Semiclassical quantization of Rotating Strings in Pilch-Warner geometry

arXiv:hep-th/0304035v2 16 Apr 2003

Semiclassical quantization of Rotating Strings in Pilch-Warner geometry H. Dimov‡ , V. Filev† , R.C. Rashkov†1 and K.S. Viswanathan♭2 † ‡

Department of Physics, Sofia University, 1164 Sofia, Bulgaria

Department of Mathematics, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgaria ♭

Department of Physics, Simon Fraser University Burnaby, BC, V5A 1S6, Canada

Abstract Some of the recent important developments in understanding string/ gauge dualities are based on the idea of highly symmetric motion of “string solitons” in AdS5 × S 5 geometry originally suggested by Gubser, Klebanov and Polyakov. In this paper we study symmetric motion of certain string configurations in so called Pilch-Warner geometry. The two-form field A2 breaks down the supersymmetry to N = 1 but for the string configurations considered in this paper the classical values of the energy and the spin are the same as for string in AdS × S 5 . Although trivial at classical level, the presence of NS-NS antisymmetric field couples the fluctuation modes that indicates changes in the quantum corrections to the energy spectrum. We compare our results with those obtained in the case of pp-wave limit in [32, 31].

1 2

e-mail: [email protected] e-mail: [email protected]

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1

Introduction

The recent research efforts and developments in string theory in the last years were focused on the understanding of string/gauge duality and especially AdS/CFT correspondence. AdS/CFT correspondence is based on the conjecture that type II B string theory in AdS5 × S 5 background with large number of fluxes turned on S 5 is dual to four dimensional N = 4 supersymmetric Yang-Mills theory (SYM). Actually this is the best studied case where the spectrum of a state of string theory on AdS5 × S 5 corresponds to the spectrum of single trace operators in the gauge theory. The great interest in AdS/CFT correspondence is inspired by the simple reason that it can be used to make predictions about N = 4 SYM at strong coupling. Until recently this conjecture has been mostly investigated in the so called zero mode approximation. In this case it was found that the supergravity modes on AdS5 ×S 5 are in one to one correspondence with the chiral operators in the gauge theory. However, this restriction means that the radius of the curvature is large and all α′ R corrections are negligible (R is the radius of AdS). Although important, these considerations are not enough to verify the correspondence in its full extent and to extract useful information about the gauge theory at strong coupling. It is important therefore to go beyond the supergravity limit and to consider at least α′ R correction. The recent progress in string propagation in pp-wave background attracted much interest [1, 2, 3]. This background possesses at least two very attractive features: it is maximally supersymmetric and at the tree level Green-Schwarz superstring is exactly solvable [4, 5]. Since this background can be obtained by taking the Penrose limit [6] of AdS5 × S 5 geometry, the next natural step then is to ask about AdS/CFT correspondence in these geometries. In a remarkable paper, Berenstein, Maldacena and Nastase (BMN) [7] have been able to connect the tree level string theory and the dual SYM theory in a beautiful way. Starting from the gauge theory side, BMN have been able to identify particular string states with gauge invariant operators with large R-charge J, relating the energy of the string states to the dimension of the operators. It has been shown that these identifications are consistent with all planar contributions [8] and non-planar diagrams for large J [9, 10]. Although this is a very important development, it describes however only one particular corner of the full range of gauge invariant operators in SYM corresponding to large J. It is desirable therefore to extend these results to 2

wider class of operators. Some ideas about the extension of AdS/CFT correspondence beyond the supergravity approximation were actually suggested by Polyakov [11] but until recently they wasn’t quite explored. In an important recent development, Gubser, Klebanov and Polyakov [12] (GKP) suggested another way of going beyond the supergravity approximation. The main idea is to consider a particular configuration of closed string in AdS5 × S 5 background executing a highly symmetric motion. The theory of a generic string in this background is highly non-linear, but semiclassical treatment will ensure the existence of globally conserved quantum numbers. AdS/CFT correspondence will allow us to relate these quantum numbers to the dimensions of particular gauge invariant operators in SYM. Gubser, Klebanov and Polyakov investigated three examples: a) rotating “string soliton” on S 5 stretched along the radial direction ρ of AdS part. This case represents string states carrying large R-charge and GKP have been able to reproduce the results obtained from string theory in pp-wave backgrounds [7]. b) rotating “string soliton” on AdS5 stretched along the radial direction ρ. This motion represents string states carrying spin S. The corresponding gauge invariant operators suggested in [12] are important for deep inelastic scattering amplitudes as discussed in [13]. c) strings spinning on S 5 and stretched along an angular direction at ρ = 0. All these states represent highly excited string states and since the analysis is semiclassical, the spin S is assumed quantized. Shortly after [12], interesting generalizations appeared. In [14, 15, 16] a more general solution which includes the above cases was investigated. Beside the solution interpolating between the cases considered by GKP, the authors of [14, 15, 16] have been able to find a general formula relating the energy E, spin S and the R-charge J. In [17] a more general solution interpolating between all the above cases was suggested. Further progress has been made for the case of black hole AdS geometry [18] and confining AdS/CFT backgrounds [19, 20]. The study of the operators with large R-charge and twist two have been initiated in [22, 23]. Further interesting developments of this approach can be found also in [24, 21]. We would note also the solution for circular configurations suggested in [25]. The string soliton in this case can be interpreted as a pulsating string. Further developments exploring this approach to the problem can be found also in [26, 28, 29]. From the preceding discussion it is clear that the extension of the semi3

