# Semiclassical relativistic strings in and long coherent operators in =4 SYM theory

###### Abstract:

We consider the low energy effective action corresponding to the 1-loop, planar, dilatation operator in the scalar sector of SYM theory. For a general class of non-holomorphic “long” operators, of bare dimension , it is a sigma model action with 8-dimensional target space and agrees with a limit of the phase-space string sigma model action describing generic fast-moving strings in the part of . The limit of the string action is taken in a way that allows for a systematic expansion to higher orders in the effective coupling . This extends previous work on rigid rotating strings in (dual to operators in the sector of the dilatation operator) to the case when string oscillations or pulsations in are allowed. We establish a map between the profile of the leading order string solution and the structure of the corresponding coherent, “locally BPS”, SYM scalar operator.

As an application, we explicitly determine the form of the non-holomorphic operators dual to the pulsating strings. Using action–angle variables, we also directly compute the energy of pulsating solutions, simplifying previous treatments.

^{†}

^{†}preprint: BRX TH-543

hep-th/0406189

## 1 Introduction

The AdS/CFT correspondence [1] gave a precise example of the conjectured relation [2] between the large limit of gauge theories and string theory. In its most well-known form it claims that SYM theory with gauge group and coupling on one hand, and type IIB string theory on with units of RR 5-form flux, and string coupling on the other, are just two different descriptions of the same theory. The string theory becomes weakly coupled, i.e. the theory becomes “stringy”, in the limit , , with the ’t Hooft coupling being fixed and large, .

It remained unclear, however, how strings “emerge” from the field theory, in particular, which (local, single-trace) gauge theory operators [3] should correspond to which “excited” string states and how one may verify the matching of their dimensions/energies beyond the well-understood BPS/supergravity sector. An important step in that direction was made in [4] where it was shown how this correspondence can be established for a class of “small” (nearly point-like) near-BPS strings which are ultrarelativistic, i.e. whose kinetic energy is much larger than their mass.

Shortly after, in [5] it was argued that at least a qualitative agreement between the non-BPS states on the two sides of the duality can be established also for certain extended string states represented by classical string solutions with one large angular momentum. The semiclassical approach of [5] was further developed and generalized to multispin string states in [6, 7]. It was proposed in [7] that, like for “small” near-BPS BMN [4] string states, a quantitative agreement between string theory and gauge theory should be found also for a class of extended classical solutions with two or three non-zero angular momenta in the (see [8] for a review). For such solutions the classical energy has a regular expansion in powers of , where is the total spin, , and the string corrections are suppressed in the limit fixed. Assuming that the large and then small limit is well-defined also on the SYM side, one should then be able to compare the classical string results for the energy to the quantum SYM results for the corresponding anomalous dimensions.

Using the crucial observation of [9] that the one-loop scalar dilatation operator can be interpreted as a Hamiltonian of an integrable spin chain and thus can be diagonalized even for large by the Bethe ansatz method, the proposal of [7] was confirmed at the leading order of expansion in in [10, 11]. There, a remarkable agreement was found between energies of various string solutions and eigenvalues of the dilatation operator representing dimensions of particular SYM operators.

The established correspondence was thus between a thermodynamic limit of the Bethe ansatz eigenstates of the integrable spin chain and a large spin, or, equivalently, large energy, limit of the classical solitonic solutions of the string sigma model action. The classical bosonic coset sigma model also has a well-known integrable structure which becomes explicit for particular rigid-shape rotating string configurations [12, 13] and which can be mapped [14, 15] to the one of the spin chain.

The next important step was made in [16] where it was shown that one can take the large energy, or ultrarelativistic, limit directly in the string action getting a reduced “non-relativistic” sigma model that describes in a universal way the leading-order corrections to the energies of all string solutions in the two-spin sector. Moreover, it was found [16] that the resulting action agrees exactly with the semiclassical coherent state action describing the (Heisenberg XXX) sector of the spin chain in the fixed limit, which turns out to be equivalent to a continuum limit in which one keeps only linear and quadratic derivative terms. This demonstrated how a string action can directly emerge from a gauge theory in the large- limit and provided a direct map between “coherent” SYM states or operators built out of two holomorphic scalars, and all two-spin classical string states, bypassing the need to apply the Bethe ansatz to find anomalous dimension for each particular state. Furthermore, the correspondence established at the level of the action implies also the matching of fluctuations around particular solutions (just as in the BMN case where one matches fluctuations near a BPS state) and thus goes beyond the case of rigidly rotating strings. This remark applies also to other subsectors which we shall describe below which therefore overlap, i.e. are related by small deformations. These subsectors are named according to a certain type of basic solutions they contain (which carry several large conserved charges) but they also describe many “nearby” states which may be labeled by higher conserved charges or oscillation numbers.

