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EPHOU-05-002 KEK-TH-1004 Imperial/TP/050402 UTHEP-502 hep-th/0504123 Quantum fluctuations of rotating strings in AdS S5 5 × 6 0 0 2 Hiroyuki Fuji † n a Department of Physics, Hokkaido University J 1 Sapporo, 060-0810, Japan 1 High Energy Accelerator Research Organization (KEK) 3 v Tsukuba, Ibaraki 305-0801, Japan 3 2 1 4 Yuji Satoh‡ 0 5 Blackett Laboratory, Imperial College 0 / London, SW7 2BZ, U.K. h t - Institute of Physics, University of Tsukuba p e Tsukuba, Ibaraki 305-8571, Japan h : v i X r a Abstract We discuss quantum fluctuations of a class of rotating strings in AdS S5. In particular, we 5 × develop a systematic method to compute the one-loop sigma-model effective actions in closed forms as expansions for large spins. As examples, we explicitly evaluate the leading terms for the constant radii strings in the SO(6) sector with two equal spins, the SU(2) sector, and the SL(2) sector. We also obtain the leading quantum corrections to the space-time energy for these sectors. April 2005 †[email protected] ‡[email protected] 1 Introduction To better understand dynamical aspects of the AdS/CFT correspondence, one may need studies beyond the BPS sectors. An important step toward this direction was made in [1]. Rotating strings in AdS S5 provide its generalizations, where deeply non-BPS sectors can 5 × be probed [2]-[9]. (For a review, see [10, 11].) In particular, for a certain class of rotating strings, one can find an exact agreement between unprotected quantities on the string and the gauge theory sides, which is not necessarily guaranteed by the AdS/CFT correspondence. Moreover, the correspondence has been extended, in a unified manner, to that of effective theories or general solutions [12]-[15]. A clue for understanding this agreement may be the integrable structures on both sides (see, e.g., [3, 8], [14]-[20]). A role of a certain asymptotic (“nearly”) BPS condition has also been pointed out [21]. For large spins, the world-sheets of the rotating strings form nearly null surfaces and the strings become effectively tensionless [21, 22], which intuitively means that the constituents of the strings appear to be free [23]. However, we do not yet know why we find the exact agreement, and why it starts to break down at a certain level [24]. In this development, the analysis on the string side tends to be classical because of difficulties in the quantization, despite that information of the quantized strings is necessary to complete the correspondence. This contrasts with detailed quantum analysis on the gauge theory side [11]. As for the quantum aspects of the rotating strings,1 the one-loop sigma- model fluctuations and their stability have been studied for the “constant radii” strings [4, 20, 29]. Based on these results, the one-loop corrections to the space-time energy have been studied numerically for the SO(6) and the SU(2) sectors with two equal spins [30], and for the SL(2) sector [31]. Furthermore, the leading correction for the SL(2) sector has been matched in a closed form with the finite size correction of the anomalous dimension on the gauge theory side [32]. This result can also be extrapolated to