classical analysis to wider class of moving strings in AdS5 × S 5 background is highly desirable. The purpose of this note is to consider a more general sigma model of closed strings moving in the above background. One of the possibilities is to include closed string with antisymmetric B-field turned on. There are several known backgrounds that are solutions to the string equations. One of them is so called Pilch-Warner (PW) geometry [30]. We will consider the case of rotating strings in PW geometry and will analyze the relation between the conserved quantum numbers in the theory - in our case the energy E and the spin S. We found that, for the configurations we investigated, the relations between the energy and the spin are the same as in the case of rotating strings in AdS5 × S 5 background. The presence of the non-trivial NS-NS two-form field manifests itself in considerations of the fluctuation modes defining the quantum corrections to the conserved quantities. The paper is organized as follows. In the next section we present a brief review of rotating “string soliton” in AdS5 × S 5 background. In Section 3, we give a short description of Pilch-Warner geometry and its basic properties, as well as we briefly comment on the dual conformal theory. Next, in analogy with [12], we find the relation between the energy E and spin S. After that,we study the quadratic fluctuations around the classical solutions and give explicit calculations of the quantum corrections to the anomalous dimensions. We conclude with summary and comparison of our results with those of taking Penrose limit and discuss some further directions of development. To make the presentation more selfcontained we review the Penrose limit of Pilch-Warner geometry in Appendix A.

2

Rotating strings in AdS5 × S 5

In this section we will review the semiclassical analysis of rotating strings [12] following mainly [14, 15, 16]. The idea is to consider rotating and boosted closed “string soliton” stretched in particular directions of AdS5 × S 5 background. The general form of the non-linear sigma model Lagrangian (bosonic part) can be written as: LB =

i h 1√ (AdS ) (S 5 ) ∂α X n ∂β X m −gg αβ Gab 5 ∂α X a ∂β X b + Gnm 2 4

(2.1)

The supergravity solutions of AdS5 × S 5 in global coordinates has the form: ds2 = −cosh2 ρ dt2 + dρ2 + sinh2 ρ dΩ2S 3 + dψ12

+ cos2 ψ1 dψ22 + cos2 ψ2 dΩ2S˜3

where, dΩ2S 3 = dβ12 + cos2 β1 dβ22 + cos2 β2 dβ32 dΩ2S˜3

=

dψ32

2

+ cos ψ3

dψ42

+ cos

2




ψ4 dψ52







(2.2)

(2.3) (2.4)

(we have incorporated R2 factor in the overall constant multiplying the action, i.e. 1/α → R2 /α). Looking for solutions with conserved energy and angular momentum, we assume that the motion is executed along φˆ = β3 direction on AdS5 and θ = ψ5 direction on S 5 . Solution for closed string folded onto itself can be found by making the following ansatz: φˆ = ωτ ; β1 = β2 = 0;

t = κτ ; ρ = ρ(σ);

θ = ντ ψi = 0 (i = 1, . . . , 4)

(2.5)

where ω, κ and ν are constants. We note that one can always use the reparametrization invariance of the worldsheet to set the time coordinate of space-time t proportional to the worldsheet proper time τ . This means that our rod-like string is rotating with constant angular velocity in the corresponding spherical parts of AdS5 and S 5 . Using the ansatz (2.5) and the Lagrangian (2.1) one can easily find the classical string equations, which in this case actually reduces to the following equation for ρ(σ):
d2 ρ 2 2 = κ − ω coshρ sinhρ (2.6) dσ 2 and the Virasoro constraints take the form: 2

ρ′ = κ2 cosh2 ρ − ω 2 sinh2 ρ − ν 2 .

(2.7)

One can obtain also the induced metric which turns out to be conformally flat as expected: 2 gαβ = ρ′ ηαβ , (2.8)

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where prime on ρ denotes derivative with respect to σ. It is straightforward to obtain the constants of motion of the theory under consideration3 : 2π 2π   √ Z dσ √ Z dσ ∂LB E= λ − = λκ cosh2 ρ 2π 2π ∂ t˙

S=

J=

√ √

λ

λ

0 Z2π

0 Z2π 0

dσ 2π dσ 2π



∂LB ˙ ∂ φˆ

!