The observation made in [16] may be viewed also as based on the fact that the spin chain has two equivalent descriptions, a Hamiltonian operator one and a Lagrangian path integral one. The corresponding Lagrangian was shown to be identical to a limit of the classical string Lagrangian. The relevant semiclassical limit of the path integral is, as usual, naturally represented by the coherent states of the operator approach. The matching demonstrated in [10, 11] can then be interpreted as an equivalence between the two descriptions of the spin chain. The remarkable fact is that the classical solutions in the path integral approach, namely the solutions of the Landau-Lifshitz (LL) equations, are in correspondence with exact eigenstates found using the thermodynamic limit of the Bethe ansatz: the energies as well as all higher conserved charges are the same. The general proof of this fact was given later in [17] using integrable models methods.

From the effective action point of view, one can also argue that, to lowest order in derivatives, there is only one unknown coefficient that can be fixed, e.g., by comparison with the BMN [4] result. Therefore, at leading order in the fixed limit, the effective action is unique and should be expected to reproduce the same limit of the exact results. This uniqueness is lost at higher orders in where more terms in the effective action are present. In that case, the coherent state approach needs to be generalized as was explained in [18]. This allowed us to verify the correspondence at the order. An equivalent general result (using the integrable spin chain embedding of [19]) was obtained also in the Bethe ansatz approach [17].

The approach of [16, 18] was also generalized (to leading order in ) to the three-spin or sector [20, 21] (as well as to the [22, 11] sector corresponding to one spin in and one in the [21]). On the Bethe ansatz side the agreement between the energies of particular 3-spin string solutions and the corresponding spin chain eigenvalues was previously shown in [15, 23].

The results reviewed above explained the matching between all rotating string solutions with 3 large angular momenta and all “long” operators constructed out of the three scalars , , (including, as mentioned above, also “near-by” states). However, there is also another interesting class of solutions for strings moving in – the so called pulsating strings [24, 15] which also have a regular expansion of their energy in terms where is a large “oscillating number”. Their energies were matched (to order ) to the energies of the SYM theory states in [10, 15] using the Bethe ansatz techniques for the spin chain of [9].

To carry out a
similar matching at the level of the effective action,
that is to match the corresponding
coherent states and not only the energy eigenvalues,^{1}^{1}1The Bethe ansatz approach [9, 10, 15] provides, in principle,
a recipe to construct the corresponding pure eigenstates or Bethe wave functions,
but this is not easy to do in practice.
we need to understand how to
extend the
ideas discussed above in the rotating string sector
to the whole spin chain [9], i.e. to
the subset of the SYM operators
constructed out of all 6 real scalars and
not limited to the holomorphic
products of , , .

The first step towards that goal was made in [21]. There, the Grassmanian was identified as the coherent state target space for the spin chain sigma model since it parametrizes the orbits of the half-BPS operator under the rotations. In a related development, motivated by the suggestions in [25] and [16], the procedure of taking the high energy or limit of the classical string theory was generalized [26, 27] to the whole bosonic action (and was later extended to include fermions [28]).

In the present paper we shall study the sector in detail, carefully working out the spin chain and the string theory sides of the correspondence. We will show that the agreement between the two effective actions extends to the whole subsector of scalar operators characterized by a “local BPS” condition, i.e. built out of products of rotations of the BPS 6-vector . It is this condition that selects the coset as the target space. This condition ensures that the corresponding anomalous dimensions on the field theory side are of order and thus can be compared to the leading order corrections to the energy on the string side. The role of this locally BPS condition was also emphasized in [27].

