the sphere. It was argued in [24, 25, 26]
that this action variable still exists in the supersymmetric case
but is not local. What survives the supersymmetric extension
is the statement that the deck transformation
in each order of the null-surface perturbation theory is
generated by a finite sum of the local conserved charges
(the classical analogues of the super-Yangian Casimirs).
Indeed, the definition of the deck transformation
does not require the splitting of the sigma-model into two parts
and the locality of the deck transformation in the perturbation
theory is manifest; it is essentially a consequence of
the worldsheet causality. I want to thank N. Beisert for
a discussion of this subject.. In each
order of the perturbation theory we can approximate
this action variable by a finite sum of the Pohlmeyer
charges.
This action variable corresponds to the length of the
spin chain on the field theory side [27].

This “length” was studied for the finite gap solutions in [28, 25]. Here we will study it for the rational solutions. We will consider the simplest case of the classical string on . This system is related [16] to the sine-Gordon model and we will actually discuss mostly the sine-Gordon model. The existence of the special action variable can be understood locally on the worldsheet, at least in the null-surface perturbation theory. Therefore to study this action variable we do not have to impose the periodicity conditions on the spacial direction of the string worldsheet; we can formally consider infinitely long strings. This allows us to use the rational solutions of the sine-Gordon equation which are probably “simpler” than the finite gap solutions studied in [29, 30, 28, 31, 25] (at least if we consider the elementary functions “simpler” than the theta functions.)

In Section 2 we discuss the relation between the classical string propagating on and the sine-Gordon model. In Section 3 we discuss the tau-function and bilinear identities. In Section 4.1 we discuss Bäcklund transformations and define the “hidden” symmetry . In Section 4.2 we consider the plane wave limit. In Section 4.3 we explain how acts on breathers. In Section 4.4 we discuss the “improved” currents and show that has a local expansion in the null-surface perturbation theory. In Section 5 we summarize our construction of the action variable and outline the analogous construction for the sigma model.

## 2 Sine-Gordon and string on .

### 2.1 Sine-Gordon equation from the classical string.

The sine-Gordon model is one of the simplest exactly solvable models of interacting relativistic fields, and the bosonic string propagating on is one of the simplest nonlinear string worldsheet theories. On the level of classical equations of motion these two models are equivalent.

Consider the classical string propagating on . Let denote the time coordinate parametrizing . We will choose the conformal coordinates on the worldsheet so that the induced metric is proportional to . We will also fix the residual freedom in the choice of the conformal coordinates by putting . We can parametrize the sphere by unit vectors ; the embedding of the classical string in is parametrized by . The worldsheet equations of motion are:

(2) |

These equations of motion follow from the constraints:

(3) | |||

(4) |

The map to the sine-Gordon model is given by [16]:

(5) |

In other words

(6) |

The Virasoro constraints (3) are equivalent to the sine-Gordon equation:

(7) |

What can we say about the inverse map, from to ? Let us consider the limit when the string moves very fast.

### 2.2 Null-surface limit, plane wave limit, free field limit.

When the string moves very fast . As in [22, 23] we replace with where is a small parameter. Because of (6) we should also replace with ; the new field will be finite in the null-surface limit:

(8) |

The string embedding satisfies:

(9) | |||

(10) |

The rescaled sine-Gordon field satisfies:

(11) |

In the strict null-surface limit and

(12) |

In this limit the string worldsheet is a collection of null-geodesics. The -part is therefore a collection of equators of . For each point on the worldsheet the intersection of with the 2-plane generated by and is the corresponding equator; this equator can be parametrized by the vector . We have

Eqs. (9) and (12) show that the one-parameter family of equators forming the null-surface is given by the equation

where and are determined from by (12). Therefore in the limit the null-surface is determined by . It should be possible in principle to extend this analysis to higher orders and find the extremal surface corresponding to the solution of the sine-Gordon equation. The extremal surface is determined by up to the rotations of .

Another important limit is the plane wave limit. To get to the plane wave limit we first go to the null-surface limit (8) and then take an additional rescaling . In the strict limit the equations for become linear:

(13) |

The plane wave limit is therefore the free field limit.