the SU(2) sector (with two equal spins), up to subtleties of instability. In this paper, we discuss quantum fluctuations of the rotating strings in AdS S5. In 5 × particular, we develop a systematic method to compute the one-loop sigma-model effective actions and the corrections to the space-time energy in such backgrounds, so that they are obtained in closed forms as expansions for large spins. As examples, we consider the constant radiistringsintheSO(6)sector with two equalspins, theSU(2) sector, andtheSL(2)sector, and explicitly evaluate the leading terms in the expansion. We note that it is in principle possible tocarry outtheexpansion up toanygiven order. The asymptotic BPSconditionand the effective tensionless limit seem to be characteristic of the correspondence and useful for understanding the string in AdS S5 itself. However, their consequences for the quantum 5 × string have not been investigated well. Through concrete computations, we can see how they work to make the quantum corrections subleading for large spins. 1For related works, see, e.g., [25]-[28]. 1 The organization of this paper is as follows. In section 2, we summarize necessary ingredients to discuss the one-loop fluctuations of strings in AdS S5. In section 3, we discuss 5 × the quadratic fluctuations of the constant radii strings in the SO(6) sector with two equal spins. We evaluate the fluctuation operators in rotated functional bases, so that the expansion for large spins becomes well-defined. In the course, we obtain the fermionic fluctuation operator for the generic SO(6) sector. In section 4, we develop a large J (total spin) expansion of the one-loop effective action. We explicitly evaluate the leading term for large J (up to and including (1/J)) in a closed form. Essentially the same procedures are applied to O other sectors in the following sections. We briefly summarize the results for the SU(2) sector in section 5, and for the SL(2) sector in section 6. At the leading order, all the one-loop effective actions take a universal form, which is proportional to a geometric invariant. In section 7, from the one-loop effective actions, we read off the corrections to the space-time energy up to and including (1/J2). Comparing the results with the finite size corrections O to the anomalous dimension on the gauge theory side [32]-[36], we find that the dependence on the winding numbers and the filling fractions agree with that of the “non-anomalous” (zero-mode) part on the gauge theory side. Relation to the earlier results in the literature is also discussed. We conclude in section 8. In the appendix, we summarize how to evaluate a constant which appears in the expression of the one-loop effective actions. 2 Preliminaries We consider one-loop sigma-model fluctuations of a certain class of rotating strings in type IIB theroy. Here, we summarize our notation and ingredients used in the following sections. Coordinates The metric of AdS S5 takes a form 5 × ds2 = G dxµdxν = ds2 +ds2 , µν AdS5 S5 ds2 = dρ2 cosh2ρ dt2 +sinh2ρ (dθ2 +cos2θdφ2 +sin2θdφ2), (2.1) − AdS5 − 4 5 ds2 = dγ2 +cos2γdφ2 +sin2γ (dψ2 +cos2ψdφ2 +sin2ψdφ2). − S5 3 1 2 To express the rotating string solutions, it is useful to introduce complex variables Z (r = r 0,...,5),sothatAdS andS5 areexpressedashypersufacesinflatspaces, Z 2 Z 2 Z 2 = 5 0 4 5 | | −| | −| | 1 and Z 2 + Z 2 + Z 2 = 1, respectively. In terms of the above coordinates, one can set 1 2 3 | | | | | | Z = a eiφr , (2.2) r r with φ = t and 0 a = sinγ cosψ, a = sinγ sinψ, a = cosγ, 1 2 3 (2.3) a = sinhρ cosθ, a = sinhρ sinθ, a = coshρ. 