=

∂LB ∂ θ˙



=

√

λω

0 2π Z

dσ sinh2 ρ 2π

(2.9)

(2.10)

0

√

λν

(2.11)

√ where we denoted R2 /α′ as λ. An immediate but important relation following from (2.9) and (2.10) connect E and S: √ κ E = λκ + S (2.12) ω Assuming that ρ(σ) has a turning point at π/2 and using the periodicity condition4 , one can relate the three free parameters κ, ω and ν as follows: coth2 ρ0 =

ω2 − ν 2 =1+η κ2 − ν 2

(2.13)

where ρ0 is the maximal value of ρ(σ). A new parameter η introduced in (2.13) is useful for studying the limiting cases of short strings (η large) and long strings (η small). Using the constraint (2.7), the following expressions for the periodicity condition and the constants of motion can be found:   √ 1 1 1 1 2 2 κ −ν = √ F , (2.14) , , 1; − η 2 2 η √   1 λκ 1 1 1 , (2.15) E=√ √ F − , , 1; − 2 2 η κ2 − ν 2 η √   1 1 1 λω 1 S=√ , (2.16) F , , 2; − 2 2 η κ2 − ν 2 2η 3/2 (2.17) 3 4

We follow the notations of [12] and [14] Since the closed string is folded onto itself ρ(σ) = ρ(σ + π).

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where F (α, β; γ; z) is the hypergeometric function. In the case of short strings 1/η is very small. Straightforward approx√ imations lead to the relation between the energy E = E/ λ and the spin √ S = S/ λ: q s 2S ν 2 + √1+ν 2 2S +q E ≈ ν2 + √ S (2.18) 2S 2 1 + ν2 √ 1 + ν + 1+ν 2 One can study several limiting cases. The simplest one is when ν ln 1/η E ≈S+J +

λ ln2 (S/J) 2π 2 J

We conclude this section by noting that more general solutions can be obtained by taking ψ1 = ψ1 (σ) [17]. In this case the solutions interpolate smoothly between the cases described above. We note that the same approach have been applied to the case of Mtheory with G2 holonomy [26] where the logarithmic dependence of the energy on the spin J for certain configurations was reproduced.

3

Rotating strings in Pilch-Warner geometry

In this Section we apply the ideology developed in the previous Section to the case of Pilch-Warner geometry. First we briefly review the form and the properties of the geometry and its gauge dual theory. Next we consider rotating closed strings in this geometry and fluctuations around particular classical configurations. We compute also the quantum corrections to the energy and other conserved quantities.

3.1

Basic facts about Pilch-Warner geometry

In this subsection we will review the basic facts about Pilch-Warner geometry. We discuss briefly also on the dual conformal theory. The so called Pilch-Warner geometry is obtained in [30] as a solution of five-dimensional N = 8 gauged supergravity lifted to ten dimensions. This geometry interpolates between the standard maximally supersymmetric AdS5 × S 5 in the ultraviolet (UV) critical point and warped AdS5 × S 5 times squashed five-sphere in the IR. In this paper we will restrict ourself to the IR critical point. An interesting feature on the gravity side is that it preserves 1/8 supersymmetry everywhere while at IR fixed point it is enhanced to 1/4. On the SYM side the IR fixed point corresponds to large N limit of the superconformal N = 1 theory of Leigh-Strassler [33]. To begin with, we discuss first the form of the metric describing so called Pilch-Warner geometry. It is warped AdS5 space times squashed S 5 given

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by5 : ds2 =

ds25

=

X L20


X 1/2 cosh χ 2A 2 4 e ds (M ) + dr 2 + ds25 , ρ

(3.1)

 ρ6 cos2 θ 2 sechχ 2 dθ + (σ1 + σ22 ) ρ3 X  2 ρ12 sin2 (2θ) (2 − ρ6 ) + σ3 + dφ 4X 2 2ρ6  2 # 2 4 cos θ ρ6 cosh2 χ (3 − cos(2θ))2 dφ + σ3 , (3.2) + 16X 2 3 − cos(2θ)

1/2

where we used X(r, θ) = cos2 θ + ρ6 sin2 θ. and the left-invariant SU(2) one-forms σi are given in (3.7) below. The global isometries of the metric can be readily read off from (3.2) and it is SU(2) × U(1)β × U(1)φ taught only a certain combinations of β and φ is preserved by the two-form fields. The IR is described by r → −∞ which means the following IR limit for the fields: √ χ → 2/ 3;

ρ → 21/6 ;

A(r) →

25/3 r 3L0

All these together gives in global coordinates: √ 2
3L0 √ 3 3 − cos 2θ −cosh2 ρ dτ 2 + dρ2 + sinh2 ρ dΩ23 ds2 (IR) = 8 + ds25 (IR), (3.3)

ds25 (IR)

5

=

√

 3L20 √ 4 cos2 θ (σ 2 + σ22 ) 3 − cos 2θ dθ2 + 4 3 − cos 2θ 1  2 # 4 sin2 (2θ) 2 2 4 cos2 θ + dφ + σ + σ3 , (3.4) (3 − cos 2θ)2 3 3 3 − cos(2θ)

We use the notations of [32].

