On the Pulsating Strings in 𝐴𝑑𝑆₄×ℂℙ³

Abstract

We study the class of pulsating strings on 𝐴𝑑𝑆₄×ℂℙ³. 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 𝒩=6 Chern-Simons theory.

1. Introduction

The attempts to establish a correspondence between the large 𝑁 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 [1]. The convincing results from the duality between type IIB string theory on 𝐴𝑑𝑆₅×𝑆⁵ and 𝒩=4 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 𝒩=6 superconformal Chern-Simons theory (ABJM theory) coupled with bi-fundamental matter, describing 𝑁 membrane on 𝑆⁷/ℤ_𝑘. This model, which was first proposed by Aharony, Bergman, Jafferis, and Maldacena [9], is believed to be dual to M-theory on 𝐴𝑑𝑆₄×𝑆⁷/ℤ_𝑘.

The ABJM theory actually consists of two Chern-Simons theories of level 𝑘 and −𝑘 correspondingly and each with gauge group 𝑆𝑈(𝑁). The two pairs of chiral superfields transform in the bi-fundamental representations of 𝑆𝑈(𝑁)×𝑆𝑈(𝑁) and the R-symmetry is 𝑆𝑈(4) as it should be for 𝒩=6 supersymmetry of the theory. It was observed in [9] that there exists a natural definition of a 't Hooft coupling −𝜆=𝑁/𝑘. It was observed that in the 't Hooft limit 𝑁→∞ with 𝜆 held fixed, one has a continuous coupling 𝜆 and the ABJM theory is weakly coupled for 𝜆≪1. The ABJM theory is conjectured to be dual to M-theory on 𝐴𝑑𝑆₄×𝑆⁷/ℤ_𝑘 with 𝑁 units of four-form flux. In the scaling limit 𝑁,𝑘→∞ with 𝑘≪𝑁≪𝑘⁵ the theory can be compactified to type IIA string theory on 𝐴𝑑𝑆₄×ℙ³. Thus, the AdS/CFT correspondence, which has led to many exciting developments in the duality between type IIB string theory on 𝐴𝑑𝑆₅×𝑆⁵ and 𝒩=4 super Yang-Mills theory, is now being extended to the 𝐴𝑑𝑆₄/𝐶𝐹𝑇₃ 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 𝐴𝑑𝑆₅/𝑆𝑌𝑀₄ 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 𝐴𝑑𝑆₄×ℂℙ³ as a coset were first studied in [32] (see also [33]) 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 𝐴𝑑𝑆₄×ℂℙ³, but only its subsector. The complete string dual of the ABJM model, that is, the complete type-IIA Green-Schwarz string action in 𝐴𝑑𝑆₄×ℂℙ³ superspace has been constructed in [34].

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 𝐴𝑑𝑆₅/𝐶𝐹𝑇₄ duality is the class of pulsating strings introduced first in [67] and studied and generalized further in [68–70]. In the case of 𝐴𝑑𝑆₄×ℂℙ³ background the pulsating strings are also expected to play an essential role. The simplest such solutions were mentioned in [57], 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 ℂℙ³ part of the 𝐴𝑑𝑆₄×ℂℙ³ 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 𝑅×𝑆³ background. Next, we proceed with the pulsating strings on 𝐴𝑑𝑆₄×ℂℙ³ background restricting the string dynamics to ℂℙ³ 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 𝐴𝑑𝑆₄×ℂℙ³ 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 𝐴𝑑𝑆₄×ℂℙ³

To find the ABJM theory one starts with analysis M2-brane dynamics governed by 11d supergravity action [9]

1𝑆=2𝜅²₁₁𝑑𝑥¹¹𝜎−𝑔(𝑅−𝜙12⋅4!𝐹_{𝜇𝜈𝜌𝜎}𝐹^{𝜇𝜈𝜌𝜎}1)−12𝜅²₁₁𝐶⁽³⁾𝐹⁽⁴⁾𝐹⁽⁴⁾, (1.1) where 𝜅²₁₁=2⁷𝜋⁸𝑙⁹_𝑝. Solving for the equations of motions

𝑅^𝜇_𝜈=12(1𝐹3!^{𝜇𝛼𝛽𝛾}𝐹_{𝜈𝛼𝛽𝛾}−1𝛿3⋅4!^𝜇𝜈𝐹_{𝛼𝛽𝜌𝜎}𝐹^{𝛼𝛽𝜌𝜎}𝜕),_𝜎(√−𝑔𝐹^{𝜎𝜇𝜈𝜉}1)=2⋅(4!)²𝜖^{𝜇𝜈𝜉𝛼1⋯𝛼8}𝐹_{𝛼1⋯𝛼4}𝐹_{𝛼5⋯𝛼8}, (1.2) one can find the M2-brane solutions whose near horizon limit becomes 𝐴𝑑𝑆₄×𝑆⁷

𝑑𝑠²=𝑅²4𝑑𝑠²_{𝐴𝑑𝑆4}+𝑅²𝑑𝑠²_𝑆7. (1.3) In addition we have 𝑁^′ units four-form flux

𝐹⁽⁴⁾=3𝑅³8𝜖_{𝐴𝑑𝑆4},𝑅=𝑙_𝑝(2⁵𝑁^′𝜋²)^1/6. (1.4) Next we consider the quotient 𝑆⁷/ℤ_𝑘 acting as 𝑧_𝑖→𝑒^{𝑖(2𝜋/𝑘)}𝑧_𝑖. It is convenient first to write the metric on 𝑆⁷ as

𝑑𝑠²_𝑆7=(𝑑𝜑^′+𝜔)²+𝑑𝑠²_𝐶𝑃3, (1.5) where

𝑑𝑠²_𝐶𝑃3=∑_𝑖𝑑𝑧_𝑖𝑑𝑧_𝑖𝑟²−|∑_𝑖𝑧_𝑖𝑑𝑧_𝑖|²𝑟⁴,𝑟²≡⁴_𝑖=1|𝑧_𝑖|²,𝑑𝜑^′𝑖+𝜔≡2𝑟²(𝑧_𝑖𝑑𝑧_𝑖−𝑧_𝑖𝑑𝑧_𝑖𝑧),𝑑𝜔=𝐽=𝑖𝑑(^𝑖𝑟)𝑑(𝑧_𝑖𝑟), (1.6) and then to perform the ℤ_𝑘 quotient identifying 𝜑^′=𝜑/𝑘 with 𝜑∼𝜑+2𝜋 (𝐽 is proportional to the Kahler form on ℂℙ³). The resulting metric becomes