### 2.3 Poisson structure.

The canonical Poisson structure of the classical string on does not agree with the canonical Poisson structure of the sine-Gordon model. But it corresponds to another Poisson structure of the sine-Gordon model, which is compatible with the canonical one [32]. Therefore, the Hamiltonian flows generated by the local conserved charges of the sine-Gordon model should differ from the flows of the Pohlmeyer charges of the classical string only by the relabelling of the charges. This means that the action variable of the sine-Gordon model which we discuss in this paper corresponds to the action variable of the classical string, which is also generated by an infinite linear combination of the local Pohlmeyer charges.

## 3 Rational solutions.

### 3.1 Tau-functions and the dependence on higher times.

In this section we will discuss the dependence of the sine-Gordon solutions on the “higher times” following mostly [33, 34, 35]. We will first introduce the tau-functions and then explain how they are related to the solutions of the sine-Gordon equations.

The tau-functions for the rational solutions are

(14) | |||||

Here and , are parameters characterizing the solution, and , are the so-called times. We identify and . The “higher” times correspond to the higher conserved charges. Changing the higher times corresponds to the motion on the “Liouville torus” in the phase space. Rational solutions correspond to finite ; the tau-functions of the rational solutions are the determinants of the matrices .

Let us consider the left and right Bäcklund transformations:

(16) | |||

(17) |

where and are constant parameters. The tau-functions satisfy the following bilinear identities:

(18) | |||

(19) | |||

(20) | |||

These bilinear identities can be derived from the free fermion representation of the tau-function as explained for example is [35]. We introduce free fermions and , . The “vacuum vectors” are labeled by so that . Let us put and . We have for

(21) | |||

Eq. (18) is Eq. (2.42) of [35] if we take into account that

(22) |

Let us study some differential equations following from the bilinear identities. From (20) we have at the first order in :

(23) | |||

(24) |

Therefore the equations of motion for the sine-Gordon model

(25) |

follow if we set

(26) |

Expanding (18) in the powers of we have

(27) |

and the same equation with and exchanged. This can be rewritten as the first order differential equation relating to :

(28) |

Expanding (20) in powers of we get:

(29) |

and the same equation with exchanged with . This gives us the second equation relating to :

(30) |

Expanding (19) in the powers of we get

(31) |

and the same equation with and exchanged. This gives us the equation relating to :

(32) |

The second equation follows from (20):

(33) |

Equations (28), (30), (32) and (33) are usually taken as the definition of the left and right Bäcklund transformations. These equations do not determine and unambiguously from because there are integration constants. Eqs. (16) and (17) provide a particular solution.

The Bäcklund transformations for the sine-Gordon field correspond to the Bäcklund transformations for the classical string. If is a string worldsheet and is the corresponding solution of the sine-Gordon model defined by Eq. (5) then

(34) |

satisfies

(35) | |||

and

(36) |

satisfies

(37) | |||

The relation between and , and between and , is given by Eq. (5). The relation between Bäcklund transformations in model and sine-Gordon model has been previously discussed in [36].

### 3.2 The reality conditions and a restriction on the class of solutions.

To get the real solutions of the sine-Gordon theory we need to be the complex conjugate of . This can be achieved if the parameters come in pairs and such that and . We want to restrict ourselves with considering only the solutions for which all have a nonzero imaginary part:

(38) |

The purely real would lead to kinks; we consider the solutions with kinks too far from being the fast moving strings.

General solutions of the sine-Gordon equations on a real line were discussed in [34] using the inverse scattering method. There is a difference in notations: our differ from of [34] by a factor of . The scattering data of the general solution includes a discrete set of real (in our notations) , . Besides that, there is a discrete set of complex conjugate pairs with and also a continuous data parametrized by a function with . General solutions can be approximated by the rational solutions, which have . Therefore rational solutions depend only on the discrete set of parameters and . It is useful to look at the asymptotic form of these rational solutions in the infinite future, when . At the rational solutions split into well-separated breathers (corresponding to ) and kinks (corresponding to ). The energy of a breather can be made very small by putting sufficiently close to the imaginary axis (see Section 4.3). This means that one can continuously create a new breather from the vacuum. In other words, creation of the new pair is a continuous operation; it changes a solution in the continuous way. But the creation of a kink is not a continuous operation. The creation of an odd number of kinks would necessarily change the topological charge of the solution. But even to create a pair of kink and anti-kink would require a finite energy. This is our justification for considering separately a sector of solutions which do not have real . We will discuss the action variable in this sector.