4 5 0 2 Bosonic fluctuation In the conformal gauge, the bosonic part of the world-sheet action is given by √λ S = dτdσηijG ∂ xµ∂ xν , (2.4) B µν i j − 4π Z where i,j = (τ,σ), and η = η = +1. A simple way to obtain the fluctuation Lagrangian ττ σσ − is the geodesic expansion [37]. Introducing a vector ya in the tangent space of the space-time, the quadratic fluctuation Lagrangian is given by 1 1 L(2) = η D yaDiyb yaybeceidR . (2.5) B −2 ab i − 2 i acbd η is the flat ten-dimensional metric with mostly minus signatures. D and ea are the ab i i projections of the covariant derivative D and the vielbein ea, respectively; for example, µ µ (D y)a = ∂ xµ(∂ +ω a)yb with ω the connection one-form. For AdS S5, the curvature i i µ µ,b µ,ba 5× R is simple: abcd R = (η η η η ), (2.6) abcd ac bd ad bc ∓ − withtheminussignifallindices(a,b,c,d)correspondtoAdS andtheplussignif(a,b,c,d)to 5 S5; otherwise it vanishes. In term of the global coordinate system in (2.1), the non-vanishing ω (= ∂ xµω ) are, up to the anti-symmetry, i,ab i µ,ab ω = coshρ∂ θ, ω = sinhρ∂ t, ω = coshρcosθ∂ φ , i,θρ i i,tρ − i i,φ4ρ i 4 ω = sinθ∂ φ , ω = coshρsinθ∂ φ , ω = cosθ∂ φ , i,φ4θ − i 4 i,φ5ρ i 5 i,φ5θ i 5 (2.7) ω = cosγ∂ ψ, ω = sinγ∂ φ , ω = cosγcosψ∂ φ , i,ψγ i i,φ3γ − i 3 i,φ1γ i 1 ω = sinψ∂ φ , ω = cosγsinψ∂ φ , ω = cosψ∂ φ . i,φ1ψ − i 1 i,φ2γ i 2 i,φ2ψ i 2 Fermionic fluctuation To study the fermionic fluctuations, we use the quadratic fermionic part of the type IIB Green-Schwarz (GS) action on AdS S5 [38]. Following the notation in [39], it is expressed 5 × as L(2) = iθIPij θK , (2.8) F IJDijJK 1 1 = σˆ (∂ ω σab)δ F σabcdeσˆ (ρ ) , ijJK i j j,ab JK abcde j 0 JK D − 4 − 4 480 (cid:20) · (cid:21) Pij = ηijδ +ǫij(ρ ) , σˆ = eaσ¯ , ǫ01 = +1, IJ IJ 3 IJ j j a 0 1 1 0 ρ = , ρ = . 0 3 1 0 ! 0 1 ! − − 3 Here, (θI)α (I = 1,2) are ten-dimensional Majorana-Weyl spinors with 16 components; (σ¯a) ,(σa)αβ are 16 16 gamma matrices in ten dimensions; their anti-symmetrization αβ × is given, e.g., by σab = (σaσ¯b σbσ¯a)/2, σ¯ab = (σ¯aσb σ¯bσa)/2. If we label the AdS and S5 5 − − parts by a,b = (0,6,7,8,9) and (1,2,3,4,5), respectively, the non-vanishing components of the five-form are F = F = 4. 06789 12345 The GS action has the κ-symmetry, the relevant part of which is now δ ϑα = (σˆ )αβηijκ , (2.9) κ i jβ where ηijκ = ǫijκ¯ , ϑ = θ1+iθ2 and κ = κ1 +iκ2 . Following [29], we fix this symmetry jβ jβ jβ jβ jβ by setting θ1 = θ2. (2.10) One can check that this gauge is actually possible for the backgrounds which we consider in the following sections. In this gauge, the form of the quadratic Lagrangian is simplified to L(2) = 2iθ1D θ1, F F 1 1 D = ηijσˆ (∂ ω σab) ǫijσˆ σ σˆ , (2.11) F i j j,ab i j − 4 − 2 ∗ where σ σ06789 = σ12345. ∗ ≡ 3 Three-spin rotating string in S5 In this paper, we consider classes of the “constant radii” solutions [10], in which a in r (2.2) are constant. Bearing in mind possible extensions to more general cases, we develop a method to compute one-loop effective actions in a large-spin expansion. In this and the next sections, we discuss it in some detail for the constant radii strings with three spins in S5, i.e., in the SO(6) sector. We apply this method to other cases later. 