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This geometry is supported by the following five- and two-form fields: 8 21/3 (1 + ⋆) ǫ(E4 ) ∧ dr, 9 L0   2 2i sin 2θ −2iφ L0 cos θ dθ − σ3 ∧ (σ1 + iσ2 ), A2 (IR) =Ae 2 3 − cos 2θ F5 (IR) = −

(3.5) (3.6)

where the left invariant SU(2) one-forms are given by 1 (sin βdα − sin α cos βdγ) 2 1 σ2 = (cos βdα + sin α sin βdγ) 2 1 σ3 = − (dβ + cos αdγ) 2 σ1 =

(3.7)

normalized by dσi = 21 εijk dσj ∧ dσk 6 . The three-form field strength is G3 = dA2 = F3 + iH3 . Pilch-Warner geometry actually represents ”holographic renormalization group flow” from N = 4 Yang-Mills theory in UV to Leigh-Strassler theory in the N = 1 IR fixed point by a deformation of AdS5 × S 5 background. In the dual gauge theory, the SU(2) doublet parametrized by    1 1 u −iφ/2 =e cos θ g , u3 = e−iφ sin θ 2 0 u   1 v −¯ v2 ∈ SU(2) g= v 2 v¯1 corresponds to two massless chiral superfields Φ1 and Φ2 while the singlet corresponds to a massive superfield Φ3 . The relevant N = 1 deformation of N = 4 SYM is define by the modified superpotential 1 U = g trΦ3 [Φ1 , Φ2 ] + m tr(Φ3 )2 . 2 The R-symmetry in this case is a direct product of SU(2) acting on Φ1 and Φ2 , and U(1)R . The U(1)R charges for the superfields Φ1,2,3 ensuring long 6

Note that our normalization is that employed in [32] with the corresponding numerical coefficients in the metric (3.4) (cf. ref.[32]).

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distance conformal dynamics are correspondingly ( 21 , 21 , 1) (for more details see [33, 34] and references therein). In the next subsection we will study the AdS/CFT correspondence beyond the supergravity limit.

3.2

Rotating strings in Pilch-Warner geometry

In this subsection we develop semiclassical analysis of a closed string in PilchWarner geometry (2.3). First of all we will consider classical solutions of closed string and their conserved quantities such as energy, R-charge and spin. To start with, let us consider an ansatz for the embedding coordinates representing string configuration of folded string executing symmetric motion in the space-time defined in (3.4). The classical configuration we will study is realized by the following ansatz: t = κτ ; ρ = ρ(σ) = ρ(σ + 2π); φ = φ(τ );

β3 = ωτ ; θ = 0; β = β(τ );

βi6=3 = 0; α = 0; γ = γ(τ ).

(3.8)

It is straightforward exercise to check that the above ansatz solves the string equations of motion providing β¨ + γ¨ + φ¨ = 0

(3.9)

We assume that the ”string soliton” defined in (3.8) is rotating in β, γ and φ directions with7 : β = cβ τ, def 2 ν = (cβ + cγ + cφ ), 3

γ = cγ τ, def

ψ = β+γ+φ

φ = cφ τ (3.10)

With the choice (3.8) and (3.10) the bosonic part of the Lagrangian becomes: √  3 3 2 2 2 LB = √ L0 κ cosh2 ρ + ρ′ − ω 2sinh2 ρ − ν 2 (3.11) 4 2

7 We note that with the choice (3.8) the metric gets degenerated and the three degrees of freedom β, γ and φ are replaced by one: ψ = β + γ + φ. The same happens in the case of taking Penrose limit in [32].

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From (3.11) it is obvious that the equations of motion for ρ(σ) and the Virasoro constraints have formally the same form as in the case of rotating strings in AdS5 × S 5 considered in the previous Section:
d2 ρ 2 2 = κ − ω coshρ sinhρ dσ 2 2 ρ′ = κ2 cosh2 ρ − ω 2 sinh2 ρ − ν 2 .