𝑑𝑠²_{𝑆7/ℤ𝑘}=1𝑘²(𝑑𝜑+𝑘𝜔)²+𝑑𝑠²_𝐶𝑃3. (1.7) One can observe that the first volume factor on the right-hand side is divided by factor of 𝑘 compared to the initial one. In order to have consistent quantized flux one must impose 𝑁^′=𝑘𝑁 where 𝑁 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 𝐴𝑑𝑆₄×𝑆⁷ onto the ℤ_𝑘 invariant states. In this setup there is a natural definition of 't Hooft coupling 𝜆≡𝑁/𝑘. Decoupling limit should be taken as 𝑁, 𝑘→∞ while 𝑁/𝑘 is kept fixed.

One can follow now [9] to make reduction to type IIA with the following final result:

𝑑𝑠²_string=𝑅³𝑘(14𝑑𝑠²_{𝐴𝑑𝑆4}+𝑑𝑠²_ℂ𝑃3𝑒),_2𝜙=𝑅³𝑘³∼𝑁^1/2𝑘^5/2=1𝑁²(𝑁𝑘)^5/2,𝐹₄=38𝑅³𝜖₄,𝐹₂=𝑘𝑑𝜔=𝑘𝐽. (1.8) We end up then with 𝐴𝑑𝑆₄×ℂℙ³ compactification of type IIA string theory with 𝑁 units of 𝐹₄ flux on 𝐴𝑑𝑆₄ and 𝑘 units of 𝐹₂ flux on the ℂ𝑃¹⊂ℂℙ³ 2-cycle.

The radius of curvature in string units is 𝑅²_str=𝑅³/𝑘=2^5/2𝜋√𝜆. It is important to note that the type IIA approximation is valid in the regime where 𝑘≪𝑁≪𝑘⁵.

In order to fix the notations, we write down the explicit form of the metric on 𝐴𝑑𝑆₄×ℂℙ³ in spherical coordinates. The metric on 𝐴𝑑𝑆₄×ℂℙ³ can be written as [71]

𝑑𝑠²=𝑅²{14[−cosh²𝜌𝑑𝑡²+𝑑𝜌²+sinh²𝜌𝑑Ω²₂]+𝑑𝜇²+sin²𝜇[𝑑𝛼²+14sin²𝛼(𝜎²₁+𝜎²₂+cos²𝛼𝜎²₃1)+4cos²𝜇(𝑑𝜒+sin²𝜇𝜎₃)²]}. (1.9) Here 𝑅 is the radius of the 𝐴𝑑𝑆₄, and 𝜎_1,2,3 are left-invariant 1-forms on an 𝑆³, parameterized by (𝜃,𝜙,𝜓),

𝜎₁𝜎=cos𝜓𝑑𝜃+sin𝜓sin𝜃𝑑𝜙,₂𝜎=sin𝜓𝑑𝜃−cos𝜓sin𝜃𝑑𝜙,₃=𝑑𝜓+cos𝜃𝑑𝜙. (1.10) The range of the coordinates is 𝜋0≤𝜇,𝛼≤2,0≤𝜃≤𝜋,0≤𝜙≤2𝜋,0≤𝜒,𝜓≤4𝜋. (1.11)

2. A Brief Review of Pulsating Strings on 𝑅×𝑆³

In this section we give a brief review of the pulsating string solutions obtained first by Minahan [67] and generalized later in [68–70]. We will concentrate only on the case of pulsating string on 𝑆⁵ part of 𝐴𝑑𝑆₅×𝑆⁵, that is, a circular string which pulsates expanding and contracting on 𝑆⁵ is considered. To fix the notations, we start with the metric of 𝑆⁵ and relevant part of 𝐴𝑑𝑆₅

𝑑𝑠²=𝑅²(cos²𝜃𝑑Ω²₃+𝑑𝜃²+sin²𝜃𝑑𝜓²+𝑑𝜌²−cosh²𝑑𝑡²), (2.1) where 𝑅²=2𝜋𝛼^′√𝜆. In order to obtain the simplest solution, we identify the target space time with the worldsheet one, 𝑡=𝜏, and use the ansatz 𝜓=𝑚𝜎, that is, the string is stretched along 𝜓 direction, 𝜃=𝜃(𝜏) and keep the dependence on 𝜌=𝜌(𝜏) for a while. The reduced Nambu-Goto action in this case is

𝑆=𝑚√𝜆𝑑𝑡sin𝜃cosh²𝜌−𝜃̇². (2.2) 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 [67])

𝐻=cosh𝜌Π²_𝜌+Π²_𝜃+𝑚²𝜆sin²𝜃. (2.3) Fixing the string to be at the origin of 𝐴𝑑𝑆₅ space (𝜌=0), 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

cosh𝜌sinh³𝜌𝑑𝑑𝜌cosh𝜌sinh³𝜌𝑑𝑑𝜌Ψ(𝜌,𝜃)−cosh²𝜌sin𝜃cos³𝜃𝑑𝑑𝜃sin𝜃cos³𝜃𝑑𝑑𝜃Ψ(𝜌,𝜃)=𝐸²Ψ(𝜌,𝜃). (2.4) The solution to the above equation is

Ψ_2𝑛(𝜌,𝜃)=(cosh𝜌)^−2𝑛−4𝑃_2𝑛(cos𝜃), (2.5) where 𝑃_2𝑛(cos𝜃) are spherical harmonics on 𝑆⁵ and the energy is given by

𝐸_2𝑛=Δ=2𝑛+4. (2.6) Since we consider highly excited states (large energies), one should take large 𝑛, so we can approximate the spherical harmonics as