## 4 Bäcklund transformations and the “hidden” symmetry .

### 4.1 Construction of .

Eq. (16) shows that the
Bäcklund transformations^{2}^{2}2more precisely, a particular solution
of the Bäcklund equations, defined as a series in or
can be understood as a -dependent
shift of times. We have two 1-parameter families of shifts
and . We have
and .
It is not true that or
is a one-parameter group of transformations, because
it is not true that is equal to
with some .

Both and preserve the symplectic structure. Therefore we can discuss the Hamiltonian vector fields and such that:

(39) |

One could imagine an ambiguity in the definition of and , but we have the continuous families connecting to and to . The existence of these continuous families allows us to define and unambiguously, see Fig. 1. The formula is:

(40) | |||

(41) |

These vector fields act on the rational solutions through the parameters :

(42) | |||

(43) |

Let us consider the limit:

(44) |

We have:

(45) |

We see that the trajectories of the vector field are periodic:

(46) |

Therefore is the Hamiltonian vector field of an action variable. We denote the corresponding hidden symmetry. Notice that exchanges and and therefore maps :

(47) |

The corresponding symmetry of the classical string is . We see that the discrete geometric -symmetry (the “reflection” ) is related to the continuous hidden symmetry (generated by the higher Hamiltonians). This example is of the same nature as the relation between the anomalous dimension and the local charges discussed in [15].

In the language of free fermions, corresponds to the creation of the free fermion from the left vacuum, and to the creation of from the right vacuum. When , the leading term in the operator product expansion of is a -number, and it cancels between and in (26). The shift of the charge of the left and right Dirac vacua leads to the exchange , and therefore Eq. (26) gives .

This construction essentially used the fact that . In fact for any real we have

(48) |

This can be understood directly from (28), (30), (32), (33). First of all we have to explain the meaning of the left hand side of (48), because we defined only for large as a series in and for small as a series in . Let us consider the null-surface limit (8) and construct and as a series in , where is the small parameter of the null-surface perturbation theory defined in (8). In this perturbation theory we have and . The zeroth approximation to (28), (30) and (32), (33) is:

(49) | |||

(50) |

We see that in the leading order of the null-surface perturbation theory and both depend on and as rational functions. The higher orders are also rational functions of and . Therefore in the null-surface perturbation theory and both have an unambiguous analytic continuation to finite values of and . Therefore we can take and (48) follows from (28), (30), (32), (33) and (49), (50). This means that the generator of which we defined in (44) as can be also defined as for any real . For the rational solutions with all having a nonzero imaginary part we have

(51) |

for any . Therefore commutes with the Lorentz boosts which transform to .

### 4.2 Free field limit.

In the limit the equations of motion become

(52) |

And the left and right Bäcklund transformations become:

(53) | |||

(54) |

This means that in the free field limit:

(55) | |||

(56) |

The generator of acts as follows:

(57) |

The free field has an oscillator expansion:

(58) |

where . Eq. (57) implies that is the oscillator number:

(59) |

This is in agreement with the results of [17] and shows that the considered here is the same as considered in [18, 17].

### 4.3 Action of on a breather.

Consider , , and and denote . We get

Therefore

(62) |

Remember that and . The limit corresponds to a circular null-string. Indeed, with Eqs. (6) and (62) imply in this limit that at we have .

The generator of acts on a breather by shifting the phase :

(63) |

The general solution without kinks can be approximated by collections of breathers. The will shift the phases of all the breathers by the same amount.

We have seen in Section 4.2 that the generator of can be also understood as the nonlinear analogue of the oscillator number. On the other hand, we can see from Eqs. (62) and (63) that in the null surface limit

(64) |

where dots denote the terms subleading in the null surface limit.
The leading term is the energy of the string, and the subleading
terms are the higher conserved charges.
The fact that the energy is the oscillator number plus corrections
was observed
already in
the work of H.J. de Vega, A.L. Larsen and N. Sanchez
[37]^{3}^{3}3I want
to thank A. Tseytlin for bringing my attention to this work. .