# On the Pulsating Strings inAdS4×ℂℙ3

Authors: H. Dimov; R. C. Rashkov
Journal: Advances in High Energy Physics
 (2009)

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## Abstract

We study the class of pulsating strings onAdS4×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.


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## Body

## 1. Introduction

The attempts to establish a correspondence between the largeN 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/ℤk. This model, which was first proposed by Aharony, Bergman, Jafferis, and Maldacena [9], is believed to be dual to M-theory on AdS4×S7/ℤk.The ABJM theory actually consists of two Chern-Simons theories of levelk 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/ℤk 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 ofAdS5/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 ofAdS5/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 ofR×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 onAdS4×ℂℙ3

To find the ABJM theory one starts with analysis M2-brane dynamics governed by 11d supergravity action [9](1.1)S=12κ112∫dx11σ-g(R-ϕ12·4!FμνρσFμνρσ)-112κ112∫C(3)⋀F(4)⋀F(4),
where κ112=27π8lp9. Solving for the equations of motions(1.2)Rνμ=12(13!FμαβγFναβγ-13·4!δνμFαβρσFαβρσ),∂σ(-gFσμνξ)=12·(4!)2ϵμνξα1⋯α8Fα1⋯α4Fα5⋯α8,
one can find the M2-brane solutions whose near horizon limit becomes AdS4×S7(1.3)ds2=R24dsAdS42+R2dsS72.
In addition we have N′ units four-form flux(1.4)F(4)=3R38ϵAdS4,R=lp(25N′π2)1/6.
Next we consider the quotient S7/ℤk acting as zi→ei(2π/k)zi. It is convenient first to write the metric on S7 as(1.5)dsS72=(dφ′+ω)2+dsCP32,
where(1.6)dsCP32=∑idzidz̅ir2-|∑izidz̅i|2r4,r2≡∑i=14|zi|2,dφ′+ω≡i2r2(zidz̅i-z̅idzi),dω=J=id(zir)d(z̅ir),
and then to perform the ℤk quotient identifying φ′=φ/k with φ~φ+2π (J is proportional to the Kahler form on ℂℙ3). The resulting metric becomes(1.7)dsS7/ℤk2=1k2(dφ+kω)2+dsCP32.
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 ℤk 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:(1.8)dsstring2=R3k(14dsAdS42+dsℂP32),e2ϕ=R3k3~N1/2k5/2=1N2(Nk)5/2,F4=38R3ϵ4,F2=kdω=kJ.
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 isRstr2=R3/k=25/2πλ. 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 onAdS4×ℂℙ3 in spherical coordinates. The metric on AdS4×ℂℙ3 can be written as [71](1.9)ds2=R2{14[-cosh2ρdt2+dρ2+sinh2ρdΩ22]+dμ2+sin2μ[dα2+14sin2α(σ12+σ22+cos2ασ32)+14cos2μ(dχ+sin2μσ3)2]}.
Here R is the radius of the AdS4, and σ1,2,3 are left-invariant 1-forms on an S3, parameterized by (θ,ϕ,ψ),(1.10)σ1=cosψdθ+sinψsinθdϕ,σ2=sinψdθ-cosψsinθdϕ,σ3=dψ+cosθdϕ.
The range of the coordinates is
(1.11)0≤μ,α≤π2,0≤θ≤π,0≤ϕ≤2π,0≤χ,ψ≤4π.


## 1.1. ABJM and Strings onAdS4×ℂℙ3

To find the ABJM theory one starts with analysis M2-brane dynamics governed by 11d supergravity action [9](1.1)S=12κ112∫dx11σ-g(R-ϕ12·4!FμνρσFμνρσ)-112κ112∫C(3)⋀F(4)⋀F(4),
where κ112=27π8lp9. Solving for the equations of motions(1.2)Rνμ=12(13!FμαβγFναβγ-13·4!δνμFαβρσFαβρσ),∂σ(-gFσμνξ)=12·(4!)2ϵμνξα1⋯α8Fα1⋯α4Fα5⋯α8,
one can find the M2-brane solutions whose near horizon limit becomes AdS4×S7(1.3)ds2=R24dsAdS42+R2dsS72.
In addition we have N′ units four-form flux(1.4)F(4)=3R38ϵAdS4,R=lp(25N′π2)1/6.
Next we consider the quotient S7/ℤk acting as zi→ei(2π/k)zi. It is convenient first to write the metric on S7 as(1.5)dsS72=(dφ′+ω)2+dsCP32,
where(1.6)dsCP32=∑idzidz̅ir2-|∑izidz̅i|2r4,r2≡∑i=14|zi|2,dφ′+ω≡i2r2(zidz̅i-z̅idzi),dω=J=id(zir)d(z̅ir),
and then to perform the ℤk quotient identifying φ′=φ/k with φ~φ+2π (J is proportional to the Kahler form on ℂℙ3). The resulting metric becomes(1.7)dsS7/ℤk2=1k2(dφ+kω)2+dsCP32.
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 ℤk 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:(1.8)dsstring2=R3k(14dsAdS42+dsℂP32),e2ϕ=R3k3~N1/2k5/2=1N2(Nk)5/2,F4=38R3ϵ4,F2=kdω=kJ.
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 isRstr2=R3/k=25/2πλ. 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 onAdS4×ℂℙ3 in spherical coordinates. The metric on AdS4×ℂℙ3 can be written as [71](1.9)ds2=R2{14[-cosh2ρdt2+dρ2+sinh2ρdΩ22]+dμ2+sin2μ[dα2+14sin2α(σ12+σ22+cos2ασ32)+14cos2μ(dχ+sin2μσ3)2]}.
Here R is the radius of the AdS4, and σ1,2,3 are left-invariant 1-forms on an S3, parameterized by (θ,ϕ,ψ),(1.10)σ1=cosψdθ+sinψsinθdϕ,σ2=sinψdθ-cosψsinθdϕ,σ3=dψ+cosθdϕ.
The range of the coordinates is
(1.11)0≤μ,α≤π2,0≤θ≤π,0≤ϕ≤2π,0≤χ,ψ≤4π.


## 2. A Brief Review of Pulsating Strings onR×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(2.1)ds2=R2(cos2θdΩ32+dθ2+sin2θdψ2+dρ2-cosh2dt2),
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(2.2)S=mλ∫dtsinθcosh2ρ-θ̇2.
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])(2.3)H=coshρΠρ2+Πθ2+m2λsin2θ.
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(2.4)coshρsinh3ρddρcoshρsinh3ρddρΨ(ρ,θ)-cosh2ρsinθcos3θddθsinθcos3θddθΨ(ρ,θ)=E2Ψ(ρ,θ).
The solution to the above equation is(2.5)Ψ2n(ρ,θ)=(coshρ)-2n-4P2n(cosθ),
where P2n(cosθ) are spherical harmonics on S5 and the energy is given by(2.6)E2n=Δ=2n+4.
Since we consider highly excited states (large energies), one should take large n, so we can approximate the spherical harmonics as(2.7)P2n(cosθ)≈4πcos(2nθ).
The correction to the energy can be obtained by using perturbation theory, which to first order is(2.8)δE2=∫0π/2dθΨ2n*(0,θ)m2λsin2θΨ2n(θ)=m2λ2.
Up to the first order in λ we find for the anomalous dimension of the corresponding YM operators (see [67] for more details)(2.9)Δ-4=2n[1+12m2λ(2n)2].
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(2.10)S=-mλ∫dtsinθ1-θ̇2-cos2θgijϕ̇iϕ̇j,
where ϕi are S3 angles and gij is the corresponding S3 metric. The Hamiltonian in this case is [68–70](2.11)H=Πθ2+gijΠiΠjcos2θ+m2λsin2θ.
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(2.12)-4ωddωΨ(ω)+J(J+1)ωΨ(ω)=L(L+4)Ψ(ω),
where we set ω=cos2θ. The solution to the Schrodinger equation is(2.13)Ψ(ω)=2(l+1)(l-j)!1ω(ddω)l-jωl+j(1-ω)l-j,j=J2;l=L2.
The first-order correction to the energy δE is(2.14)δE2=m2λ2(l+1)2-(j+1)2-j2(2l+1)(2l+3),
or, up to first order in λ(2.15)E2=L(L+4)+m2λL2-J22L2.
The anomalous dimension then is given by(2.16)γ=m2λ4Lα(2-α),
where α=1-J/L.We conclude this section referring for more details to [67–70].


## 3. Pulsating Strings onAdS4×ℂℙ3

In this section we consider a circular pulsating string expanding and contracting only onℂℙ3 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(3.1)dst×ℂℙ32=R2(-14dt2+dsℂℙ32),R2=32π2λ.
Having in mind the explicit form of the ℂℙ3 metric (1.9), one can write it as(3.2)dsℂℙ32=gijdxidxj+ĝpqdypdyq,
where the part gij is defined as(3.3)gij=diag(1,sin2μ,14sin2μsin2α),i,j=1,2,3,x1=μ,x2=α,x3=θ,
while ĝpq=ĝpq(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:(3.4)x1=x1(τ)=μ(τ),x2=x2(τ)=α(τ),x3=x3(τ)=θ(τ),y1=ϕ=m1σ+h1(τ),y2=ψ=m2σ+h2(τ),y3=χ=m3σ+h3(τ).
We are interested in the induced worldsheet metric which in our case has the form(3.5)dsws2=R2{(-14+gijẋiẋj+ĝpqḣpḣq)dτ2+(ĝpqmpmq)dσ2+2(ĝpqmpḣq)dτdσ}.
The Nambu-Goto action is(3.6)SNG=-T∫dτdσ-det(Gμν∂αXμ∂βXν),
in this ansatz then reduces to expression(3.7)SNG=-TR2∫dτdσ(14-gijẋiẋj-ĝpqḣpḣq)(ĝpqmpmq)+(ĝpqmpḣq)(ĝpqmqḣp),
where TR2=22λ, 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(3.8)Πi=22λ(ĝpqmpmq)gijẋj(1/4-gijẋiẋj-ĝpqḣpḣq)(ĝpqmpmq)+(ĝpqmpḣq)(ĝpqmqḣp),i=1,2,3,Π̂p=22λ(ĝpqmpmq)ĝpqḣq-(ĝpqmpḣq)ĝpqmq(1/4-gijẋiẋj-ĝpqḣpḣq)(ĝpqmpmq)+(ĝpqmpḣq)(ĝpqmqḣp),p=1,2,3.
Solving for the derivatives in terms of the canonical momenta and substituting back into the Legendre transform of the Lagrangian, we find the Hamiltonian(3.9)H2=14(gijΠiΠj+ĝpqΠ̂pΠ̂q)+(22λ)2(ĝpqmpmq)4.
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(3.10)V(μ,α,θ)=2λĝpq(μ,α,θ)mpmq.
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(3.11)V(μ,α,θ)=λ2{sin2μsin2α[sin2θ+cos2αcos2θ(1+sin2α)]m12+sin2μsin2αcos2α(1+sin2α)m22+sin2μcos2αm32+2sin2μsin2αcos2αcosθ(1+sin2α)m1m2+2sin2μsin2αcos2αcosθm1m3+2sin2μsin2αcos2αm2m3}.
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 onℂℙ3. 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 onℂℙ3 and then to obtain the correction to the energy due to the induced potential.


### 4.1. Theℂℙ3 Metric and the Associated Laplace-Beltrami Operator

We start with the Laplace-Beltrami operator in coordinates parameterizingℂℙ3 as follows. First we define coordinates(4.1)y1+iy2=μsinαcosθ2e(i/2)(ϕ+ψ+χ),y3+iy4=μsinαsinθ2e(i/2)(ϕ-ψ-χ),y5+iy6=μcosαe(i/2)χ.
The line element of ℂℙ3 in these coordinates is explicitly given by(4.2)dsℂℙ32=dμ2+sin2μdα2+14sin2μsin2αdθ2+14sin2μsin2α[sin2θ+cos2αcos2θ(1+sin2α)]dϕ2+14sin2μsin2αcos2α(1+sin2α)dψ2+14sin2μcos2αdχ2+24sin2μsin2αcos2αcosθ(1+sin2α)dψdϕ+24sin2μsin2αcos2αcosθdχdϕ+24sin2μsin2αcos2αdψdχ.


#### 4.1.1. The Laplace-Beltrami Operator onℂℙ3

Using the standard definition of the Laplace-Beltrami operator in global coordinates we find(4.3)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ[sin5μ∂∂μ]+1sin2μsin3αcos2α∂∂α[sin3αcos2α∂∂α]+4sin2μsin2αsinθ∂∂θ[sinθ∂∂θ]+4sin2μsin2αsin2θ∂2∂ϕ2-8cosθsin2μsin2αsin2θ∂2∂ϕ∂ψ+4(sin2θ+cos2αcos2θ)sin2μsin2αsin2θcos2α∂2∂ψ2-8sin2μcos2α∂2∂χ∂ψ+4(1+sin2α)sin2μcos2α∂2∂χ2.
Since we are going to separate the variables, it is useful to arrange the terms as follows:(4.4)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ(sin5μ∂∂μ)+1sin2μ{1sin3αcos2α∂∂α(sin3αcos2α∂∂α)+4sin2αcos2α(∂∂ψ-sin2α∂∂χ)2+4cos2α∂2∂χ2+4sin2α[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ(∂∂ϕ-cosθ∂∂ψ)2]}.
Then the operator Δ(μ,α,θ,ϕ,ψ,χ) can be written as(4.5)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ(sin5μ∂∂μ)+1sin2μΔ(α,θ,ϕ,ψ,χ),
where(4.6)Δ(α,θ,ϕ,ψ,χ)=1sin3αcos2α∂∂α(sin3αcos2α∂∂α)+4sin2αcos2α(∂∂ψ-sin2α∂∂χ)2+4cos2α∂2∂χ2+4sin2αΔ(θ,ϕ,ψ),Δ(θ,ϕ,ψ)=1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ(∂∂ϕ-cosθ∂∂ψ)2.
The full measure on ℂℙ3 is(4.7)dΩ(μ,α,θ)=116sin5μsin3αcos2αsinθdμdαdθ.


#### 4.1.2. The Wave Function

The Schrödinger equation for the wave function is(4.8)Δ(μ,α,θ,ϕ,ψ,χ)Ψ(μ,α,θ,ϕ,ψ,χ)=-4E2Ψ(μ,α,θ,ϕ,ψ,χ).
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(4.9)f4(ϕ,ψ,χ)=exp(il1ϕ)exp(il2ψ)exp(il3χ),l1,l2,l3∈ℤ.
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(4.10)[Δ(θ,ϕ,ψ)+L2]f3(θ)≡{1sinθddθ(sinθddθ)-1sin2θ(l1-cosθl2)2+L2}f3(θ)=0.
It is convenient to define a new variable z=cosθ in which the equation can be written as(4.11){(1-z2)d2dz2-2zddz-11-z2(l1-zl2)2+L2}f3(z)=0.
The solution to this equation is(4.12)f3(z)=(1-z)|l1-l2|/2(1+z)|l1+l2|/2×F21[12(|l1-l2|+|l1+l2|+1-1+4(l22+L2)),12(|l1-l2|+|l1+l2|+1+1+4(l22+L2));1+|l1+l2|,1+z2].
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:(4.13)1+4(l22+L2)-|l1-l2|-|l1+l2|-1=2n,n∈ℕ.
Introducing new parameters α=|l1+l2| and β=|l1-l2| the solution can be written in the form(4.14)f3(z)=(1-z)α/2(1+z)β/2n!Γ(α+1)Γ(α+1+n)Pn(α,β)(z).
The equation for f2(α) can be treated analogously and has the form
(4.15)[Δ(α,θ,ϕ,ψ,χ)+M2]f2(α)≡{1sin3αcos2αddα(sin3αcos2αddα)-4sin2αcos2α(l2-sin2αl3)2-4cos2αl32-4sin2αL2+M2}f2(α)=0.
With the help of the new variable y=sin2α, the above equation can be written as(4.16){4y(1-y)d2dy2+2(4-7y)ddy-4y(1-y)(l2-yl3)2-41-yl32-4yL2+M2}f2(y)=0.
Recognizing the hypergeometric equation mentioned previously we find the solution as(4.17)f2(y)=y(1+4(L2+l22)-1)/2(1-y)2((l2-l3)2+l32)×F21[12(32+1+4(L2+l22)+4((l2-l3)2+l32)-25+16(M2+4l32)2),12(32+1+4(L2+l22)+4((l2-l3)2+l32)+25+16(M2+4l32)2);1+1+4(L2+l22),y25+16(M2+4l32)2].In addition we have to ensure that the solutionf2(α) is square integrable with respect to the measure in α. This requirement imposes the following restriction on the parameters:(4.18)25+16(M2+4l32)2-32-1+4(L2+l22)-4((l2-l3)2+l32)=2m,m∈ℕ.
It is useful to introduce new parameters p=1+4(L2+l22) and q=1/2+4((l2-l3)2+l32), in terms of which the solution can be written in the form(4.19)f2(y)=y(p-1)/2(1-y)(q-1/2)/2m!Γ(p+1)Γ(p+1+m)Pm(p,q)(1-2y).
Then, the equation for f1(μ) we have to solve becomes(4.20){1sin5μddμ(sin5μddμ)-M2sin2μ+4E2}f1(μ)=0.
We change the variable defining x=cosμ(4.21){(1-x2)d2dx2-6xddx-11-x2M2+4E2}f1(x)=0.
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(4.22)f1(x)=(1-x2)-1Pkl(x).
In terms of the new variables the measure (4.7) becomes(4.23)dΩ(x,y,z)=116(1-x2)2y(1-y)1/22dxdydz,0≤x,y≤1,-1≤z≤1.
The normalized wave functions of (4.14), (4.19), (4.22) have a form(4.24)Ψnα,β(z)=(2n+α+β+1)n!Γ(n+α+β+1)2α+β+1Γ(n+α+1)Γ(n+β+1)(1-z)α/2(1+z)β/2Pn(α,β)(z),Ψmp,q(y)=2(2m+p+q+1)m!Γ(m+p+q+1)Γ(m+p+1)Γ(m+q+1)y(p-1)/2(1-y)(q-1/2)/2Pm(p,q)(1-2y),Ψkl(x)=4(2k+1)(k-l)!(k+l)!(1-x2)-1Pkl(x).


### 4.2. Corrections to the Energy

There is potential (3.11) in terms x,y,z(4.25)V(x,y,z)=λ2(1-x2){m12y(1-z2)+y(1-y2)[m12z2+2m1m2z+m22]+m32(1-y)+2y(1-y)[m1m3z+m2m3]}.
There is correction to the energy(4.26)δE2=∫01∫01∫-11dΩ(x,y,z)V(x,y,z)Ψkl2(x)Ψmp,q2(y)Ψnα,β2(z).The explicit form of the previous formula using the explicit form of the various wave functions is(4.27)δE2=λ2116∫01(1-x2)Ψkl2(x)(1-x2)2dx×{m1212∫01yΨmp,q2(y)y(1-y)1/2dy∫-11(1-z2)Ψnα,β2(z)dz+12∫01y(1-y2)Ψmp,q2(y)y(1-y)1/2dy×[m12∫-11z2Ψnα,β2(z)dz+2m1m2∫-11zΨnα,β2(z)dz+m22]+m3212∫01(1-y)Ψmp,q2(y)y(1-y)1/2dy+12∫01y(1-y)Ψmp,q2(y)y(1-y)1/2dy×[m1m3∫-11zΨnα,β2(z)dz+m2m3]}.In short notations, it looks as(4.28)δE2=λ2I0{m12I1·I2+I5·[m12(1-I2)+2m1m2I6+m22]+m32I3+I4[m1m3I6+m2m3]},
where the integrals Ik 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(4.29)I0=12+12(l2-14)1k2+O(1k3),I1=12+18(p2-q2)1m2+O(1m3),I2=18+132(2α2+2β2-1)1n2+O(1n3),I3=12+18(q2-p2)1m2+O(1m3)I4=18+132(2p2+2q2-1)1m2+O(1m3),I5=336+164(5p2+7q2-3)1m2+O(1m3),I6=1+14(β2-α2)1n2+O(1n3).
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(4.30)δE2≈λ2[12+12(l2-14)1k2]×{29128m12+316m22+12m32+38m1m2+18m1m3+18m2m3-[21512m12+364m22+332m1m2+132m1m3+132m2m3]1m2-5512m121n2+[43512m12+564m22-18m32+532m1m2+116m1m3+116m2m3]p2m2+[41512m12+764m22+18m32+732m1m2+116m1m3+116m2m3]q2m2+[5256m12-332m1m2-132m1m3]α2n2+[5256m12+332m1m2+132m1m3]β2n2}.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 onℂP1.


#### 4.2.1. Pulsating inℂP1

The case of pulsating onℂP1 subspace of the spacetime can be defined as follows. The metric can be written as(4.31)dsℂℙ32=dμ2+sin2μ[14cos2μ(δχ+sin2μσ3)2+dsℂP22],
where(4.32)dsℂP22=dα2+14sin2α[σ12+σ22︸dsℂP12+cos2ασ32],
and as before σ1,2,3 are left-invariant 1-form on an S3. Since α=|l1+l2| and β=|l1-l2| it follows that α=β=|l1| and(4.33)L2=(n+|l1|)[(n+|l1|)+1]=(n+α)[(n+α)+1]~(n+α)2=(n+|l1|)2.
The potential in this case becomes(4.34)V=λ2m12sin2θ=λ2m12(1-z2).
The correction to the energy can be computed using the same technology as before and it gives the expression(4.35)δEℂP12=λ2m12∫-11(1-z2)Ψnα,α2(z)dz.
To obtain an explicit expression; we have to calculate the following integral:(4.36)I2αβ=∫-11(1-z2)Ψnα,β2(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+12n+α+β+2-n+β2n+α+β)2+n(n-1)(n+α)(n+β)(2n+α+β+1)(2n+α+β)2(2n+α+β-1).
In (4.35) α=β=|l1| and therefore we end up with(4.37)I2αα=116(n+2α+1)(n+2α+2)(n+α+1)(n+α+3)+116n(n-1)(n+α+1)(n+α-1).
On other hand we need the limit of large L(4.38)L~(n+α)=(n+|l1|),n=L-α,
which gives the following approximation of the integral I2αα:(4.39)I2=116(L+α+1)(L+α+2)(L+1)(L+3)+116n(n-1)(L+1)(L-1)~116(1+αL)2+116(L-α)2-L+αL2-1~18[1+(αL)2+α2L2].
Substituting into the expression for the correction to that energy, we find(4.40)δEℂP12=λ2m1218[1+(|l1|L)2+|l1|2L2].
We note that in the limit of large L we reproduce the of [68].


## 4.1. Theℂℙ3 Metric and the Associated Laplace-Beltrami Operator

We start with the Laplace-Beltrami operator in coordinates parameterizingℂℙ3 as follows. First we define coordinates(4.1)y1+iy2=μsinαcosθ2e(i/2)(ϕ+ψ+χ),y3+iy4=μsinαsinθ2e(i/2)(ϕ-ψ-χ),y5+iy6=μcosαe(i/2)χ.
The line element of ℂℙ3 in these coordinates is explicitly given by(4.2)dsℂℙ32=dμ2+sin2μdα2+14sin2μsin2αdθ2+14sin2μsin2α[sin2θ+cos2αcos2θ(1+sin2α)]dϕ2+14sin2μsin2αcos2α(1+sin2α)dψ2+14sin2μcos2αdχ2+24sin2μsin2αcos2αcosθ(1+sin2α)dψdϕ+24sin2μsin2αcos2αcosθdχdϕ+24sin2μsin2αcos2αdψdχ.


### 4.1.1. The Laplace-Beltrami Operator onℂℙ3

Using the standard definition of the Laplace-Beltrami operator in global coordinates we find(4.3)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ[sin5μ∂∂μ]+1sin2μsin3αcos2α∂∂α[sin3αcos2α∂∂α]+4sin2μsin2αsinθ∂∂θ[sinθ∂∂θ]+4sin2μsin2αsin2θ∂2∂ϕ2-8cosθsin2μsin2αsin2θ∂2∂ϕ∂ψ+4(sin2θ+cos2αcos2θ)sin2μsin2αsin2θcos2α∂2∂ψ2-8sin2μcos2α∂2∂χ∂ψ+4(1+sin2α)sin2μcos2α∂2∂χ2.
Since we are going to separate the variables, it is useful to arrange the terms as follows:(4.4)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ(sin5μ∂∂μ)+1sin2μ{1sin3αcos2α∂∂α(sin3αcos2α∂∂α)+4sin2αcos2α(∂∂ψ-sin2α∂∂χ)2+4cos2α∂2∂χ2+4sin2α[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ(∂∂ϕ-cosθ∂∂ψ)2]}.
Then the operator Δ(μ,α,θ,ϕ,ψ,χ) can be written as(4.5)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ(sin5μ∂∂μ)+1sin2μΔ(α,θ,ϕ,ψ,χ),
where(4.6)Δ(α,θ,ϕ,ψ,χ)=1sin3αcos2α∂∂α(sin3αcos2α∂∂α)+4sin2αcos2α(∂∂ψ-sin2α∂∂χ)2+4cos2α∂2∂χ2+4sin2αΔ(θ,ϕ,ψ),Δ(θ,ϕ,ψ)=1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ(∂∂ϕ-cosθ∂∂ψ)2.
The full measure on ℂℙ3 is(4.7)dΩ(μ,α,θ)=116sin5μsin3αcos2αsinθdμdαdθ.


### 4.1.2. The Wave Function

The Schrödinger equation for the wave function is(4.8)Δ(μ,α,θ,ϕ,ψ,χ)Ψ(μ,α,θ,ϕ,ψ,χ)=-4E2Ψ(μ,α,θ,ϕ,ψ,χ).
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(4.9)f4(ϕ,ψ,χ)=exp(il1ϕ)exp(il2ψ)exp(il3χ),l1,l2,l3∈ℤ.
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(4.10)[Δ(θ,ϕ,ψ)+L2]f3(θ)≡{1sinθddθ(sinθddθ)-1sin2θ(l1-cosθl2)2+L2}f3(θ)=0.
It is convenient to define a new variable z=cosθ in which the equation can be written as(4.11){(1-z2)d2dz2-2zddz-11-z2(l1-zl2)2+L2}f3(z)=0.
The solution to this equation is(4.12)f3(z)=(1-z)|l1-l2|/2(1+z)|l1+l2|/2×F21[12(|l1-l2|+|l1+l2|+1-1+4(l22+L2)),12(|l1-l2|+|l1+l2|+1+1+4(l22+L2));1+|l1+l2|,1+z2].
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:(4.13)1+4(l22+L2)-|l1-l2|-|l1+l2|-1=2n,n∈ℕ.
Introducing new parameters α=|l1+l2| and β=|l1-l2| the solution can be written in the form(4.14)f3(z)=(1-z)α/2(1+z)β/2n!Γ(α+1)Γ(α+1+n)Pn(α,β)(z).
The equation for f2(α) can be treated analogously and has the form
(4.15)[Δ(α,θ,ϕ,ψ,χ)+M2]f2(α)≡{1sin3αcos2αddα(sin3αcos2αddα)-4sin2αcos2α(l2-sin2αl3)2-4cos2αl32-4sin2αL2+M2}f2(α)=0.
With the help of the new variable y=sin2α, the above equation can be written as(4.16){4y(1-y)d2dy2+2(4-7y)ddy-4y(1-y)(l2-yl3)2-41-yl32-4yL2+M2}f2(y)=0.
Recognizing the hypergeometric equation mentioned previously we find the solution as(4.17)f2(y)=y(1+4(L2+l22)-1)/2(1-y)2((l2-l3)2+l32)×F21[12(32+1+4(L2+l22)+4((l2-l3)2+l32)-25+16(M2+4l32)2),12(32+1+4(L2+l22)+4((l2-l3)2+l32)+25+16(M2+4l32)2);1+1+4(L2+l22),y25+16(M2+4l32)2].In addition we have to ensure that the solutionf2(α) is square integrable with respect to the measure in α. This requirement imposes the following restriction on the parameters:(4.18)25+16(M2+4l32)2-32-1+4(L2+l22)-4((l2-l3)2+l32)=2m,m∈ℕ.
It is useful to introduce new parameters p=1+4(L2+l22) and q=1/2+4((l2-l3)2+l32), in terms of which the solution can be written in the form(4.19)f2(y)=y(p-1)/2(1-y)(q-1/2)/2m!Γ(p+1)Γ(p+1+m)Pm(p,q)(1-2y).
Then, the equation for f1(μ) we have to solve becomes(4.20){1sin5μddμ(sin5μddμ)-M2sin2μ+4E2}f1(μ)=0.
We change the variable defining x=cosμ(4.21){(1-x2)d2dx2-6xddx-11-x2M2+4E2}f1(x)=0.
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(4.22)f1(x)=(1-x2)-1Pkl(x).
In terms of the new variables the measure (4.7) becomes(4.23)dΩ(x,y,z)=116(1-x2)2y(1-y)1/22dxdydz,0≤x,y≤1,-1≤z≤1.
The normalized wave functions of (4.14), (4.19), (4.22) have a form(4.24)Ψnα,β(z)=(2n+α+β+1)n!Γ(n+α+β+1)2α+β+1Γ(n+α+1)Γ(n+β+1)(1-z)α/2(1+z)β/2Pn(α,β)(z),Ψmp,q(y)=2(2m+p+q+1)m!Γ(m+p+q+1)Γ(m+p+1)Γ(m+q+1)y(p-1)/2(1-y)(q-1/2)/2Pm(p,q)(1-2y),Ψkl(x)=4(2k+1)(k-l)!(k+l)!(1-x2)-1Pkl(x).


## 4.1.1. The Laplace-Beltrami Operator onℂℙ3

Using the standard definition of the Laplace-Beltrami operator in global coordinates we find(4.3)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ[sin5μ∂∂μ]+1sin2μsin3αcos2α∂∂α[sin3αcos2α∂∂α]+4sin2μsin2αsinθ∂∂θ[sinθ∂∂θ]+4sin2μsin2αsin2θ∂2∂ϕ2-8cosθsin2μsin2αsin2θ∂2∂ϕ∂ψ+4(sin2θ+cos2αcos2θ)sin2μsin2αsin2θcos2α∂2∂ψ2-8sin2μcos2α∂2∂χ∂ψ+4(1+sin2α)sin2μcos2α∂2∂χ2.
Since we are going to separate the variables, it is useful to arrange the terms as follows:(4.4)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ(sin5μ∂∂μ)+1sin2μ{1sin3αcos2α∂∂α(sin3αcos2α∂∂α)+4sin2αcos2α(∂∂ψ-sin2α∂∂χ)2+4cos2α∂2∂χ2+4sin2α[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ(∂∂ϕ-cosθ∂∂ψ)2]}.
Then the operator Δ(μ,α,θ,ϕ,ψ,χ) can be written as(4.5)Δ(μ,α,θ,ϕ,ψ,χ)=1sin5μ∂∂μ(sin5μ∂∂μ)+1sin2μΔ(α,θ,ϕ,ψ,χ),
where(4.6)Δ(α,θ,ϕ,ψ,χ)=1sin3αcos2α∂∂α(sin3αcos2α∂∂α)+4sin2αcos2α(∂∂ψ-sin2α∂∂χ)2+4cos2α∂2∂χ2+4sin2αΔ(θ,ϕ,ψ),Δ(θ,ϕ,ψ)=1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ(∂∂ϕ-cosθ∂∂ψ)2.
The full measure on ℂℙ3 is(4.7)dΩ(μ,α,θ)=116sin5μsin3αcos2αsinθdμdαdθ.


## 4.1.2. The Wave Function

The Schrödinger equation for the wave function is(4.8)Δ(μ,α,θ,ϕ,ψ,χ)Ψ(μ,α,θ,ϕ,ψ,χ)=-4E2Ψ(μ,α,θ,ϕ,ψ,χ).
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(4.9)f4(ϕ,ψ,χ)=exp(il1ϕ)exp(il2ψ)exp(il3χ),l1,l2,l3∈ℤ.
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(4.10)[Δ(θ,ϕ,ψ)+L2]f3(θ)≡{1sinθddθ(sinθddθ)-1sin2θ(l1-cosθl2)2+L2}f3(θ)=0.
It is convenient to define a new variable z=cosθ in which the equation can be written as(4.11){(1-z2)d2dz2-2zddz-11-z2(l1-zl2)2+L2}f3(z)=0.
The solution to this equation is(4.12)f3(z)=(1-z)|l1-l2|/2(1+z)|l1+l2|/2×F21[12(|l1-l2|+|l1+l2|+1-1+4(l22+L2)),12(|l1-l2|+|l1+l2|+1+1+4(l22+L2));1+|l1+l2|,1+z2].
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:(4.13)1+4(l22+L2)-|l1-l2|-|l1+l2|-1=2n,n∈ℕ.
Introducing new parameters α=|l1+l2| and β=|l1-l2| the solution can be written in the form(4.14)f3(z)=(1-z)α/2(1+z)β/2n!Γ(α+1)Γ(α+1+n)Pn(α,β)(z).
The equation for f2(α) can be treated analogously and has the form
(4.15)[Δ(α,θ,ϕ,ψ,χ)+M2]f2(α)≡{1sin3αcos2αddα(sin3αcos2αddα)-4sin2αcos2α(l2-sin2αl3)2-4cos2αl32-4sin2αL2+M2}f2(α)=0.
With the help of the new variable y=sin2α, the above equation can be written as(4.16){4y(1-y)d2dy2+2(4-7y)ddy-4y(1-y)(l2-yl3)2-41-yl32-4yL2+M2}f2(y)=0.
Recognizing the hypergeometric equation mentioned previously we find the solution as(4.17)f2(y)=y(1+4(L2+l22)-1)/2(1-y)2((l2-l3)2+l32)×F21[12(32+1+4(L2+l22)+4((l2-l3)2+l32)-25+16(M2+4l32)2),12(32+1+4(L2+l22)+4((l2-l3)2+l32)+25+16(M2+4l32)2);1+1+4(L2+l22),y25+16(M2+4l32)2].In addition we have to ensure that the solutionf2(α) is square integrable with respect to the measure in α. This requirement imposes the following restriction on the parameters:(4.18)25+16(M2+4l32)2-32-1+4(L2+l22)-4((l2-l3)2+l32)=2m,m∈ℕ.
It is useful to introduce new parameters p=1+4(L2+l22) and q=1/2+4((l2-l3)2+l32), in terms of which the solution can be written in the form(4.19)f2(y)=y(p-1)/2(1-y)(q-1/2)/2m!Γ(p+1)Γ(p+1+m)Pm(p,q)(1-2y).
Then, the equation for f1(μ) we have to solve becomes(4.20){1sin5μddμ(sin5μddμ)-M2sin2μ+4E2}f1(μ)=0.
We change the variable defining x=cosμ(4.21){(1-x2)d2dx2-6xddx-11-x2M2+4E2}f1(x)=0.
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(4.22)f1(x)=(1-x2)-1Pkl(x).
In terms of the new variables the measure (4.7) becomes(4.23)dΩ(x,y,z)=116(1-x2)2y(1-y)1/22dxdydz,0≤x,y≤1,-1≤z≤1.
The normalized wave functions of (4.14), (4.19), (4.22) have a form(4.24)Ψnα,β(z)=(2n+α+β+1)n!Γ(n+α+β+1)2α+β+1Γ(n+α+1)Γ(n+β+1)(1-z)α/2(1+z)β/2Pn(α,β)(z),Ψmp,q(y)=2(2m+p+q+1)m!Γ(m+p+q+1)Γ(m+p+1)Γ(m+q+1)y(p-1)/2(1-y)(q-1/2)/2Pm(p,q)(1-2y),Ψkl(x)=4(2k+1)(k-l)!(k+l)!(1-x2)-1Pkl(x).


## 4.2. Corrections to the Energy

There is potential (3.11) in terms x,y,z(4.25)V(x,y,z)=λ2(1-x2){m12y(1-z2)+y(1-y2)[m12z2+2m1m2z+m22]+m32(1-y)+2y(1-y)[m1m3z+m2m3]}.
There is correction to the energy(4.26)δE2=∫01∫01∫-11dΩ(x,y,z)V(x,y,z)Ψkl2(x)Ψmp,q2(y)Ψnα,β2(z).The explicit form of the previous formula using the explicit form of the various wave functions is(4.27)δE2=λ2116∫01(1-x2)Ψkl2(x)(1-x2)2dx×{m1212∫01yΨmp,q2(y)y(1-y)1/2dy∫-11(1-z2)Ψnα,β2(z)dz+12∫01y(1-y2)Ψmp,q2(y)y(1-y)1/2dy×[m12∫-11z2Ψnα,β2(z)dz+2m1m2∫-11zΨnα,β2(z)dz+m22]+m3212∫01(1-y)Ψmp,q2(y)y(1-y)1/2dy+12∫01y(1-y)Ψmp,q2(y)y(1-y)1/2dy×[m1m3∫-11zΨnα,β2(z)dz+m2m3]}.In short notations, it looks as(4.28)δE2=λ2I0{m12I1·I2+I5·[m12(1-I2)+2m1m2I6+m22]+m32I3+I4[m1m3I6+m2m3]},
where the integrals Ik 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(4.29)I0=12+12(l2-14)1k2+O(1k3),I1=12+18(p2-q2)1m2+O(1m3),I2=18+132(2α2+2β2-1)1n2+O(1n3),I3=12+18(q2-p2)1m2+O(1m3)I4=18+132(2p2+2q2-1)1m2+O(1m3),I5=336+164(5p2+7q2-3)1m2+O(1m3),I6=1+14(β2-α2)1n2+O(1n3).
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(4.30)δE2≈λ2[12+12(l2-14)1k2]×{29128m12+316m22+12m32+38m1m2+18m1m3+18m2m3-[21512m12+364m22+332m1m2+132m1m3+132m2m3]1m2-5512m121n2+[43512m12+564m22-18m32+532m1m2+116m1m3+116m2m3]p2m2+[41512m12+764m22+18m32+732m1m2+116m1m3+116m2m3]q2m2+[5256m12-332m1m2-132m1m3]α2n2+[5256m12+332m1m2+132m1m3]β2n2}.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 onℂP1.


### 4.2.1. Pulsating inℂP1

The case of pulsating onℂP1 subspace of the spacetime can be defined as follows. The metric can be written as(4.31)dsℂℙ32=dμ2+sin2μ[14cos2μ(δχ+sin2μσ3)2+dsℂP22],
where(4.32)dsℂP22=dα2+14sin2α[σ12+σ22︸dsℂP12+cos2ασ32],
and as before σ1,2,3 are left-invariant 1-form on an S3. Since α=|l1+l2| and β=|l1-l2| it follows that α=β=|l1| and(4.33)L2=(n+|l1|)[(n+|l1|)+1]=(n+α)[(n+α)+1]~(n+α)2=(n+|l1|)2.
The potential in this case becomes(4.34)V=λ2m12sin2θ=λ2m12(1-z2).
The correction to the energy can be computed using the same technology as before and it gives the expression(4.35)δEℂP12=λ2m12∫-11(1-z2)Ψnα,α2(z)dz.
To obtain an explicit expression; we have to calculate the following integral:(4.36)I2αβ=∫-11(1-z2)Ψnα,β2(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+12n+α+β+2-n+β2n+α+β)2+n(n-1)(n+α)(n+β)(2n+α+β+1)(2n+α+β)2(2n+α+β-1).
In (4.35) α=β=|l1| and therefore we end up with(4.37)I2αα=116(n+2α+1)(n+2α+2)(n+α+1)(n+α+3)+116n(n-1)(n+α+1)(n+α-1).
On other hand we need the limit of large L(4.38)L~(n+α)=(n+|l1|),n=L-α,
which gives the following approximation of the integral I2αα:(4.39)I2=116(L+α+1)(L+α+2)(L+1)(L+3)+116n(n-1)(L+1)(L-1)~116(1+αL)2+116(L-α)2-L+αL2-1~18[1+(αL)2+α2L2].
Substituting into the expression for the correction to that energy, we find(4.40)δEℂP12=λ2m1218[1+(|l1|L)2+|l1|2L2].
We note that in the limit of large L we reproduce the of [68].


## 4.2.1. Pulsating inℂP1

The case of pulsating onℂP1 subspace of the spacetime can be defined as follows. The metric can be written as(4.31)dsℂℙ32=dμ2+sin2μ[14cos2μ(δχ+sin2μσ3)2+dsℂP22],
where(4.32)dsℂP22=dα2+14sin2α[σ12+σ22︸dsℂP12+cos2ασ32],
and as before σ1,2,3 are left-invariant 1-form on an S3. Since α=|l1+l2| and β=|l1-l2| it follows that α=β=|l1| and(4.33)L2=(n+|l1|)[(n+|l1|)+1]=(n+α)[(n+α)+1]~(n+α)2=(n+|l1|)2.
The potential in this case becomes(4.34)V=λ2m12sin2θ=λ2m12(1-z2).
The correction to the energy can be computed using the same technology as before and it gives the expression(4.35)δEℂP12=λ2m12∫-11(1-z2)Ψnα,α2(z)dz.
To obtain an explicit expression; we have to calculate the following integral:(4.36)I2αβ=∫-11(1-z2)Ψnα,β2(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+12n+α+β+2-n+β2n+α+β)2+n(n-1)(n+α)(n+β)(2n+α+β+1)(2n+α+β)2(2n+α+β-1).
In (4.35) α=β=|l1| and therefore we end up with(4.37)I2αα=116(n+2α+1)(n+2α+2)(n+α+1)(n+α+3)+116n(n-1)(n+α+1)(n+α-1).
On other hand we need the limit of large L(4.38)L~(n+α)=(n+|l1|),n=L-α,
which gives the following approximation of the integral I2αα:(4.39)I2=116(L+α+1)(L+α+2)(L+1)(L+3)+116n(n-1)(L+1)(L-1)~116(1+αL)2+116(L-α)2-L+αL2-1~18[1+(αL)2+α2L2].
Substituting into the expression for the correction to that energy, we find(4.40)δEℂP12=λ2m1218[1+(|l1|L)2+|l1|2L2].
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 inAdS4×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 inAdS4×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 theℂℙ1 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 fromAdS/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.



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*Source: 953987-2009-12-01.xml*