𝑃_2𝑛(cos𝜃)≈4𝜋cos(2𝑛𝜃). (2.7) The correction to the energy can be obtained by using perturbation theory, which to first order is

𝛿𝐸²=₀^𝜋/2𝑑𝜃Ψ^∗_2𝑛(0,𝜃)𝑚²𝜆sin²𝜃Ψ_2𝑛𝑚(𝜃)=²𝜆2. (2.8) Up to the first order in 𝜆 we find for the anomalous dimension of the corresponding YM operators (see [67] for more details)

1Δ−4=2𝑛[1+2𝑚²𝜆(2𝑛)²]. (2.9) It should be noted that in this case the 𝑅-charge is zero. In order to include it, we consider pulsating string on 𝑆⁵ which has a center of mass that is moving on the 𝑆³ subspace of 𝑆⁵ [68–70]. While in the previous example 𝑆³ part of the metric was assumed trivial, now we consider all the 𝑆³ angles to depend on 𝜏 (only). The corresponding Nambu-Goto action now is

𝑆=−𝑚√𝜆𝑑𝑡sin𝜃1−𝜃̇²−cos²𝜃𝑔_𝑖𝑗𝜙̇^𝑖𝜙̇^𝑗, (2.10) where 𝜙_𝑖 are 𝑆³ angles and 𝑔_𝑖𝑗 is the corresponding 𝑆³ metric. The Hamiltonian in this case is [68–70]

𝐻=Π²_𝜃+𝑔^𝑖𝑗Π_𝑖Π_𝑗cos²𝜃+𝑚²𝜆sin²𝜃. (2.11) 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 𝑆³ and 𝑆⁵ by 𝐽 and 𝐿 correspondingly, one can write the Schrodinger equation

−4𝜔𝑑𝑑𝜔Ψ(𝜔)+𝐽(𝐽+1)𝜔Ψ(𝜔)=𝐿(𝐿+4)Ψ(𝜔), (2.12) where we set 𝜔=cos²𝜃. The solution to the Schrodinger equation is

Ψ(𝜔)=2(𝑙+1)1(𝑙−𝑗)!𝜔(𝑑)𝑑𝜔^𝑙−𝑗𝜔^𝑙+𝑗(1−𝜔)^𝑙−𝑗𝐽,𝑗=2𝐿;𝑙=2. (2.13) The first-order correction to the energy 𝛿𝐸 is

𝛿𝐸²=𝑚²𝜆2(𝑙+1)²−(𝑗+1)²−𝑗²,(2𝑙+1)(2𝑙+3) (2.14) or, up to first order in 𝜆

𝐸²=𝐿(𝐿+4)+𝑚²𝜆𝐿²−𝐽²2𝐿². (2.15) The anomalous dimension then is given by

𝑚𝛾=²𝜆4𝐿𝛼(2−𝛼), (2.16) where 𝛼=1−𝐽/𝐿.

We conclude this section referring for more details to [67–70].

3. Pulsating Strings on 𝐴𝑑𝑆₄×ℂℙ³

In this section we consider a circular pulsating string expanding and contracting only on ℂℙ³ part of 𝐴𝑑𝑆₄×ℂℙ³. We will consider the string dynamics confined in the ℂℙ³ 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

𝑑𝑠²_{𝑡×ℂℙ3}=𝑅²1(−4𝑑𝑡²+𝑑𝑠²_ℂℙ3),𝑅²=√32𝜋²𝜆. (3.1) Having in mind the explicit form of the ℂℙ³ metric (1.9), one can write it as

𝑑𝑠²_ℂℙ3=𝑔_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗+𝑔_𝑝𝑞𝑑𝑦^𝑝𝑑𝑦^𝑞, (3.2) where the part 𝑔_𝑖𝑗 is defined as

𝑔_𝑖𝑗=diag(1,sin²1𝜇,4sin²𝜇sin²𝛼),𝑖,𝑗=1,2,3,𝑥¹=𝜇,𝑥²=𝛼,𝑥³=𝜃, (3.3) while 𝑔_𝑝𝑞=𝑔_𝑝𝑞(𝜇,𝛼,𝜃) is the remaining part of metric associated with 𝜙,𝜓,𝜒 coordinates, denoted here as 𝑝,𝑞=1,2,3, 𝑦¹=𝜙, 𝑦²=𝜓, 𝑦³=𝜒. The residual worldsheet symmetry allows us to identify 𝑡 with 𝜏 and to obtain a classical pulsating string solution. To do that we use the following ansatz:

𝑥¹=𝑥¹(𝜏)=𝜇(𝜏),𝑥²=𝑥²(𝜏)=𝛼(𝜏),𝑥³=𝑥³𝑦(𝜏)=𝜃(𝜏),₁=𝜙=𝑚₁𝜎+ℎ¹(𝜏),𝑦²=𝜓=𝑚₂𝜎+ℎ²(𝜏),𝑦³=𝜒=𝑚₃𝜎+ℎ³(𝜏). (3.4) We are interested in the induced worldsheet metric which in our case has the form

𝑑𝑠²_𝑤𝑠=𝑅²1{(−4+𝑔_𝑖𝑗𝑥̇^𝑖𝑥̇^𝑗+𝑔_𝑝𝑞ℎ̇^𝑝ℎ̇^𝑞)𝑑𝜏²+(𝑔_𝑝𝑞𝑚_𝑝𝑚_𝑞)𝑑𝜎²+2(𝑔_𝑝𝑞𝑚_𝑝ℎ̇^𝑞)𝑑𝜏𝑑𝜎}. (3.5) The Nambu-Goto action is

𝑆_NG=−𝑇𝑑𝜏𝑑𝜎−det(𝐺_𝜇𝜈𝜕_𝛼𝑋^𝜇𝜕_𝛽𝑋^𝜈), (3.6) in this ansatz then reduces to expression

