Research Article | Open Access
On the Pulsating Strings in
We study the class of pulsating strings on . Using a generalized ansatz for pulsating string configurations we find new solutions of this class. Further we quasiclassically quantize the theory and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual Chern-Simons theory.
The attempts to establish a correspondence between the large limit of gauge theories and string theory have more than 30 years history and over the years they showed different faces. Recently an explicit realization of this correspondence was provided by the Maldacena conjecture about AdS/CFT correspondence . The convincing results from the duality between type IIB string theory on and super Yang-Mills theory [1–3] made this subject a major research area and many fascinating new features have been established.
An important part of the current understanding of the duality between gauge theories and strings (M-theory) is the world-volume dynamics of the branes. Recently there has been a number of works focused on the understanding of the world-volume dynamics of multiple M2-branes—an interest inspired by Bagger, Lambert, and Gustavsson [4–7] investigations based on the structure of Lie 3-algebra.
Recently, motivated by the possible description of the world-volume dynamics of coincident membranes in M-theory, a new class of conformal invariant, maximally supersymmetric field theories in 2 + 1 dimensions has been found [8, 9]. The main feature of these theories is that they contain gauge fields with Chern-Simons-like kinetic terms. Based on this development, Aharony, Bergman, Jafferis, and Maldacena proposed a new gauge/string duality between an superconformal Chern-Simons theory (ABJM theory) coupled with bi-fundamental matter, describing membrane on . This model, which was first proposed by Aharony, Bergman, Jafferis, and Maldacena , is believed to be dual to M-theory on .
The ABJM theory actually consists of two Chern-Simons theories of level and correspondingly and each with gauge group . The two pairs of chiral superfields transform in the bi-fundamental representations of and the R-symmetry is as it should be for supersymmetry of the theory. It was observed in  that there exists a natural definition of a 't Hooft coupling . It was observed that in the 't Hooft limit with held fixed, one has a continuous coupling and the ABJM theory is weakly coupled for . The ABJM theory is conjectured to be dual to M-theory on with units of four-form flux. In the scaling limit with the theory can be compactified to type IIA string theory on . Thus, the AdS/CFT correspondence, which has led to many exciting developments in the duality between type IIB string theory on and super Yang-Mills theory, is now being extended to the and is expected to constitute a new example of exact gauge/string theory duality.
The semiclassical strings have played, and still play, an important role in studying various aspects of correspondence [10–31]. The development in this subject gives a strong hint about how the new emergent duality can be investigated. An important role in these studies plays the integrability. The superstrings on as a coset were first studied in  (see also ) which opens the door for investigation of the integrable structures in the theory. To pursue these issues it is necessary to have the complete superstring action. It was noticed that the string supercoset model does not describe the entire dynamics of type IIA superstring in , but only its subsector. The complete string dual of the ABJM model, that is, the complete type-IIA Green-Schwarz string action in superspace has been constructed in .
Various properties on gauge theory side and tests on string theory side as rigid rotating strings, pp-wave limit, relation to spin chains, and certain limiting cases as well as pure spinor formulation have been considered [32–66].
In these intensive studies many properties were uncovered and impressive results obtained, but still the understanding of this duality is far from completeness.
Another class of string solutions which proved its importance in the case of duality is the class of pulsating strings introduced first in  and studied and generalized further in [68–70]. In the case of background the pulsating strings are also expected to play an essential role. The simplest such solutions were mentioned in , but thorough analysis and quasiclassical quantization are still missing. The purpose of this paper is to analyze and quasiclassically quantize the class of pulsating strings on part of the background. The first corrections to the energy, which according to the AdS/CFT conjecture give the anomlaous dimensions of gauge theory operators, will be the main subject of these considerations.
The paper is organized as follows. In the Introduction we briefly present the basic facts about ABJM theory. To explain the method and fix the procedure, in the next section we give short review of the pulsating strings and their quasi-classical quantization for the case of background. Next, we proceed with the pulsating strings on background restricting the string dynamics to part of the spacetime. The next section is devoted to the derivation of the correction to the energy. We find the wave function associated with the Laplace-Beltrami operator of first and then compute the first correction to the energy. The latter is supposed to give the anomalous dimension of operator of the dual theory. We conclude with a brief discussion on the results.