(3.12)

Therefore, all the conclusions concerning the classical values of the conserved quantities including the limiting cases of short and long strings remain unchanged (up to numerical coefficients). This is because for the chosen configuration the CS term vanishes and the Lagrangian takes formally the same form. The difference from the case studied in Sec 1. is that in our case we can have different momenta along the angles φ, β, γ which will correspond to different operators in SYM. We should note that the famous logarithmic dependence for long strings also take place but it is not a priori obvious that there will be no higher power logarithmic terms coming from the quantum corrections8 . We will return to this question in the Concluding section. It is worth to note that for point-like string the Lagrangian becomes √  3 3 2 2 2 ˙ (3.13) LB = √ L0 −τ˙ + ψ 4 2

and the result should be compared with those of taking Penrose limit [32] (see the discussion in the Concluding section and the Appendix).

3.3

Quadratic fluctuations

In this Section we study the quadratic fluctuations around the classical solutions of previous subsection. The aim is to find the leading quantum corrections to the energy spectrum which determines the anomalous dimensions in SYM. ¯ µ is the use A standard method for studying small fluctuations around X of Riemann normal coordinates developed in [35]. We start with the sigma model string action: Z √ αβ 1 2 µ ν αβ µ ν S=− d σ gg G ∂ X ∂ X + ε B ∂ X ∂ X . (3.14) µν α β µν α b 4πα′ 8

The absence of higher powers of logarithmic corrections in the case of rotating strings in AdS5 × S 5 is discussed in [16].

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¯ µ generates the following transformations The small fluctuations around X for the entries of (3.14) (see for instance [35]) ¯ µ + Dα ξ µ + 1 Rµ ξ λ ξ κ ∂α X ¯ν + · · · ∂α X µ = ∂α X λκν 3 1 ¯ − Rµρνκ ξ ρ ξ κ + · · · Gµν (X) = Gµν (X) 3 ¯ + Dρ Bµν (X)ξ ¯ ρ + 1 Dλ Dρ Bµν (X)ξ ¯ λ ξ ρ− Bµν (X) = Bµν (X) 2 1 λ ¯ ρ κ ¯ ξ + 1 Rλ Bλµ (X)ξ ¯ ρξ κ + · · · , − Rρµκ (X)Bλν (X)ξ ρνκ 6 6 where

¯ ν, Dα ξ µ = ∂α ξ µ + Γµλν ξ λ ∂α X

µ and Rνρλ is the usual Riemann tensor for the ambient space. The quadratic part can be easily extracted from (3.14) (2)

(2)

S (2) = SG + SB , where (2) SG (2)

SB

Z   1 √ ¯ α ξ µ Dβ ξ ν + Aµν;αβ ξ µ ξ ν d2 σ gg αβ Gµν (X)D (3.15) =− ′ 2πα   Z 1 1 δ ν µ 2 αβ λ δ µ ν ¯ ¯ ¯ =− ∂α X Hδµν ξ Dβ ξ + Dλ Hδµν ξ ξ ∂α X ∂β X . d σε 2πα′ 2 (3.16)

In (3.15) Aµν;αβ is defined as9 : ¯ αX ¯ δ ∂β X ¯ κ. Aµν;αβ = Rµδνκ (X)∂

(3.17)

(2)

It is convenient to rewrite SG in Lorentz frame Z
 1 √  (2) d2 σ g g αβ ηAB Dα ξ A Dβ ξ B + g αβ RACBD YαC YβD ξ A ξ B , SG = − ′ 4πα (3.18) 9

The expressions for the AdS part of the metric are analogous to those in [36] for the open superstring. In our case there are additional contributions from the squashed sphere.

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where Dα ξ A = ∂a ξ A + ωαAB ξ B ¯ µ. YαA = EµA ∂α X

(3.19)

In (3.19) the coefficients of the projected spin connection are
A ωBµ = EνA ∂µ EBν + Γνµλ EBλ ¯ µ. ω AB = ω AB ∂α X α

µ

Now we have to apply the above general scheme to the case of PilchWarner geometry (3.3-3.6) and string configuration (3.8), (3.10). It is tedious but straightforward to calculate the components of the projected spin connection for our concrete configuration and the result is given as follows10 a) for AdS part ω¯001 = κ sinhρ;

ω¯014 = ω coshρ.

(3.20)

b) for S5 part ˜˜

ω¯003 =

1 (4cφ + cβ + cγ ) ; 3

˜˜

ω¯042 =

1 (2cφ + 2cβ − cγ ) . 3

(3.21)

The mass terms can be computed by using the following expressions a) for AdS part √ 3 6 2 L0 RACBD = −ηAB ηCD + ηAD ηCB . (3.22) 8 b) for S5 part, after projection on θ = α = 0, the only non-vanishing components are Rµ˜φ˜ν φ ξ µ˜ ξ ν˜ = Rµ˜β ν˜β ξ µ˜ ξ ν˜ = Rµ˜γ ν˜γ ξ µ˜ ξ ν˜ = Rµ˜φ˜ν β ξ µ˜ ξ ν˜ = Rµ˜φ˜ν γ ξ µ˜ ξ ν˜ = Rµ˜β ν˜γ ξ µ˜ ξ ν˜, with Rµ˜φ˜ν φ eµA˜ eνB˜ = 10

i 4h i 13 h ˜0 ˜0 ˜ ˜ ˜ ˜ ˜ ˜ δA δB + δA3 δB3 + δA2 δB2 + δA4 δB4 . 9 9

(3.23)

(3.24)

We will use indices with tilde for S5 part and without tilde for AdS part. The same will be applied to Lorentz indices denoted by numbers.