On the string side, we need to find a “reduced” sigma model by taking a large energy limit of the classical string action. We shall essentially follow [16, 18, 21] but improve the derivation of the reduced action in two ways. First, we will clarify the gauge fixing procedure by using an alternative, 2-d dual (or “T-dual”) action where the linear in time derivative “Wess-Zumino” term appears more naturally from the usual –field coupling term. Second, we will use canonical transformations to systematize the change of variables that was previously needed [18] to eliminate terms of higher than first power of time derivatives. In this way will we find a completely systematic and universal procedure to derive higher order in corrections on the string theory side. The procedure is independent of a particular solution one may consider and, moreover, it should be possible to generalize this procedure to the full case.

We should mention that the method of canonical perturbations was already applied to this problem in ref. [27] which computed the leading order action for a generic fast motion in using a similar but somewhat different approach based on consideration of near light-like surfaces. The advantage of our procedure is in its systematic nature which allows us to compute higher order corrections with relative ease.

On the spin chain side, we will use a variation of the coherent state approach that was successful in the and case. In the coherent state approach, one first reformulates the quantum mechanical spin chain problem in terms of a coherent state path integral [29] and then observes that in the limit we are interested in, i.e. fixed, one can take the continuum limit and all quantum corrections can be ignored. This is essentially equivalent to ignoring quantum mechanical correlations between different sites of the spin chain, and, as a result, we are lead to a classical action for the system. The variation of the coherent state approach we shall use is simply to ignore quantum correlations from the very beginning by considering states which are the product of independent states at each site. The classical action can then be thought of as the action that leads to the Heisenberg equations of motion in this restricted subspace. This method, which is equivalent to the one used in [18], leads to the same result at leading order but has some practical advantages when applied at the next order.

In this paper we will only consider the leading order term in the spin chain effective action which turns out to be in perfect agreement with the leading order action obtained on the string side.

An important consistency check is that, starting with the reduced sigma model, we should be able to reproduce the leading order results for all pulsating and rotating string solutions in . The map that emerges between the field theory and the string theory [16, 26, 27] is that each portion of the string that moves approximately along a maximum circle carrying one unit of R-charge, corresponds to a site of the spin chain at which there is a half-BPS operator with the same R-charge. In particular, this map allows us to find the coherent operators corresponding to the pulsating strings of [24, 15]. Since the pulsating solutions are a particular case, the agreement between the actions that we find here explains the agreement already observed in [10, 15] between the string result for the energy and the eigenvalues obtained using the Bethe ansatz for the chain. This also shows that in this case there is an exact agreement between the energy as a function of the conserved quantities as obtained from the solutions of our sigma model and the one obtained from the Bethe ansatz. This leads us to conjecture that one should be able to prove the agreement in general as was done in the case in [17].

As a side but interesting result, we also find, using action–angle variables, an exact classical relation between the energy and the constants of motion of pulsating solutions. This simplifies previous treatments, putting these solutions at the same level as the rotating ones of [7, 12, 13].

The organization of this paper is as follows. In section 2 we discuss the most general case of rotating strings in and the gauge fixing procedure. In the next section 3 we do the same for the most general fast motion on which includes also pulsating solutions. We also explain there a systematic procedure to compute higher orders in from the string side. In section 4 we obtain the same leading order sigma model starting from the field theory side and studying the action of the dilatation operator on operators constructed out of scalars, i.e. starting from the spin chain Hamiltonian of [9]. We give examples of the string–spin chain correspondence in the following section 5 where we verify that it includes all known solutions in which the motion is on the . Finally, in section 6 we give a detailed analysis of the pulsating solution of [15] using action–angle variables.

In section 7 we make some concluding remarks, and in the Appendix we give a brief introduction to the subject of canonical perturbation theory to make the paper self-contained and as a reference for the reader.

## 2 String theory side: rotating strings ( sector)

In preparation for studying the full sector, which is done in the next section, let us start here by describing the general procedure to derive the effective action for a rotating string in the limit of large semiclassical rotation parameters. This connects and generalizes previous partial results of [16, 18, 20, 21]. We shall isolate and gauge-fix a “fast” collective coordinate and get an action for “slow” variables as an expansion in , where is the total angular momentum. This action will thus reproduce the expression for the energy of rotating string solutions expanded in powers of [7, 12, 13, 8]. In the two-spin or “” sector it will coincide with the action found in [18], while in the most-general pure-rotation three-spin or “” sector it will generalize the leading-order action found in [21, 20] (see also [27, 28]) to all orders in . Here we shall explicitly consider only the case when string moves on but a generalization to the case when there is also a motion in is possible too (see [21] and references there).