𝑆_NG=−𝑇𝑅²𝑑𝜏𝑑𝜎(14−𝑔_𝑖𝑗𝑥̇^𝑖𝑥̇^𝑗−𝑔_𝑝𝑞ℎ̇^𝑝ℎ̇^𝑞)(𝑔_𝑝𝑞𝑚_𝑝𝑚_𝑞)+(𝑔_𝑝𝑞𝑚_𝑝ℎ̇^𝑞)(𝑔_𝑝𝑞𝑚_𝑞ℎ̇^𝑝), (3.7) where 𝑇𝑅²=2√2𝜆, and we set for simplicity 𝛼^′=1. 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

Π_𝑖=2√(𝑔2𝜆^𝑝𝑞𝑚_𝑝𝑚_𝑞)𝑔_𝑖𝑗𝑥̇^𝑗(1/4−𝑔_𝑖𝑗𝑥̇^𝑖𝑥̇^𝑗−𝑔_𝑝𝑞ℎ̇^𝑝ℎ̇^𝑞)(𝑔_𝑝𝑞𝑚_𝑝𝑚_𝑞)+(𝑔_𝑝𝑞𝑚_𝑝ℎ̇^𝑞)(𝑔_𝑝𝑞𝑚_𝑞ℎ̇^𝑝)Π,𝑖=1,2,3,_𝑝=2√(𝑔2𝜆^𝑝𝑞𝑚_𝑝𝑚_𝑞)𝑔_𝑝𝑞ℎ̇^𝑞−(𝑔_𝑝𝑞𝑚_𝑝ℎ̇^𝑞)𝑔_𝑝𝑞𝑚_𝑞(1/4−𝑔_𝑖𝑗𝑥̇^𝑖𝑥̇^𝑗−𝑔_𝑝𝑞ℎ̇^𝑝ℎ̇^𝑞)(𝑔_𝑝𝑞𝑚_𝑝𝑚_𝑞)+(𝑔_𝑝𝑞𝑚_𝑝ℎ̇^𝑞)(𝑔_𝑝𝑞𝑚_𝑞ℎ̇^𝑝),𝑝=1,2,3. (3.8) Solving for the derivatives in terms of the canonical momenta and substituting back into the Legendre transform of the Lagrangian, we find the Hamiltonian

𝐻²=14(𝑔^𝑖𝑗Π_𝑖Π_𝑗+𝑔^𝑝𝑞Π_𝑝Π_𝑞√)+(22𝜆)²(𝑔_𝑝𝑞𝑚_𝑝𝑚_𝑞)4. (3.9) The interpretation of this relation is as in the case of 𝐴𝑑𝑆₅×𝑆⁵, namely, the first term in the brackets is the kinetic term while the second one is considered as a potential 𝑉, which in our case has the form

𝑉(𝜇,𝛼,𝜃)=2𝜆𝑔_𝑝𝑞(𝜇,𝛼,𝜃)𝑚_𝑝𝑚_𝑞. (3.10) 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

𝜆𝑉(𝜇,𝛼,𝜃)=2{sin²𝜇sin²𝛼[sin²𝜃+cos²𝛼cos²𝜃(1+sin²𝛼)]𝑚²₁+sin²𝜇sin²𝛼cos²𝛼(1+sin²𝛼)𝑚²₂+sin²𝜇cos²𝛼𝑚²₃+2sin²𝜇sin²𝛼cos²𝛼cos𝜃(1+sin²𝛼)𝑚₁𝑚₂+2sin²𝜇sin²𝛼cos²𝛼cos𝜃𝑚₁𝑚₃+2sin²𝜇sin²𝛼cos²𝛼𝑚₂𝑚₃}. (3.11) 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 ℂℙ³ 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 ℂℙ³. 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 ℂℙ³ and then to obtain the correction to the energy due to the induced potential.

4.1. The ℂℙ³ Metric and the Associated Laplace-Beltrami Operator

We start with the Laplace-Beltrami operator in coordinates parameterizing ℂℙ³ as follows. First we define coordinates

𝑦₁+𝑖𝑦₂𝜃=𝜇sin𝛼cos2𝑒^{(𝑖/2)(𝜙+𝜓+𝜒)},𝑦₃+𝑖𝑦₄𝜃=𝜇sin𝛼sin2𝑒^{(𝑖/2)(𝜙−𝜓−𝜒)},𝑦₅+𝑖𝑦₆=𝜇cos𝛼𝑒^(𝑖/2)𝜒. (4.1) The line element of ℂℙ³ in these coordinates is explicitly given by

𝑑𝑠²_ℂℙ3=𝑑𝜇²+sin²𝜇𝑑𝛼²+14sin²𝜇sin²𝛼𝑑𝜃²+14sin²𝜇sin²𝛼[sin²𝜃+cos²𝛼cos²𝜃(1+sin²𝛼)]𝑑𝜙²+14sin²𝜇sin²𝛼cos²𝛼(1+sin²𝛼)𝑑𝜓²+14sin²𝜇cos²𝛼𝑑𝜒²+24sin²𝜇sin²𝛼cos²𝛼cos𝜃(1+sin²+2𝛼)𝑑𝜓𝑑𝜙4sin²𝜇sin²𝛼cos²2𝛼cos𝜃𝑑𝜒𝑑𝜙+4sin²𝜇sin²𝛼cos²𝛼𝑑𝜓𝑑𝜒. (4.2)

4.1.1. The Laplace-Beltrami Operator on ℂℙ³

Using the standard definition of the Laplace-Beltrami operator in global coordinates we find

1Δ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒)=sin⁵𝜇𝜕𝜕𝜇[sin⁵𝜇𝜕1𝜕𝜇]+sin²𝜇sin³𝛼cos²𝛼𝜕𝜕𝛼[sin³𝛼cos²𝛼𝜕]+4𝜕𝛼sin²𝜇sin²𝜕𝛼sin𝜃𝜕𝜕𝜃[sin𝜃4𝜕𝜃]+sin²𝜇sin²𝛼sin²𝜃𝜕²𝜕𝜙²−8cos𝜃sin²𝜇sin²𝛼sin²𝜃𝜕²+𝜕𝜙𝜕𝜓4(sin²𝜃+cos²𝛼cos²𝜃)sin²𝜇sin²𝛼sin²𝜃cos²𝛼𝜕²𝜕𝜓²−8sin²𝜇cos²𝛼𝜕²+𝜕𝜒𝜕𝜓4(1+sin²𝛼)sin²𝜇cos²𝛼𝜕²𝜕𝜒². (4.3) Since we are going to separate the variables, it is useful to arrange the terms as follows:

=1Δ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒)sin⁵𝜇𝜕𝜕𝜇(sin⁵𝜇𝜕)+1𝜕𝜇sin²𝜇{1sin³𝛼cos²𝛼𝜕𝜕𝛼(sin³𝛼cos²𝛼𝜕4𝜕𝛼)+sin²𝛼cos²𝛼(𝜕𝜕𝜓−sin²𝛼𝜕)𝜕𝜒²+4cos²𝛼𝜕²𝜕𝜒²+4sin²𝛼[1𝜕sin𝜃𝜕𝜕𝜃(sin𝜃1𝜕𝜃)+sin²𝜃(𝜕𝜕𝜕𝜙−cos𝜃)𝜕𝜓²]}. (4.4) Then the operator Δ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒) can be written as

1Δ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒)=sin⁵𝜇𝜕𝜕𝜇(sin⁵𝜇𝜕1𝜕𝜇)+sin²𝜇Δ(𝛼,𝜃,𝜙,𝜓,𝜒), (4.5) where

1Δ(𝛼,𝜃,𝜙,𝜓,𝜒)=sin³𝛼cos²𝛼𝜕𝜕𝛼(sin³𝛼cos²𝛼𝜕4𝜕𝛼)+sin²𝛼cos²𝛼(𝜕𝜕𝜓−sin²𝛼𝜕)𝜕𝜒²+4cos²𝛼𝜕²𝜕𝜒²+4sin²𝛼1Δ(𝜃,𝜙,𝜓),Δ(𝜃,𝜙,𝜓)=𝜕sin𝜃𝜕𝜕𝜃(sin𝜃1𝜕𝜃)+sin²𝜃(𝜕𝜕𝜕𝜙−cos𝜃)𝜕𝜓². (4.6) The full measure on ℂℙ³ is

1𝑑Ω(𝜇,𝛼,𝜃)=16sin⁵𝜇sin³𝛼cos²𝛼sin𝜃𝑑𝜇𝑑𝛼𝑑𝜃. (4.7)

4.1.2. The Wave Function

The Schrödinger equation for the wave function is

Δ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒)Ψ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒)=−4𝐸²Ψ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒). (4.8) To separate the variables, we define Ψ as Ψ(𝜇,𝛼,𝜃,𝜙,𝜓,𝜒)=𝑓₁(𝜇)𝑓₂(𝛼)𝑓₃(𝜃)𝑓₄(𝜙,𝜓,𝜒), where

𝑓₄(𝜙,𝜓,𝜒)=exp(𝑖𝑙₁𝜙)exp(𝑖𝑙₂𝜓)exp(𝑖𝑙₃𝜒),𝑙₁,𝑙₂,𝑙₃∈ℤ. (4.9) With this choice we can solve for the eigenfunctions replacing the derivatives along Killing directions (𝜕_𝜙,𝜕_𝜓,𝜕_𝜒) by (𝑖𝑙₁,𝑖𝑙₂,𝑖𝑙₃) correspondingly. The equation for 𝑓₃(𝜃) then separates from the rest and has the form

[Δ(𝜃,𝜙,𝜓)+𝐿²]𝑓₃1(𝜃)≡{𝑑sin𝜃𝑑𝑑𝜃(sin𝜃1𝑑𝜃)−sin²𝜃(𝑙₁−cos𝜃𝑙₂)²+𝐿²}𝑓₃(𝜃)=0. (4.10) It is convenient to define a new variable 𝑧=cos𝜃 in which the equation can be written as

{(1−𝑧²)𝑑²𝑑𝑧²𝑑−2𝑧−1𝑑𝑧1−𝑧²(𝑙₁−𝑧𝑙₂)²+𝐿²}𝑓₃(𝑧)=0. (4.11) The solution to this equation is

𝑓₃(𝑧)=(1−𝑧)^{|𝑙1−𝑙2|/2}(1+𝑧)^{|𝑙1+𝑙2|/2}×₂𝐹₁[12(|𝑙₁−𝑙₂|+|𝑙₁+𝑙₂|+1−1+4(𝑙²₂+𝐿²1)),2(|𝑙₁−𝑙₂|+|𝑙₁+𝑙₂|+1+1+4(𝑙²₂+𝐿²));1+|𝑙₁+𝑙₂|,1+𝑧2]. (4.12) Our wave function must be square integrable with respect to the measure on ℂℙ³ (4.7). In separated variables this means that 𝑓₃(𝜃) must be square integrable with respect to the measure on 𝜃. This condition imposes the following restriction on the parameters:

1+4(𝑙²₂+𝐿²)−|𝑙₁−𝑙₂|−|𝑙₁+𝑙₂|−1=2𝑛,𝑛∈ℕ. (4.13) Introducing new parameters 𝛼=|𝑙₁+𝑙₂| and 𝛽=|𝑙₁−𝑙₂| the solution can be written in the form

𝑓₃(𝑧)=(1−𝑧)^𝛼/2(1+𝑧)^𝛽/2𝑛!Γ(𝛼+1)𝑃Γ(𝛼+1+𝑛)^{𝑛(𝛼,𝛽)}(𝑧). (4.14) The equation for 𝑓₂(𝛼) can be treated analogously and has the form [Δ(𝛼,𝜃,𝜙,𝜓,𝜒)+𝑀²]𝑓₂1(𝛼)≡{sin³𝛼cos²𝛼𝑑𝑑𝛼(sin³𝛼cos²𝛼𝑑4𝑑𝛼)−sin²𝛼cos²𝛼(𝑙₂−sin²𝛼𝑙₃)²−4cos²𝛼𝑙²₃−4sin²𝛼𝐿²+𝑀²}𝑓₂(𝛼)=0. (4.15) With the help of the new variable 𝑦=sin²𝛼, the above equation can be written as