3.1 Solution The solution which we consider is Z = eiκτ , Z = a ei(wsτ+msσ), (3.1) 0 s s where s = 1,2,3; κ,a ,w are constant with Σ a2 = 1; m are integers (when the period s s s s s of σ is 2π). Other fields including fermionic ones vanish. The equations of motion and the Virasoro constraints give w2 = ν2 +m2 (if a = 0), s s s 6 κ2 = a2(w2 +m2), 0 = a2w m , (3.2) s s s s s s X X 4 with ν a constant. Classically, the space-time energy and the three spins in S5 are given by E = √λκ, J = √λa2w , (3.3) s s s where √λ = R2/α and R is the radius of AdS S5. ′ 5 × When J = J and m = 0, the parameters of the solution become 1 2 3 w = w w, m = m m, a = a = s /√2, 1 2 1 2 1 2 γ ≡ − ≡ w = ν, m = 0, a = c , (3.4) 3 3 3 γ w2 = ν2 +m2, κ2 = ν2 +2m2s2 , γ where s sinx,c cosx. In the following, we call this simplified solution the J = J x x 1 2 ≡ ≡ three-spin solution. The point-like (BMN) solution in [1] is also obtained by setting a = 1,2 w = m = m = 0 in (3.1) and (3.2). 1,2 1,2 3 3.2 Bosonic fluctuation Now, let us consider the bosonic fluctuations around the three-spin solution in (3.1). Substituting the solution into (2.5), one finds that the fluctuations in the AdS and the S5 5 parts decouple. The contribution to the one-loop effective action from the AdS part is then 5 represented by the determinant of the quadratic operator, DB = η ∂2 +κ2R , (3.5) pq − pq ptqt where p,q = (t,ρ,θ,φ ,φ ) or (0,6,7,8,9), and ∂2 = ηij∂ ∂ . This operator describes one 4 5 i j time-like massless boson and four space-like bosons with mass squared κ2, as in the BMN case. Taking the determinant with respect to the tangent space indices gives detDB = ∂2(∂2 +κ2)4. (3.6) pq − For the S5 part, the quadratic term of the connection one-form and the curvature term cancel each other, to give DB = δ ∂2 +2(ω ∂ ω ∂ ), (3.7) mn mn τ,mn τ − σ,mn σ detDB = ∂2 (∂2)4 + Σ3 (1 a2)Ω2 (∂2)2 +(a2Ω2Ω2 +a2Ω2Ω2 +a2Ω2Ω2) , mn s=1 − s s 1 2 3 2 3 1 3 1 2 h (cid:16) (cid:17) i where m,n = (φ ,φ ,φ ,ψ,γ) or (1,2,3,4,5), and 1 2 3 Ω = 2(w ∂ m ∂ ). (3.8) s s τ s σ − This determinant has been obtained in [20]. Note that we have assumed here that none of a vanishes in order to use the constraints w2 = ν2 +m2. s s s Change of functional bases 5 Later, we explicitly evaluate the functional determinant in a large J(= J ) expansion. s It turns out that detDB in (3.7) has an inappropriate infrared behavior for this purpose. mn P Here, we make anSO(5)rotationto avoid it, concentrating ontheJ = J three-spin solution 1 2 with (3.4). First, we take an orthogonal matrix, s ν v 0 c w 0 γ γ   s ν v 0 c w 0 γ γ 1  −  Qm =  √2c w 0 0 √2s ν 0  , (3.9) n √2v  γ − γ     0 0 √2v 0 0       0 0 0 0 √2v     −    with v2 = w2 M2 and M2 = s2m2. By a change of bases using Qm, the connection one-form − γ n is brought into a standard form. Namely, defining ωˆ Qk ω Ql , we find that i,mn ≡ m i,kl n 0   0 x ωˆ = p( w) , p(x) = . (3.10) τ,mn  −  x 0 !    