### 2.1 Isolating the “fast” angular coordinate

In the case of generic rotating strings it is natural to follow [7, 18, 21] and parametrize the metric in terms of 3 complex coordinates ()

(1) |

where is the time direction of . In terms of 6 real coordinates or standard angles of one has

(2) |

In this parametrization string states that carry three independent (Cartan) components of angular momentum should be rotating in the three orthogonal planes [7, 8]. To consider the limit of large total spin we would like to isolate the corresponding collective coordinate, i.e. the common phase of . In the familiar case of fast motion of the center of mass the role of is played by linear momentum or . Here, however, represents the sum of an “orbital” and “internal” angular momenta and thus does not correspond simply to the center of mass motion. This is thus a generalization of the limit considered in [4]: we are interested in “large” extended string configurations and not in nearly point-like strings.

Isolating the common phase in the three orthogonal planes by introducing the new coordinates and

(3) |

one finds that the metric (1) becomes

(4) |

where

(5) |

Here belongs to : the metric is invariant under a simultaneous shift of and a rotation of . In general, this parametrization corresponds to a Hopf fibration of over : is the Fubini-Study metric and is the covariantly constant Kähler form on . In the two-spin or sector () where the motion is within of the two coordinates of can be replaced by a unit 3-vector [18]

(6) |

and has a non-local WZ-type representation

The general form of the string action in (the string is positioned at the center of with being the time) with the metric (4) is then (we use world-sheet signature ))

(7) |

(8) |

The crucial point is that one should view and as “longitudinal” coordinates that reflect the redundancy of the reparametrization-invariant string description: they are not “seen” on the gauge theory side, and should be gauged away (or eliminated using the constraints). At the same time, the vector describing string profile should be interpreted as a “transverse” or physical coordinate which should thus have a counterpart on the spin chain side, with an obvious candidate being a vector parametrizing the coherent state [16, 18, 21]. The conserved charges corresponding to translations in time and are

(9) |

(10) |

where the effective coupling constant is directly related to the (rescaled) charge in (10)

(11) |

The “fast motion” expansion in powers of which we will be interested in is, thus, the same as the expansion in powers of .

What remains is to do the following three steps: (i) proper gauge fixing; (ii) expansion of the action at large , suppressing time derivatives of in favor of spatial derivatives; (iii) field redefinitions to eliminate from the expanded action terms of higher order than first in time derivatives.

To leading order in one may simply use the standard conformal gauge (as was done in the sector in [16, 18] and in the sector in [21, 20]). As was pointed out in [18], to get the full action to all orders in one should use a special “adapted” gauge. Here we shall make this first gauge fixing step particularly transparent by explaining that the gauge fixing procedure used in [18] amounts simply to the standard static gauge for the coordinate which is 2-d dual (or “T-dual”) to .

Having in mind comparison with the spin chain side it is natural to request that translations in time in the target space and on the world sheet should be related. Also, we should ensure that the angular momentum is homogeneously distributed along the string so that its density in (10), i.e. the momentum conjugate to , is constant. Therefore, one should fix the following gauge [18]

(12) |

As was shown in [18], starting with the phase-space form of the string action and imposing this “non-conformal” gauge one finds the following effective Lagrangian for

(13) |

(14) |

In the case [18] this gives an equivalent action for (6) with . As was noted in [18], apart from the WZ-type term , the Lagrangian (13) looks like a Nambu Lagrangian in a static gauge, suggesting that there may be a more direct way of deriving it. This is indeed the case as we shall explain below.