𝑑{4𝑦(1−𝑦)²𝑑𝑦²𝑑+2(4−7𝑦)−4𝑑𝑦𝑦(1−𝑦)(𝑙₂−𝑦𝑙₃)²−4𝑙1−𝑦²³−4𝑦𝐿²+𝑀²}𝑓₂(𝑦)=0. (4.16) Recognizing the hypergeometric equation mentioned previously we find the solution as

𝑓₂(𝑦)=𝑦^{(1+4(𝐿2+𝑙22)−1)/2}(1−𝑦)^{2((𝑙2−𝑙3)2+𝑙23)}×₂𝐹₁[12(32+1+4(𝐿²+𝑙²₂)+4((𝑙₂−𝑙₃)²+𝑙²₃)−25+16(𝑀²+4𝑙²₃)21),2(32+1+4(𝐿²+𝑙²₂)+4((𝑙₂−𝑙₃)²+𝑙²₃)+25+16(𝑀²+4𝑙²₃)2);1+1+4(𝐿²+𝑙²₂),𝑦]. (4.17)

In addition we have to ensure that the solution 𝑓₂(𝛼) is square integrable with respect to the measure in 𝛼. This requirement imposes the following restriction on the parameters:

25+16(𝑀²+4𝑙²₃)2−32−1+4(𝐿²+𝑙²₂)−4((𝑙₂−𝑙₃)²+𝑙²₃)=2𝑚,𝑚∈ℕ. (4.18) It is useful to introduce new parameters 𝑝=1+4(𝐿²+𝑙²₂) and 𝑞=1/2+4((𝑙₂−𝑙₃)²+𝑙²₃), in terms of which the solution can be written in the form

𝑓₂(𝑦)=𝑦^(𝑝−1)/2(1−𝑦)^{(𝑞−1/2)/2}𝑚!Γ(𝑝+1)𝑃Γ(𝑝+1+𝑚)^{𝑚(𝑝,𝑞)}(1−2𝑦). (4.19) Then, the equation for 𝑓₁(𝜇) we have to solve becomes

{1sin⁵𝜇𝑑𝑑𝜇(sin⁵𝜇𝑑𝑀𝑑𝜇)−²sin²𝜇+4𝐸²}𝑓₁(𝜇)=0. (4.20) We change the variable defining 𝑥=cos𝜇

{(1−𝑥²)𝑑²𝑑𝑥²𝑑−6𝑥−1𝑑𝑥1−𝑥²𝑀²+4𝐸²}𝑓₁(𝑥)=0. (4.21) It is convenient to define new parameters 𝑘=(25+16𝐸²−1)/2, 𝑘=2,3,…, and 𝑙=4+𝑀², 𝑙=2,3,…, and then the solution can be written in the terms of Legendre polynomials

𝑓₁(𝑥)=(1−𝑥²)⁻¹𝑃^𝑙_𝑘(𝑥). (4.22) In terms of the new variables the measure (4.7) becomes

1𝑑Ω(𝑥,𝑦,𝑧)=16(1−𝑥²)²𝑦(1−𝑦)^1/22𝑑𝑥𝑑𝑦𝑑𝑧,0≤𝑥,𝑦≤1,−1≤𝑧≤1. (4.23) The normalized wave functions of (4.14), (4.19), (4.22) have a form

Ψ_𝑛^𝛼,𝛽(𝑧)=(2𝑛+𝛼+𝛽+1)𝑛!Γ(𝑛+𝛼+𝛽+1)2^𝛼+𝛽+1Γ(𝑛+𝛼+1)Γ(𝑛+𝛽+1)(1−𝑧)^𝛼/2(1+𝑧)^𝛽/2𝑃_𝑛^(𝛼,𝛽)Ψ(𝑧),_{𝑚𝑝,𝑞}(𝑦)=2(2𝑚+𝑝+𝑞+1)𝑚!Γ(𝑚+𝑝+𝑞+1)𝑦Γ(𝑚+𝑝+1)Γ(𝑚+𝑞+1)^(𝑝−1)/2(1−𝑦)^{(𝑞−1/2)/2}𝑃_𝑚^(𝑝,𝑞)Ψ(1−2𝑦),_𝑙𝑘(𝑥)=4(2𝑘+1)(𝑘−𝑙)!(𝑘+𝑙)!(1−𝑥²)⁻¹𝑃^𝑙_𝑘(𝑥). (4.24)

4.2. Corrections to the Energy

There is potential (3.11) in terms 𝑥,𝑦,𝑧

𝜆𝑉(𝑥,𝑦,𝑧)=2(1−𝑥²){𝑚²₁𝑦(1−𝑧²)+𝑦(1−𝑦²)[𝑚²₁𝑧²+2𝑚₁𝑚₂𝑧+𝑚²₂]+𝑚²₃(1−𝑦)+2𝑦(1−𝑦)[𝑚₁𝑚₃𝑧+𝑚₂𝑚₃]}. (4.25) There is correction to the energy

𝛿𝐸²=¹₀¹₀¹₋₁𝑑Ω(𝑥,𝑦,𝑧)𝑉(𝑥,𝑦,𝑧)Ψ^𝑙2_𝑘(𝑥)Ψ^𝑝,𝑞2_𝑚(𝑦)Ψ^𝛼,𝛽2_𝑛(𝑧). (4.26)

The explicit form of the previous formula using the explicit form of the various wave functions is

𝛿𝐸²=𝜆2116¹₀(1−𝑥²)Ψ^𝑙2_𝑘(𝑥)(1−𝑥²)²𝑑𝑥×{𝑚²₁12¹₀𝑦Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2𝑑𝑦¹₋₁(1−𝑧²)Ψ^𝛼,𝛽2_𝑛+1(𝑧)𝑑𝑧2¹₀𝑦(1−𝑦²)Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2𝑑𝑦×[𝑚²₁¹₋₁𝑧²Ψ^𝛼,𝛽2_𝑛(𝑧)𝑑𝑧+2𝑚₁𝑚₂¹₋₁𝑧Ψ^𝛼,𝛽2_𝑛(𝑧)𝑑𝑧+𝑚²₂]+𝑚²₃12¹₀(1−𝑦)Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/21𝑑𝑦+2¹₀𝑦(1−𝑦)Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2𝑑𝑦×[𝑚₁𝑚₃¹₋₁𝑧Ψ^𝛼,𝛽2_𝑛(𝑧)𝑑𝑧+𝑚₂𝑚₃]}. (4.27)

In short notations, it looks as

𝛿𝐸²=𝜆2𝐼₀{𝑚²₁𝐼₁⋅𝐼₂+𝐼₅⋅[𝑚²₁(1−𝐼₂)+2𝑚₁𝑚₂𝐼₆+𝑚²₂]+𝑚²₃𝐼₃+𝐼₄[𝑚₁𝑚₃𝐼₆+𝑚₂𝑚₃]}, (4.28) where the integrals 𝐼_𝑘 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 𝑛,𝑚≫(𝑝,𝑞), (𝛼,𝛽)≫1,2 and 𝑘≫𝑙≫1,2. Within this approximation the integrals behave like

𝐼₀=12+12(𝑙²−14)1𝑘²1+𝑂(𝑘³𝐼),₁=12+18(𝑝²−𝑞²)1𝑚²1+𝑂(𝑚³𝐼),₂=18+132(2𝛼²+2𝛽²1−1)𝑛²1+𝑂(𝑛³𝐼),₃=12+18(𝑞²−𝑝²)1𝑚²1+𝑂(𝑚³)𝐼₄=18+132(2𝑝²+2𝑞²1−1)𝑚²1+𝑂(𝑚³𝐼),₅=3+13664(5𝑝²+7𝑞²1−3)𝑚²1+𝑂(𝑚³𝐼),₆1=1+4(𝛽²−𝛼²)1𝑛²1+𝑂(𝑛³). (4.29) Ignoring the terms of high order ((𝑝²/𝑚²)(𝛼²/𝑛²), (𝑞²/𝑚²)(𝛼²/𝑛²), (𝑝²/𝑚²)(𝛽²/𝑛²), (𝑞²/𝑚²)(𝛽²/𝑛²), (1/𝑚²)(1/𝑛²)…) we obtain

𝛿𝐸²≈𝜆2[12+12(𝑙²−14)1𝑘²]×{29𝑚128²¹+3𝑚16²²+12𝑚²₃+38𝑚₁𝑚₂+18𝑚₁𝑚₃+18𝑚₂𝑚₃−[21𝑚512²¹+3𝑚64²²+3𝑚32¹𝑚₂+1𝑚32¹𝑚₃+1𝑚32²𝑚₃]1𝑚²−5𝑚512²¹1𝑛²+[43𝑚512²¹+5𝑚64²²−18𝑚²₃+5𝑚32¹𝑚₂+1𝑚16¹𝑚₃+1𝑚16²𝑚₃]𝑝²𝑚²+[41𝑚512²¹+7𝑚64²²+18𝑚²₃+7𝑚32¹𝑚₂+1𝑚16¹𝑚₃+1𝑚16²𝑚₃]𝑞²𝑚²5+[𝑚256²¹−3𝑚32¹𝑚₂−1𝑚32¹𝑚₃]𝛼²𝑛²5+[𝑚256²¹+3𝑚32¹𝑚₂+1𝑚32¹𝑚₃]𝛽²𝑛²}. (4.30)

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 ℂ𝑃¹.

4.2.1. Pulsating in ℂ𝑃¹

The case of pulsating on ℂ𝑃¹ subspace of the spacetime can be defined as follows. The metric can be written as

𝑑𝑠²_ℂℙ3=𝑑𝜇²+sin²1𝜇[4cos²𝜇(𝛿𝜒+sin²𝜇𝜎₃)²+𝑑𝑠²_ℂ𝑃2], (4.31) where

𝑑𝑠²_ℂ𝑃2=𝑑𝛼²+14sin²𝛼[𝜎²₁+𝜎²_{2𝑑𝑠21ℂ𝑃}+cos²𝛼𝜎²₃], (4.32) and as before 𝜎_1,2,3 are left-invariant 1-form on an 𝑆³. Since 𝛼=|𝑙₁+𝑙₂| and 𝛽=|𝑙₁−𝑙₂| it follows that 𝛼=𝛽=|𝑙₁| and

𝐿²=(𝑛+|𝑙₁|)[(𝑛+|𝑙₁|)+1]=(𝑛+𝛼)[(𝑛+𝛼)+1]∼(𝑛+𝛼)²=(𝑛+|𝑙₁|)². (4.33) The potential in this case becomes

𝜆𝑉=2𝑚²₁sin²𝜆𝜃=2𝑚²₁(1−𝑧²). (4.34) The correction to the energy can be computed using the same technology as before and it gives the expression

𝛿𝐸²_ℂ𝑃1=𝜆2𝑚²₁¹₋₁(1−𝑧²)Ψ^𝛼,𝛼2_𝑛(𝑧)𝑑𝑧. (4.35) To obtain an explicit expression; we have to calculate the following integral:

𝐼₂^𝛼𝛽=¹₋₁(1−𝑧²)Ψ^𝛼,𝛽2_𝑛=(𝑧)𝑑𝑧(𝑛+𝛼+𝛽+1)(𝑛+𝛼+𝛽+2)(𝑛+𝛼+1)(𝑛+𝛽+1)(2𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽+2)²+(2𝑛+𝛼+𝛽+3)𝑛(𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽+1)²(𝑛+𝛼+1−2𝑛+𝛼+𝛽+2𝑛+𝛽)2𝑛+𝛼+𝛽²+𝑛(𝑛−1)(𝑛+𝛼)(𝑛+𝛽)(2𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽)².(2𝑛+𝛼+𝛽−1) (4.36) In (4.35) 𝛼=𝛽=|𝑙₁| and therefore we end up with

𝐼₂^𝛼𝛼=116(𝑛+2𝛼+1)(𝑛+2𝛼+2)+1(𝑛+𝛼+1)(𝑛+𝛼+3)16𝑛(𝑛−1).(𝑛+𝛼+1)(𝑛+𝛼−1) (4.37) On other hand we need the limit of large 𝐿