p( v)  −    −    We further introduce 1   cosx sinx Rm(τ) = P(τw) , P(x) = , (3.11) n   sinx cosx !    P(τv)  −       so that Rm satisfies n 0 = ∂ R ωˆ Rk . (3.12) τ mn − τ,mk n Then, by an transformation of the form, ˆ (QR) 1 (QR), (3.13) − O ≡ O the quadratic operator becomes DˆB = Rk′ Qk DBQl Rl′ mn m k′ kl l′ n = δ ∂2 +M 2ρ ∂ , (3.14) mn mn σ,mn σ − where M = diag(0,w2,w2,v2,v2) and ρ = Rk ωˆ Rl . After some algebra, we also mn σ,mn m σ,kl n find that detDˆB = (∂2 +w2)2(∂2 +v2)2∂2 mn w2 w +4c2k2∂2∂2 (1+ )(∂2 +w2)(∂2 +v2)+4c2k2( )2∂2 (3.15) γ σ v2 γ v σ ν h i +4s2k2( )2∂2(∂2 +v2) (∂2 +w2)(∂2 +v2)+4c2k2∂2 . γ v σ γ σ h i 6 3.3 Fermionic fluctuation Now, let us move onto the fermionic part. In order to evaluate the one-loop determinant, it is useful to make a rotation of D , so that the kinetic term is simplified to take the form F for two-dimensional fermions [40]. For this purpose, we introduce an element of SO(1,9), κ W s −   qa 0 1 0 M Qa = b , qa =  s  , (3.16) b 0 1 ! b M  W κW /W  6×6  − s   0 l M   s      where a,b = (t,φ ,φ ,φ ,...); W a w ,M = a m , W2 W2,M2 M2; 1 is 1 2 3 s ≡ s s s s s ≡ s s ≡ s s 6×6 the 6 6 unit matrix; l = ǫ M W /MW. This was chosen so that Q a = e ϕea for × s1 s1s2s3 s2 s3 P P˜i − ˜i ˜i = (0,1), where ϕ is defined by the proportionality coefficient between the induced metric, h = eaebη , and the world-sheet metric through h = e2ϕη . In our case, e2ϕ = M2. We ij i j ab ij ij then transform D by the element of SO(1,9) corresponding to Qa , which we denote by F b S(Q). Since S 1σ S = σ Qb and hence − a b a S 1σˆ S = eϕδ˜iσ¯ , (3.17) − i i ˜i this transformation replaces σˆ with σ¯ in the quadratic operator, to give a desired form. In i i the following, we do not distinguish the indices ˜i and i. After some algebra, we then find that S 1(Q)D S(Q) = eϕDˆ , (3.18) − F F where 1 1 Dˆ = ηijσ¯ ∂ ω (σab) ǫijeϕσ¯ σ σ¯ , F i j j,ab ′ i ′ j − 4 − 2 ∗ h 1 i = σ¯i∂ +Wσ¯345 + α σ¯abc, (3.19) i abc 2MW a=0,1b=2,3c=4,5 X X X (σab) = S 1σabS,σ = S 1σ S, and ′ − ′ − ∗ ∗ α = s c κ(c2m2 +s2m2 m2), α = (c /s )κm w , 024 − γ γ ψ 1 ψ 2 − 3 124 γ γ 3 3 α = s s c w (m w m w ), α = s s c m (m w m w ), 034 γ ψ ψ 3 1 2 2 1 134 γ ψ ψ 3 1 2 2 1 − − α = s s c κ(m2 m2), α = s s c κ(m w m w ), 025 γ ψ ψ 1 − 2 125 γ ψ ψ 1 1 − 2 2 α = s c [m w w w (c2m w +s2m w )], 035 − γ γ 3 1 2 − 3 ψ 1 2 ψ 2 1 α = s c [m m w m (s2m w +c2m w )]. (3.20) 135 γ γ 1 2 3 − 3 ψ 1 2 ψ 2 1 In the above, we have assumed that none of a vanishes. s 7 Since eϕ is constant, we have now only to evaluate the determinant (pfaffian) of Dˆ . To F proceed, we adopt the following explicit realization of the gamma matrices (just to evaluate the determinants with respect to the spinor indices): σ1 = τ 1 1 1, σ4 = τ τ 1 1, σ7 = τ τ τ τ , 3 2 1 2 2 2 3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ2 = τ 1 1 1, σ5 = τ τ τ 1, σ8 = τ τ τ τ , (3.21) 1 2 2 3 2 2 2 1 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ3 = τ τ 1 1, σ6 = τ τ τ 1, σ9 = τ τ τ τ , 2 3 2 2 1 2 2 2 2 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ0 = σ¯0 = 1, and σ¯a = σa (a = 1,...,9), where τ are the Pauli matrices. With these a − gamma matrices, Dˆ takes the form F ∆+ 0 Dˆ = F 1. (3.22) F 0 ∆ !