### 2.2 2-d duality transformation and “static” gauge fixing

Let us make few remarks on
interpretation of string action in connection with spin chain
on the SYM side
and for simplicity consider the case.
In the semiclassical coherent state description
of the spin chain [16], one has a circular direction
along the chain at each point of which one has a
classical spin vector belonging to a 2-sphere.
In other words, if we
combine operator ordering direction under the trace
with an “internal” direction
we get type of geometry.
A similar geometry indeed
emerges on the string sigma model side – we have
fibered by with base .
However, in (4) the direction is that of the angle .
Fast rotation corresponds to time-dependent , so
it is not quite appropriate
to call a “longitudinal” coordinate since after
we have chosen gauge we have
already fixed a time-like coordinate.
A natural idea is that the true longitudinal coordinate
should be “T-dual” counterpart
of , i.e. one should apply 2-d duality to
the scalar field in (8).^{2}^{2}2We are grateful
to N. Nekrasov for emphasizing
to us the potential importance of applying T-duality in .

There is, however, an important subtlety. The standard discussions of T-duality are usually done in conformal gauge, but if we would fix the conformal gauge and then also we would no longer have a freedom to fix . Actually, will not, in general, be a solution of the equations for in the conformal gauge. The correct procedure is not to impose the conformal gauge; we should first apply the 2-d duality, then go from the Polyakov to the Nambu form of the action by solving for the 2-d metric , and finally fix the static gauge . Remarkably, this turns out to be equivalent to the gauge fixing procedure used in [18], leading directly to (13). This explains that the non-diagonal gauge used in [18] is nothing but the standard static gauge in the Nambu action for the dual coordinate . Not imposing the conformal gauge allows one to have solutions consistent with the above static gauge choice.

Let us first note that the equation for following from the action (8), i.e. =0, can be solved by setting

(15) |

where should then satisfy

(16) |

Comparing (15) to (10) we observe that

(17) |

which is satisfied, in particular, if one fixes the gauge by setting

(18) |

The limit of small or large (cf. (11))
is then the limit of large winding number of the dual coordinate .
^{3}^{3}3A comment on the quantization condition. In the standard case
of circular coordinate with radius the dual coordinate
has radius . In the present case
and so the period of
should be . This implies
that should indeed be an integer winding number.

Let us now apply the 2-d duality systematically at the level of the string action (8). Replacing by where is an auxiliary 2-d vector field, adding the “Lagrange multiplier” term , and then integrating out we end up with the 2-d dual counterpart of (8)

(19) |

where was given in (14). Thus the “T-dual” background has no off-diagonal metric component but has a non-trivial NS-NS 2-form coupling in the sector. Eliminating the 2-d metric we then get the Nambu form of the 2-d dual counterpart of the action (7),(8)

(20) |

(21) |

If we now fix the static gauge

(22) |

we finish with the action equivalent to (13),(14)

(23) |

(24) |

We thus uncover the origin of the string-theory counterpart of the WZ term in the spin-chain coherent state effective action: it comes from the 2-d NS-NS WZ term upon the static gauge fixing in the T-dual action.

### 2.3 Eliminating time derivatives

The remaining steps are as in [16, 18]. We assume that higher powers of time derivatives are suppressed. To define a consistent expansion, we may then redefine the time coordinate so that the leading order approximation does not involve :

(25) |

thus getting the string action (23) with

(26) |

Expanding in powers of
(and omitting the constant term)
this gives^{4}^{4}4In the case one finds [18]

(27) |

(28) |

Finally, we should eliminate higher than first time derivatives in by field redefinitions order by order in . This was explained in detail in the case in [18]. For example, this amounts to eliminating from using leading-order (Landau-Lifshitz type) equation for following from . Note that the leading term in the action (27) is the same as found by choosing the conformal gauge and eliminating from the action using the constraints [21]. Also, in the next section we will again obtain an equivalent result by a different but related method which relies on the use of canonical transformations.

To summarize, the main lesson of the above derivation is that the dual angular coordinate is just a replacement for the momentum of the “fast” coordinate . Applying T-duality allows one to fix the static gauge which thus identifies the spin chain direction with direction. Applied to sector the above procedure determines the subleading terms in the effective action. in (28) may, in principle, be compared with the 2-loop effective action on the spin chain side generalizing the comparison in the case in [18]. Indeed, the corresponding 2-loop dilatation operator (for the superspin chain containing bosonic sector) was found in [30], and, as was recently pointed out in [31], the mixing of bosonic operators from the sector with the fermionic operators at the two (and higher) loop order is effectively suppressed in the long spin chain (large ) limit.

## 3 String theory side: general fast motion ( sector)

Now we can proceed to analyze the sector.
On the string side we have a string that moves almost at the
speed of light^{5}^{5}5The corresponding
limit can also be seen as
a small tension limit [25].
along the . It is then natural to start by
isolating a coordinate that describes the fast motion in order to use
the approximation that the velocities of all other
coordinates
are small. In previous work
[16, 18, 20, 21]^{6}^{6}6See also [27] for a
different approach.
this was achieved by means of an appropriate change of coordinates.
However, this can be done only
if we already know the particular string configuration
we are aiming to describe.
Here, instead, we would like to isolate a fast coordinate
independently of the type of solution, i.e.
in a universal way that will apply to
rotating [7, 32, 11], pulsating [24, 15] and
other similar solutions (see e.g. [33, 34]) that
describe fast moving strings on .

The common feature to all of them is that each piece of the string is moving along a maximum circle almost at the speed of light. Since each point of the string moves, in general, along a different massless geodesic in , to define the fast coordinate we need to know the position and velocity of each point of the string. Therefore, we can isolate the fast coordinate only if we use the phase space description.

What we find is that the circle along which each piece of string moves, slowly changes in time due to interactions with its neighbors along the string (represented by the terms containing sigma derivatives). This is in exact agreement with the picture on the spin chain side. In an appropriate gauge, each piece of string moving along a circle carries one unit of R-charge and corresponds to a half-BPS operator on the spin chain carrying the same charge. On the spin chain, the operator we have at each site (given, for example, by the mean value of the spin in the case) also changes in time precisely as a result of the interaction with its neighbors. Notice that here we are mapping a time dependent classical string into a time dependent coherent state on the spin chain side and not into an energy eigenstate. Energy eigenstates of the spin chain or SYM theory should correspond to single-string eigenstates in the bulk, while classical string solutions should be represented by coherent states.

In the rest of this section we do the following steps. First, we isolate the fast coordinate which we shall call as in section 2. Then we obtain the leading order terms in the action by considering the limit in which changes much faster than all other coordinates. One problem is that the Lagrangian contains terms which oscillate in (proportional to ). Since they average to zero, at leading order they can be discarded. However, when we go to higher orders, they give a contribution. The simplest way to treat them is to eliminate them order by order using coordinate transformations. In this way we end up with a Lagrangian that, to the order considered, is independent of the fast coordinate, i.e., like (8), has an isometry . We can then perform a T-duality and fix a static gauge as in (18),(22), i.e. , where is essentially the momentum conjugate to . We end up with the standard Nambu action with an extra term due to the presence of a –field. This Wess-Zumino type term ensures the correct phase space structure of the action. After that the large energy or large expansion is simply a Taylor expansion of the square root in the Nambu action.

### 3.1 Isolating the “fast” angular coordinate

Consider the Lagrangian for the string on written in terms of time and 6 real coordinates

(29) |

The corresponding conformal gauge constraints are (we choose as an additional gauge fixing condition)

(30) |

In the simplest case when does not depend on , i.e. the string is point-like, we get the geodesic equations , . These are solved by

(31) |

or, equivalently, by

(32) |

where

(33) |

The constant 6-vectors and parametrize the space of massless
geodesics in , i.e. the space of maximum circles in .
Equivalently, they
parameterize the Grassmanian – the space of planes in that go through the
origin.^{7}^{7}7In general, the Grassmanian is the space of all dimensional
hyperplanes
of the vector space .
Thus, as was noted in [27],
the coset
is the moduli space of geodesics in .
On the SYM side the same space appeared
as a natural coherent state space for spin chain in [21].
Let us mention also
that describing the Grassmanian
may be interpreted as coordinates of subject
to an additional condition , i.e.
they define an 8-dimensional subspace of .

If we now consider an extended relativistic closed string, the trajectory of each piece of the string can be described as a circular trajectory but with the parameters and (or ) which determine its orientation, slowly changing in time and in (the coordinate along the string). Also, the speed along the circle is time dependent in this case. A familiar analogy is an orbital motion of a planet in a solar system. It moves along an ellipse around the Sun but the parameters of the ellipse slowly change in time due to perturbations by other planets, moons, etc. In both cases, the fast coordinate is an angle in the plane of motion determined by the position and the velocity.