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We will use the above expressions to write the Lagrangian describing fluctuations in S5 part. c) for the non-vanishing mixed terms Rµ˜µν ν˜ , µ, ν ∈ AdS, µ ˜, ν˜ ∈ S5 we find only two non-vanishing terms √   3 6L20 µ ˜ ν˜ Rt˜µt˜ν ξ ξ = cosh2 ρ ζ˜02 + ζ˜32 16 √   3 6L20 (3.25) sinh2 ρ ζ˜02 + ζ˜32 . Rβ3 µ˜β3 ν˜ ξ µ˜ ξ ν˜ = − 16 The final form of LB describing quadratic fluctuations in Lorentz frame is  ˜4  4 π ζ ˜ ˜ 0 3 ˆ LB = − √ (cφ + cβ + cγ ) (∂1 ζ , ∂1 ζ ) A(δ − ) , (3.26) 2 −ζ ˜2 3 where δ = 2φ + β + χ and Aˆ ∈ SU(2) is   π π π cos(δ − ) − sin(δ − ) 2 2 ˆ − )= A(δ . (3.27) sin(δ − π2 ) cos(δ − π2 ) 2 We find for the Lagrangian describing fluctuating modes in AdS part the expression

˜ α φ˜ + ∂α β˜i ∂ α β˜i − m2 t˜2 + m2 ρ˜2 + m2 β˜2 LAdS = −∂α t˜∂ α t˜ + ∂α ρ˜∂ α ρ˜ + ∂α φ∂ t ρ β i ˜ (3.28) + 4˜ ρ(κ sinh¯ ρ∂0 t˜ − ω cosh¯ ρ∂0 φ),

with

t˜ = ζ 0 , φ˜ = ζ 4,

β˜2 = ζ 3

ρ˜ = ζ 1 , β˜1 = ζ 2 2

m2t = 2ρ¯′ + ν 2 − κ2 , 2 m2ρ = 2ρ¯′ + 2ν 2 − κ2 − ω 2 ,

(3.29) 2

m2φ = 2ρ¯′ + ν 2 − ω 2 2 m2β = 2ρ¯′ + ν 2 .

(3.30)

Let us turn now to the quadratic fluctuations in S5 part. By making use of (3.24) we find for LS5 the following expression: 2  2  1 1 1 ∂0 ζ˜3 − (4cφ + cβ + cγ )ζ˜5 LS5 = − ∂0 ζ˜5 + (4cφ + cβ + cγ )ζ˜3 − 3 3 3  2  2 1 1 − ∂0 ζ˜2 − (2cφ + 2cβ − cγ )ζ˜4 − ∂0 ζ˜4 + (2cφ + 2cβ − cγ )ζ˜2 3 3     3 (3.31) − (∂0 ζ˜1 )2 + (∂1 ζa )2 + ν 2 ζ˜22 + ζ˜42 + ( ρ¯′ + 4ν 2 ) ζ˜52 + ζ˜32 . 2 15

To diagonalize the terms in the square brackets we perform the following rotations     4c + c + c ζ˜5 ζ˜˜ φ β γ ˆ = A( τ ) ˜5 , (3.32) ζ˜3 3 ζ˜3     2cφ + 2cβ − cγ ζ˜˜ ζ˜4 ˆ (3.33) τ ) ˜4 , = A( ζ˜2 ζ˜2 3 which leaves the rest of LS5 unchanged. The final from is given by i h i 3 2 h LS5 = ∂α ζ˜a ∂ α ζ˜a + (4M 2 + ρ¯′ ) ζ˜˜52 + ζ˜˜32 + M 2 ζ˜˜22 + ζ˜˜42 , 2

(3.34)

where we defined

4 M 2 = ν 2 = (cφ + cβ + cγ )2 . (3.35) 9 The transformations (3.32),(3.33) however affect LB since B-field is coupled to S5 part. Choosing the arbitrary constant χ = −π/2 as in [31], the expression for LB actually simplifies i h √ (3.36) LB = 2 3M ∂1 ζ˜˜5 ζ˜˜4 − ∂1 ζ˜˜3 ζ˜˜2 .