⊗ −F Denoting by the same symbols the matrices which are obtained by extracting the first two matrices in the tensor products in (3.21), for example, σ1 τ 1, one finds that 3 → ⊗ ∆ = σ¯i∂ Wσ¯012 +β σ¯024 +β σ¯124 +β σ¯034 +β σ¯134, (3.23) ±F i ∓ 1± 2± 3± 4± where 1 1 β = (α α ), β = (α α ), 1 024 135 2 124 035 ± 2MW ± ± 2MW ± 1 1 β = (α α ), β = (α α ). (3.24) 3 034 125 4 134 025 ± 2MW ∓ ± 2MW ∓ From this, it follows that 4 det∆ = (∂2)2 +2W2∂2 +2( β2 )(∂2 +∂2)+4(β β +β β )∂ ∂ (3.25) ±F n τ σ 1 2 3 4 τ σ ± ± ± ± ± n=1 X 4 + ( β2 )2 4(β β +β β )2 +2(β2 β2 β2 +β2 )W2 +W4. n± − 1± 2± 3± 4± 1± − 2± − 3± 4± n=1 X The final one-loop contribution of the fermionic sector is then represented by pfDˆ = det∆+det∆ . (3.26) F F −F Given this result, it would be interesting to generalize the analysis in [30, 32] to the generic three-spin constant radii solution. Change of functional bases As in the case of DB , it turns out that the form of det∆ in (3.25) is not convenient mn −F for our purpose. Thus, we make a change of bases again. To this end, we introduce a 4 4 × matrix R(α) = 1 P(τα), (3.27) ⊗ 8 with P(x) given in (3.11). One then finds that detR 1(α)∆ R(α) is given just by replacing − ±F β in (3.25) with β α. With this in mind, we define 3 3 ± ± − ∆ˆ R 1(β )∆ R(β ), (3.28) ±F ≡ − 3± ±F 3± so that β in (3.25) are set to be zero. This transformation for ∆+ is not inevitable for our 3 F ± purpose, but simplifies later computations. Now, we focus on the J = J three-spin solution. In this case, 1 2 c m(κ ν) w(ν κ) γ β = ± , β = ∓ , β = β = 0, (3.29) 1 3 2 4 ± − 2W ± 2W ± ± and hence det∆ˆ = (∂2)2 +2W2∂2 +2β2 (∂2 +∂2)+(β2 +W2)2. (3.30) ±F 1 τ σ 1 ± ± One can confirm that the original determinants in (3.25) with (3.29) reproduce the characteristic frequencies obtained in [29].2 4 One-loop effective action and large J expansion Inthissection, basedontheresultsinsection3,weconsidertheone-loopeffectiveactionof the GS string in the J = J three-spin background. For this background, the corresponding 1 2 gauge theory operators have been identified [9], and there exit some parameter regions where thefluctuationsarestable[20,29]. WedevelopalargeJ (totalspin)expansionoftheone-loop effective action, and compute it in a closed form up to and including (1/J). O Collecting the contributions from the bosonic, the fermionic and the ghost sectors, the one-loop effective action is given by Det(∆ˆ+)Det(∆ˆ )Det( ∂2) eiΓ(1) = F −F − , (4.1) Det21(DB)Det21(DˆB ) pq mn where Det stands for the functional determinant. To develop a large J expansion, we expand the fluctuation operators with respect to ν with ∂ and m fixed: i ∂2 +κ2 = (∂2 +ν2)+2M2, ∂2 detDˆB /∂2 = (∂2 +ν2)4 +2 (2 s2)+2s2 σ m2(∂2 +ν2)3 (4.2) mn − γ γ∂2 h i + 8c2m2∂2(∂2 +ν2)2 + , γ σ ··· det∆ˆ = (∂2 +ν2)2 +2M2∂2 +2β2 (∂2 +∂2)+2ν2(M2 +β2 )+ . ±F 1 τ σ 1 ··· ± ± Note that J = Σ J √λν and M2 2m2J /J for large ν. This expansion is also regarded s s 1 ∼ ∼ as that with respect to the power of ∂2+ν2, or the power of winding number m. The validity 2Precisely,theworld-sheetmomentahereareintegermoded,whereasthosein[29]arehalf-integermoded: We have started with an su(2) rotated coordinates [20], and do not need a σ-dependent rotation. 9