To obtain the leading order result for an effective Lagrangian of “slow” coordinates it is sufficient to work in the conformal gauge and use again the additional gauge condition . Then the phase space Lagrangian is simply

(34) |

Here and below we define the momenta (, etc.) as derivatives of the Lagrangian, i.e. do not include the tension factor in the action (7) in their definition. To isolate the coordinate we introduce its conjugate momentum and do the following coordinate transformation

(35) |

where, as in (31),
, , .
This can be conveniently written as^{8}^{8}8For constant , this is actually quite analogous
to the usual transformation
to “coherent coordinates” for a harmonic oscillator.

(36) |

where as in (33)

(37) |

The constraints on ensure that and (which is a consequence of and in the conformal gauge). There is a redundant degree of freedom which is obvious from the gauge invariance ( is an arbitrary function):

(38) |

We could fix this gauge invariance by choosing e.g. or , but it is more convenient not to do this to preserve the rotational symmetry. It is then natural to introduce the covariant derivatives defined by

(39) |

where as in (5) the gauge field is not an independent variable but is given by

(40) |

Now it is easy to derive the following useful
identities^{9}^{9}9
We shall often omit index on in quadratic relations
assuming summation over .

(41) |

which allow us to rewrite the Lagrangian (34) as

(42) | |||||

where is a real and is a complex Lagrange multiplier field. The conformal gauge constraints are then

(43) |

Notice that we could use that (since and ) to simplify these expressions but we prefer to keep the gauge invariance (38) manifest.

Another point we note for later reference is that the second constraint in (43) implies . Since the string is closed, the coordinates in (36) must be periodic in . We may assume then that (an a priori possible “winding” part can be absorbed into , cf. (38)). Then we obtain the following constraint

(44) |

On the field theory side, the fact that the string is closed is related to the fact that the dual operators are given by the single trace which is invariant under cyclic permutations of local fields under the trace. In the spin chain description such operators are represented by states of a closed chain which are invariant under cyclic permutations. This invariance gives rise to a condition equivalent to (44) (see section 4).

### 3.2 Leading order in large energy expansion

We can take into account that is a fast variable by setting

where and are the new variables,
and then taking the limit
while
keeping , and
fixed.^{10}^{10}10See [26, 27] for related ideas.
This limit is also reminiscent of the so called wrapped or
non-relativistic limit [35].
The only difference with the previous
rotating string
cases is that here there are terms in (42),(43) proportional
to . It is
clear that for large these terms
can be ignored since they average to zero.^{11}^{11}11An equivalent averaging
was also considered in [27].
Then, the conformal gauge constraints (43)
determine and as
(to leading order in ):

(45) |

Using that in the Lagrangian (42) and taking the same limit we find

(46) |

The first term here is a total derivative and may be omitted, so that the corresponding action takes the form similar to (23),(27)

(47) |

(48) |

where in (48) we rescaled the time coordinate as in (25) (and did not explicitly include the Lagrange multiplier terms).

We thus find the “Landau-Lifshitz” version of the
sigma model, i.e.
with the first time derivative WZ-type term instead of the usual
quadratic time-derivative term.^{12}^{12}12Note that this model is integrable
since it is obtained as a limit of
an integrable sigma model (in general,
Grassmanian cosets are classically integrable [36]).
This Lagrangian (46) is the same
as found in [27] through a different
procedure.^{13}^{13}13One difference compared to [27]
is that we write in gauge invariant form.
The expression in [27] should be considered as a
gauge fixed version of (46).

A Lagrangian equivalent to (46) was found also on the spin chain side in [21] (see also the next section for the derivation). As was shown in [21], the corresponding Hamiltonian represents a low-energy (continuum) limit of the expectation value of the spin chain Hamiltonian of [9] in the natural coherent state obtained by applying the transformation from to the BPS ground state at each site of the spin chain. This state may be parametrized by a real antisymmetric matrix m, where is a hermitian generator, satisfying the constraints [21] mm, tr(m. We can see the relation to the present discussion if we solve these constraints by introducing a complex 6-vector subject to , with m. Going back to the string theory side, let us note that the generators corresponding to the Lagrangian (29) written in terms of become, after using (36),