1.1. ABJM and Strings on
To find the ABJM theory one starts with analysis M2-brane dynamics governed by 11d supergravity action 
where . Solving for the equations of motions
one can find the M2-brane solutions whose near horizon limit becomes
In addition we have units four-form flux
Next we consider the quotient acting as . It is convenient first to write the metric on as
and then to perform the quotient identifying with ( is proportional to the Kahler form on ). The resulting metric becomes
One can observe that the first volume factor on the right-hand side is divided by factor of compared to the initial one. In order to have consistent quantized flux one must impose where is the number of quanta of the flux on the quotient. We note that the spectrum of the supergravity fields of the final theory is just the projection of the initial onto the invariant states. In this setup there is a natural definition of 't Hooft coupling . Decoupling limit should be taken as , while is kept fixed.
One can follow now  to make reduction to type IIA with the following final result:
We end up then with compactification of type IIA string theory with units of flux on and units of flux on the 2-cycle.
The radius of curvature in string units is . It is important to note that the type IIA approximation is valid in the regime where .
In order to fix the notations, we write down the explicit form of the metric on in spherical coordinates. The metric on can be written as 
Here is the radius of the , and are left-invariant 1-forms on an , parameterized by ,
The range of the coordinates is
2. A Brief Review of Pulsating Strings on
In this section we give a brief review of the pulsating string solutions obtained first by Minahan  and generalized later in [68–70]. We will concentrate only on the case of pulsating string on part of , that is, a circular string which pulsates expanding and contracting on is considered. To fix the notations, we start with the metric of and relevant part of
where . In order to obtain the simplest solution, we identify the target space time with the worldsheet one, , and use the ansatz , that is, the string is stretched along direction, and keep the dependence on for a while. The reduced Nambu-Goto action in this case is
For our considerations it is useful to pass to Hamiltonian formulation. For this purpose we find the canonical momenta and find the Hamiltonian in the form (for more details see )
Fixing the string to be at the origin of space (), we see that the squared Hamiltonian have a form very similar to a point particle. The last term in (2.3) can be considered as a perturbation, so first we find the wave function for a free particle in the above geometry
The solution to the above equation is
where are spherical harmonics on and the energy is given by
Since we consider highly excited states (large energies), one should take large , so we can approximate the spherical harmonics as
The correction to the energy can be obtained by using perturbation theory, which to first order is
Up to the first order in we find for the anomalous dimension of the corresponding YM operators (see  for more details)
It should be noted that in this case the -charge is zero. In order to include it, we consider pulsating string on which has a center of mass that is moving on the subspace of [68–70]. While in the previous example part of the metric was assumed trivial, now we consider all the angles to depend on (only). The corresponding Nambu-Goto action now is
Again, we see that the squared Hamiltonian looks like the point particle one, however, now the potential has angular dependence. Denoting the quantum number of and by and correspondingly, one can write the Schrodinger equation
where we set . The solution to the Schrodinger equation is
The first-order correction to the energy is
or, up to first order in
The anomalous dimension then is given by
3. Pulsating Strings on
In this section we consider a circular pulsating string expanding and contracting only on part of . We will consider the string dynamics confined in the part of the spacetime with no extension of the string in the remaining spatial coordinates. Then the relevant metric we will work with is given by
Having in mind the explicit form of the metric (1.9), one can write it as
where the part is defined as
while is the remaining part of metric associated with coordinates, denoted here as , , , . The residual worldsheet symmetry allows us to identify with and to obtain a classical pulsating string solution. To do that we use the following ansatz:
We are interested in the induced worldsheet metric which in our case has the form
The Nambu-Goto action is
in this ansatz then reduces to expression
where , and we set for simplicity . To follow the procedure described in the previous section, we need a Hamiltonian formulation of the problem. For this purpose, we have to find first the canonical momenta of our dynamical system. Straightforward calculations give for the momenta
Solving for the derivatives in terms of the canonical momenta and substituting back into the Legendre transform of the Lagrangian, we find the Hamiltonian
The interpretation of this relation is as in the case of , namely, the first term in the brackets is the kinetic term while the second one is considered as a potential , which in our case has the form
The approximation where our considerations are valid assumes high energies, which suggests that one can think of this potential term as a perturbation. For latter use we write down the explicit form of the potential
The above perturbation to the free action will produce the corrections to the energy and therefore to the anomalous dimension. In order to calculate the corrections to the energy as a perturbation due to the above potential however, we need the normalized wave function associated with space and all these issues are subject to the next section.
4. Semiclassical Correction to the Energy
In this section we will compute the quasi-classical correction to the energy of the pulsating string on . As we discussed in the previous section, the Hamiltonian of the pulsating string is interpreted as a dynamical system with high energy described by free theory Schrödinger equation and perturbed by a potential (3.11).
First of all, we have to find the wave function associated with the Laplace-Beltrami operator on and then to obtain the correction to the energy due to the induced potential.