Adding (3.36) to LS5 , we find for quadratic fluctuations in S5 part the expression Ltot S5

h i 3 ¯′ 2 h ˜2 ˜2 i A α ˜A 2 ˜2 2 2 ˜ ˜ = ∂α ζ ∂ ζ + M ζ˜2 + ζ˜4 + (4M + ρ ) ζ˜3 + ζ˜5 2   √ + 2 3M ζ˜˜4 ∂1 ζ˜˜5 − ζ˜˜2 ∂1 ζ˜˜3 . (3.37)

It is straightforward now to write the equations of motion: a) for S5 part ∇2 ζ˜˜1 = 0 √ ∇2 ζ˜2˜ − M 2 ζ˜4˜ + 3M ∂1 ζ˜˜3 = 0 √ 3 2 ∇2 ζ˜˜3 − (4M 2 + ρ¯′ )ζ˜˜3 − 3M ∂1 ζ˜˜2 = 0 2 √ ∇2 ζ˜˜4 − M 2 ζ˜˜4 − 3M ∂1 ζ˜˜5 = 0 √ 3 2 ∇2 ζ˜˜5 − (4M 2 + ρ¯′ )ζ˜˜5 + 3M ∂1 ζ˜˜4 = 0 2 16

(3.38)

b) for AdS part 2

∇2 t˜ − (2ρ¯′ + ν 2 − κ2 )t˜ + 2κ sinh¯ ρ ∂0 ρ˜ = 0 2 ˜ =0 ρ − 2(κ sinh¯ ρ ∂0 t˜ − ω cosh¯ ρ ∂0 φ) ∇2 ρ˜ − (2ρ¯′ + 2ν 2 − κ2 − ω 2 )˜ 2 ρ ∂0 ρ˜ = 0 ∇2 φ˜ − (2ρ¯′ + ν 2 − ω 2 )φ˜ − 2ω cosh¯ 2

∇2 β˜i − (2ρ¯′ + ν 2 )β˜i = 0.

(3.39)

Now we are in a position to calculate the contribution of the massive modes to the classical energy. To do that we will use the approximations in the two limiting cases – short and long strings. The problem is that in eqs. (3.38) and (3.39) the mass terms contain σ-dependent part through ρ¯′ which can be accounted for by using long and short string approximations. Let us consider first the case of short strings. In this case an approximate 2 solution for ρ¯′ (σ) is cos2 σ 2 ρ¯′ ≈ η 2 It is clear that the contribution from ρ¯′ is very small (η is very large) and it can be replaced by its average < cos2 σ >= 1/2 and the mass terms contain2 ing ρ¯′ will be corrected by terms ∼ 1/2η. The analysis of the contributions from AdS5 part follows as in [14]. The contribution of β-fields can be calculated by solving the corresponding eigenvalue problem and the solution is given by X βi = βni exp (−iωn + inσ) n

ωn2 ≈ n2 + ν 2 +

1 η

(3.40)

Due to the choice t = κτ the space-time energy is proportional to the worldsheet one and we can write a) for small ν, ν 2 ≪ S ≪ 1    ∞  √ √ ν2 1 X ν2 S +√ 1− + |n|Nn + · · · E≈ λ 2S + √ 4Sn2 n2 2 2S 2S n=−∞ (3.41) b) for ν ≫ 1    ∞  X n2 n2 S S 1 + 2 − 5 Nn + · · · (3.42) E ≈λ ν +S + 2 + 2ν 2ν ν n=−∞ 17

2

Taking into account that 2ρ¯′ ≈ 1/η, κ2 − ν 2 ≈ 1/η and ω 2 − κ2 ≈ 1 the equations for t-fluctuations becomes trivial, ∇2 t = 0 while the frequencies for ρ and ϕ are (ωn± )2

 1 1 1 =n −1+ 4ω 2(1 + )2 − ± 2 4η η r  1 1 1 [4ω 2 (1 + )2 − ]2 + 16ω 2(1 + )2 (n2 − 1) , (3.43) 4η η 4η 2

where ω 2 ≈ 1 + 1/η + ν 2 . Let us turn now to the more involving S 5 part. P ˜˜i ˜ The mode expansion ζ i = n ζn exp(−iωn + inσ) and the approximation 2 ′ ¯ ρ ≈ 1/(2η) gives the following equations √  2  ˜ ˜ −ωn + n2 + M 2 ζ˜n2 − in 3M ζ˜n3 = 0   √ 3 ˜˜3 ˜ 2 2 2 −ωn + n + 4M + ζn + in 3M ζ˜n2 = 0 4η √  2  ˜ ˜ (3.44) −ωn + n2 + M 2 ζ˜n4 + in 3M ζ˜n5 = 0   √ 3 ˜˜5 ˜ ζn − in 3M ζ˜n4 = 0, −ωn2 + n2 + 4M 2 + 4η where M 2 = ν 2 . The solutions for the frequencies are r   1 3 1 ± 2 2 2 (ωn ) = 5M + 2n + ± 12n2 M 2 + 9(M 2 + )2 , 2 4η 4η r 3 ω0− = M. ω0+ = 4M 2 + , 4η

(3.45)

We find for the mode expansion of the fluctuations the expressions r

r 3 α′ 3 A ζ (τ, σ) = cos sin 4M 2 + τ pA + τ ζ0 + q 3 4η 4η 0 4M 2 + 4η r  α′ X 1  A inσ ˜A −inσ  −iωn+ τ +i ζ e + ζn e e 2 n6=0 ωn+ n  1  A+2 inσ ˜A+2 −inσ  −iωn− τ (3.46) e + − ζn e + ζn e ωn A

4M 2

18

α′ ζ A+2(τ, σ) = cos Mτ ζ0A+2 + sin Mτ pA+2 0 M r    α ′ X c+ n A inσ ˜A e−inσ e−iωn+ τ ζ e − ζ +i n 2 n6=0 ωn+ n    − c− n −iω τ A+2 inσ A+2 −inσ + − ζn e − ζ˜n e (3.47) e n ωn where i √ c± n = 2n 3M

r   3 3 2 2 2 2 −3M − ± 12n M + (3M + ) ; 4η 4η

− c+ n cn = 1.

(3.48) To write the Hamiltonian in canonical form we must first normalize the oscillators so that they satisfy the canonical oscillator algebra. This can be achieved by rescaling the oscillators as follows s s − c− ¯ c A ζnA = ωn+ − n + ζˆnA ; ζ−n = ωn+ − n + ζˆnA cn − cn cn − cn s s c+ c+ ¯ A+2 ζ−n = ωn− + n − ζˆnA+2 (3.49) ζnA+2 = ωn− + n − ζˆnA+2; cn − cn cn − cn and similarly for ζ˜nA . We also redefine the zero modes as r   1 1 3 A A ′ A ζˆ0 = √ α p0 − 4M 2 + ζ0 3 14 4η 2α′ (4m2 + 4η ) r   1 3 A ¯ˆA √1 ′ A 2 ζ0 = α p0 + 4M + ζ0 3 14 4η 2α′ (4m2 + 4η )
1 ζˆ0A+2 = √ α ′ pA − Mζ0A+2 0 2α′ M
1 ¯ A ζˆ0A+2 = √ α′ pA+2 + Mζ 0 0 2α′ M

The commutation relations for this oscillator basis are h i h i ¯ ¯B = δnm δ AB , ζˆ0A , ζˆ0B = δ AB ; ζˆnA , ζˆm 19

(3.50)

(3.51)

and the Hamiltonian takes the standard form r X (A) 3 X (A) H = E + 4M 2 + N0 + M N0 4η A=1,2 A=3,4 X  + ωn+ Nn(A) + ωn− Nn(A+2) . (3.52) n>0

The occupation number operators are defined as usually by ¯ ¯ Nn(A) = ζˆnA ζˆnA + ζ˜nA ζ˜nA ;

(A)

N0

¯ = ζˆ0A ζˆ0A .

(3.53)

It is straightforward now √ to expand the frequencies by making use that 1/η is small. Since 1/η ≈ 2S/ ν 2 + 1, the expansion of (3.45) can be written as   3 3ν 2 1 S ± (0)± √ ωn ≈ ωn + , (3.54) 1± √ (0)± 8 12n2 ν 2 + 9ν 4 1 + ν 2 ωn where ωn(0)±

i1/2 √ 1 h 2 2 2 2 4 = sign(n) √ 5ν + 2n ± 12n ν + 9ν . 2

(3.55)

In the case of ν >> 1 we get n2 3 + 2S 2ν 8ν 3 ω0+ = 2ν + 2 S 8ν

ωn+ = 2ν +

ωn− = ν +

n4 n2 + S 3ν 3 4ν 4

ω0− = ν.

(3.56)

When ν 0




(A.19) where ∆E is the zero point energy and the occupation numbers are defined in a standard way ˜¯pn a Nn(a) = a ¯pn apn + a ˜pn , (b)

N0i = a ¯i0 ain

(A.20)

(c)

and similarly for Nn and Nn . The frequencies ωn and ωn± are given in (A.12, A.14). The difference with the maximally supersymmetric case is encoded in the complicated form of the frequencies ωn± . We note that the same analysis can be applied to the case of pointlike string (see section Conclusions). In this case ρ → ρ0 , ρ′ = 0 and the equations of motion becomes the same as (A.9). Therefore, all the results and conclusions concerning quantum corrections to the classical energy are the same as in the above.

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