𝐿∼(𝑛+𝛼)=(𝑛+|𝑙₁|),𝑛=𝐿−𝛼, (4.38) which gives the following approximation of the integral 𝐼₂^𝛼𝛼:

𝐼₂=116(𝐿+𝛼+1)(𝐿+𝛼+2)+1(𝐿+1)(𝐿+3)16𝑛(𝑛−1)∼1(𝐿+1)(𝐿−1)𝛼16(1+𝐿)²+116(𝐿−𝛼)²−𝐿+𝛼𝐿²∼1−18𝛼[1+(𝐿)²+𝛼2𝐿²]. (4.39) Substituting into the expression for the correction to that energy, we find

𝛿𝐸²_ℂ𝑃1=𝜆2𝑚²₁18[1+(|𝑙₁|𝐿)²+|𝑙₁|2𝐿²]. (4.40) We note that in the limit of large 𝐿 we reproduce the of [68].

5. Conclusions

In this section we summarize the results of our study. The purpose of this paper was to look for pulsating string solutions in 𝐴𝑑𝑆₄×ℂℙ³ background. The class of pulsating strings have been used to study AdS/CFT correspondence in the case of 𝐴𝑑𝑆₅×𝑆⁵ [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 𝐴𝑑𝑆₄×ℂℙ³ attracted much attention promising to be exact and with number of applications. Here we consider a generalized string ansatz for pulsating string in the ℂℙ³ 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 ℂℙ¹ 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 𝐴𝑑𝑆₅×𝑆⁵ 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 𝑆⁵. 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 ℂℙ¹ subsector and its embedding in ℂℙ³.

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.

Appendix

Some Integrals Appearing in the Main Text

The integrals appeared in (4.28) are defined as follows:

𝐼₀=116¹₀(1−𝑥²)Ψ^𝑙2_𝑘(𝑥)(1−𝑥²)²𝑑𝑥=1−(𝑘−𝑙+1)(𝑘+𝑙+1)−(2𝑘+1)(2𝑘+3)(𝑘−𝑙)(𝑘+𝑙),(2𝑘+1)(2𝑘−1)𝐼₁=12¹₀𝑦Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2=𝑑𝑦(𝑚+𝑝)(𝑚+𝑝+𝑞)+(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞)(𝑚+1)(𝑚+𝑞+1),(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+2)𝐼₃=12¹₀(1−𝑦)Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2=𝑑𝑦(𝑚+𝑞)(𝑚+𝑝+𝑞)+(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞)(𝑚+1)(𝑚+𝑝+1),(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+2)𝐼₄=12¹₀𝑦(1−𝑦)Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2=𝑑𝑦(𝑚+𝑝+𝑞+1)(𝑚+𝑝+𝑞+2)(𝑚+𝑝+1)(𝑚+𝑞+1)(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+2)²+(2𝑚+𝑝+𝑞+3)𝑚(𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+1)²(𝑚+𝑝+1−2𝑚+𝑝+𝑞+2𝑚+𝑞)2𝑚+𝑝+𝑞²+𝑚(𝑚−1)(𝑚+𝑝)(𝑚+𝑞)(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞)²,(2𝑚+𝑝+𝑞−1) (A.1)(A.2)(A.3)(A.4)

𝐼₅=12¹₀𝑦(1−𝑦²)Ψ^𝑝,𝑞2_𝑚(𝑦)𝑦(1−𝑦)^1/2𝑑𝑦=𝐼₁−[(𝑚+𝑝+𝑞+1)(𝑚+𝑝+𝑞+2)(𝑚+𝑝+𝑞+3)(𝑚+𝑝+3)(𝑚+𝑝+2)(𝑚+𝑝+1)(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+2)²(2𝑚+𝑝+𝑞+3)²(2𝑚+𝑝+𝑞+4)+9(𝑚+𝑝+𝑞+1)(𝑚+𝑝+𝑞+2)(𝑚+𝑞)𝑚(𝑚+𝑝+2)(𝑚+𝑝+1)(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+2)(2𝑚+𝑝+𝑞+3)²(2𝑚+𝑝+𝑞)²+9(𝑚+𝑝+𝑞+1)(𝑚+𝑞−1)(𝑚+𝑞)𝑚(𝑚−1)(𝑚+𝑝+1)(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞+2)²(2𝑚+𝑝+𝑞−1)²+(2𝑚+𝑝+𝑞)(𝑚+𝑞−2)(𝑚+𝑞−1)(𝑚+𝑞)𝑚(𝑚−1)(𝑚−2)(2𝑚+𝑝+𝑞+1)(2𝑚+𝑝+𝑞)²(2𝑚+𝑝+𝑞−1)²(2𝑚+𝑝+𝑞−2)],𝐼₂=¹₋₁(1−𝑧²)Ψ^𝛼,𝛽2_𝑛=(𝑧)𝑑𝑧(𝑛+𝛼+𝛽+1)(𝑛+𝛼+𝛽+2)(𝑛+𝛼+1)(𝑛+𝛽+1)(2𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽+2)²+(2𝑛+𝛼+𝛽+3)𝑛(𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽+1)²(𝑛+𝛼+1−2𝑛+𝛼+𝛽+2𝑛+𝛽)2𝑛+𝛼+𝛽²+𝑛(𝑛−1)(𝑛+𝛼)(𝑛+𝛽)(2𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽)²,(2𝑛+𝛼+𝛽−1)₁₋₁𝑧²Ψ^𝛼,𝛽2_𝑛(𝑧)𝑑𝑧=1−𝐼₂,𝐼₆=¹₋₁𝑧Ψ^𝛼,𝛽2_𝑛=(𝑧)𝑑𝑧2(𝑛+𝛽)(𝑛+𝛼+𝛽)+(2𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽)2(𝑛+1)(𝑛+𝛼+1).(2𝑛+𝛼+𝛽+1)(2𝑛+𝛼+𝛽+2) (A.5)(A.6)(A.7)(A.8)

Acknowledgment

This work was supported in part by the Austrian Research Fund FWF I192, NSFB VU-F-201/06, and DO 02-257.