4.1. The Metric and the Associated Laplace-Beltrami Operator
We start with the Laplace-Beltrami operator in coordinates parameterizing as follows. First we define coordinates
The line element of in these coordinates is explicitly given by
4.1.1. The Laplace-Beltrami Operator on
Using the standard definition of the Laplace-Beltrami operator in global coordinates we find
Since we are going to separate the variables, it is useful to arrange the terms as follows:
Then the operator can be written as
The full measure on is
4.1.2. The Wave Function
The Schrödinger equation for the wave function is
To separate the variables, we define as , where
With this choice we can solve for the eigenfunctions replacing the derivatives along Killing directions () by () correspondingly. The equation for then separates from the rest and has the form
It is convenient to define a new variable in which the equation can be written as
The solution to this equation is
Our wave function must be square integrable with respect to the measure on (4.7). In separated variables this means that must be square integrable with respect to the measure on . This condition imposes the following restriction on the parameters:
Introducing new parameters and the solution can be written in the form
The equation for can be treated analogously and has the form With the help of the new variable , the above equation can be written as
Recognizing the hypergeometric equation mentioned previously we find the solution as
In addition we have to ensure that the solution is square integrable with respect to the measure in . This requirement imposes the following restriction on the parameters:
It is useful to introduce new parameters and , in terms of which the solution can be written in the form
Then, the equation for we have to solve becomes
We change the variable defining
It is convenient to define new parameters , , and , , and then the solution can be written in the terms of Legendre polynomials
In terms of the new variables the measure (4.7) becomes
4.2. Corrections to the Energy
There is potential (3.11) in terms
There is correction to the energy
The explicit form of the previous formula using the explicit form of the various wave functions is
In short notations, it looks as
where the integrals are explicitly calculated in the appendix. The expression for the correction to the energy looks very complicated. Therefore, to make conclusions we use the fact that the approximation we work in is for large quantum numbers, say , and . Within this approximation the integrals behave like
Ignoring the terms of high order (, , , , ) we obtain
We see that the corrections to the energy have complicated form even at this level. Nevertheless, one can recognize some structures of the quantum numbers that appear in the energy corrections. In order to make comparison with the much well-known case of pulsating strings, we present in what follows the case of pulsating strings on .
4.2.1. Pulsating in
The case of pulsating on subspace of the spacetime can be defined as follows. The metric can be written as
and as before are left-invariant 1-form on an . Since and it follows that and
The potential in this case becomes
The correction to the energy can be computed using the same technology as before and it gives the expression
To obtain an explicit expression; we have to calculate the following integral:
In (4.35) and therefore we end up with
On other hand we need the limit of large
which gives the following approximation of the integral :
Substituting into the expression for the correction to that energy, we find
We note that in the limit of large we reproduce the of .
In this section we summarize the results of our study. The purpose of this paper was to look for pulsating string solutions in background. The class of pulsating strings have been used to study AdS/CFT correspondence in the case of [67, 69, 70] and the corrections to the string energy has been associated with anomalous dimensions of certain type operators.
The recently suggested duality between string theory in attracted much attention promising to be exact and with number of applications. Here we consider a generalized string ansatz for pulsating string in the part of the geometry. Next we considered the corrections to the classical energy. From AdS/CFT point of view the corrections to the classical energy give the anomalous dimensions of the operators in SYM theory and therefore they are of primary interest.
For this purpose, we consider the Nambu-Goto actions and find the Hamiltonian. After that we quantize the resulting theory semiclassically and obtain the corrections to the energy. Since we consider highly excited system, the kinetic term is dominating and the effective potential term serves for a small perturbation. Note that this semiclassical treatement of the problem can be interpreted as an effective summation over all classical solutions.
The obtained corrections to the classical energy look complicated, but in certain limit one can find relatively simple expressions. To identify the contribution of the different terms, it is instructive to look at the solutions for the case. At the end of the last section we present these solutions thus providing a basis for comparison. Since they correspond to a subsector of the well known from considerations one can identify the origin of the various contributions. One can see that the corrections to the energy have analogous structure to the case of pulsating, say in . The mixing between quantum numbers of the different isometry directions shows up in analogous, but slightly more complicated way. This can be seen using the result from subsector and its embedding in .
As a final comment, we note that in order to complete the analysis from AdS/CFT point of view, it is of great interest to perform an analysis comparing our result to that in SYM side. We leave this important question for future research.
Some Integrals Appearing in the Main Text
The integrals appeared in (4.28) are defined as follows: