Quantum AdS_5 x S^5 superstring in the AdS light-cone gauge
aa r X i v : . [ h e p - t h ] D ec Imperial-TP-RR-02-2009
Quantum
AdS × S superstringin the AdS light-cone gauge
S. Giombi, a, R. Ricci, b, R. Roiban, c, A.A. Tseytlin b, and C. Vergu d, a Center for the Fundamental Laws of Nature, Jefferson Physical Laboratory,Harvard University, Cambridge, MA 02138 USA b The Blackett Laboratory, Imperial College, London SW7 2AZ, U.K. c Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA d Physics Department, Brown University, Providence, RI 02912, USA
Abstract
We consider the
AdS × S superstring in the light-cone gauge adapted to a mass-less geodesic in AdS in the Poincar´e patch. The resulting action has a relativelysimple structure which makes it a natural starting point for various perturbativequantum computations. We illustrate the utility of this AdS light-cone gauge ac-tion by computing the 1-loop and 2-loop corrections to the null cusp anomalousdimension reproducing in a much simpler and efficient way earlier results obtainedin conformal gauge. This leads to a further insight into the structure of the super-string partition function in non-trivial background. [email protected] [email protected] [email protected] Also at Lebedev Institute, Moscow. [email protected] Cristian [email protected]
Introduction
The superstring theory in
AdS × S [1] has a complicated form and the problem of findingits exact quantum spectrum appears to be a non-trivial one. This is a Green-Schwarz [2]type theory, so a natural way to address the question of its quantization is to use, as inflat space, a light-cone type gauge.There are two natural choices of light-cone gauge in AdS corresponding to the twoinequivalent choices of a massless geodesic: (1) the one running entirely within AdS or(2) the one wrapping a big circle of S . The latter choice corresponds to expanding nearthe “plane-wave” vacuum [3, 4, 5, 6, 7, 8] and it was widely used in recent studies of theAdS/CFT duality as it is related to the natural “ferromagnetic” (or “magnon”) spin chainvacuum on the gauge theory side (see, e.g., [9, 10] for reviews). The resulting superstringaction has a rather involved non-polynomial form and thus is not a simple starting pointfor “first-principles” quantization.The former choice of light-cone gauge [11, 12], in which the massless geodesic runsentirely within the (Poincar´e patch of) AdS , leads to a simpler action containing termsat most quartic in fermions. However, the importance of the corresponding light-conevacuum state on the gauge-theory side is not immediately clear; for this reason this light-cone gauge choice received previously less attention (see, however, [13, 14, 15, 16, 17, 18]).In particular, there were practically no studies of the corresponding quantum theory.The aim of the present paper is to initiate the exploration of the AdS light-cone gaugeaction at the quantum level. We shall demonstrate that it leads to a consistent definitionof the quantum superstring theory by repeating the computations of the 1-loop [6, 19]and 2-loop [20, 21] corrections to the anomaly of the null cusp Wilson line [22] or to theleading term in the large spin expansion of the energy of the folded spinning string in AdSspace in global coordinates. We shall reproduce the previous results in a much simplerway, thus providing evidence for the utility of this light-cone gauge. We shall verify, inparticular, the cancellation of the 1-loop and 2-loop UV divergences. The cusp anomalycoefficients we will find match the strong-coupling Bethe ansatz [23] predictions [24, 25];this provides evidence for the quantum integrability of the superstring formulated in the The previous computations for the null cusp were carried out in the conformal gauge where, incontrast to the AdS light-cone gauge, the bosonic fluctuations mix in a nontrivial way leading to anoff-diagonal propagator and thus complicating the analysis beyond the 1-loop level. In the companion paper [27] we shall use this approach to evaluate the 2-loop correctionto the energy of the folded spinning string with an extra orbital momentum J in S [6] inthe scaling limit when ln S ≫ J √ λ ln S =fixed). Our light-cone gauge result, which isdifferent from the one found using the conformal gauge [28], turns out to be in completeagreement with the Asymptotic Bethe Ansatz calculation of [29] and with the result of[30] which generalizes the connection [31] between the scaling function and the O (6) sigmamodel.A potential future application of the AdS light-cone gauge approach, which motivatesour present interest in it, is the study of its near-flat-space expansion aimed at construct-ing the inverse string tension expansion of the energies of quantum string states withfinite quantum numbers (cf. [32]). There are several complications along the way. Oneof them is the mixing of the “center-of-mass” or superparticle [33, 18] modes with theoscillation string modes (they do not decouple in curved space). Another thorny issueis the realization of the superconformal algebra on excited quantum string states. Whilerepresentations will be classified by the same quantum numbers as for string in AdS globalcoordinates, the use of Poincar´e coordinates provides new possibilities for constructing theexcited string modes and realize the superconformal algebra. Indeed, the AdS light-conegauge action of [11, 12] is constructed in the Poincar´e patch and its Hamiltonian P − ,whose eigenstates are the quantum string states, is not directly related to the global AdSenergy E (moreover, the two operators do not commute). We hope to return to theseissues in the future.The main part of this paper is organized as follows. In section 2 we shall review the light-cone gauge action of [11, 12] and discuss its simplest “ground state” solution correspondingto the massless geodesic in
AdS in Poincar´e patch. Expanding near this ground state onefinds the same small fluctuation spectrum – 8+8 massless bosonic+fermionic degrees offreedom – as in flat space and the corresponding partition function is trivial.In section 3 we shall consider a non-trivial solution of the light-cone gauge actionrepresenting an open-string euclidean world surface ending on a null cusp on the boundary The classical integrability of the
AdS × S superstring [26] on the space of physical degrees of freedomin the AdS light-cone gauge was discussed in [15]. A relation between the descriptions of the quantum particle states based on P − and based on E wasdiscussed in [34, 18]. AdS [22]. This is still a simple “homogeneous” solution – the coefficients in thelight-cone gauge action expanded near it turn out to be constant. As was argued in[19], this world surface is the same (up to an SO (2 ,
4) transformation and a euclideancontinuation) as the one describing the asymptotic large spin limit [6, 35] of the foldedspinning string [5] when the folds approach the boundary. This relation identifies, from astring theory standpoint, the anomaly of a null cusp Wilson line and the large spin limitof the anomalous dimension of twist-2 operators, the former being given by the stringpartition function in the null cusp background. Expanding the (euclidean analog of the)light-cone gauge action near this null cusp solution we find the same small fluctuationspectrum as in [6, 19] and thus the same 1-loop correction to the cusp anomaly function.In section 4 we shall extend the computation of the string partition function in the nullcusp background to 2-loop order. An important difference compared to the correspondingconformal gauge computation [20, 21] is that one of the massive bosonic fluctuations ac-quires a nontrivial (and divergent) 1-point function through a tadpole graph with a singlefermionic loop. As a result, there are non-vanishing connected but non-1PI contributionsto the 2-loop partition function. Summing them together with the 1PI contributionsleads to cancellation of all UV divergences and reproduces the Catalan’s constant coeffi-cient in the 2-loop cusp anomaly found earlier [20, 21] by a substantially more involvedcomputation in the conformal gauge.
Let us begin with a review of the structure of the
AdS × S action in the AdS light-conegauge [11, 12].We will use the AdS × S metric in the Poincar´e patch ( m = 0 , , , M = 1 , ..., ds = z − ( dx m dx m + dz M dz M ) = z − ( dx m dx m + dz ) + du M du M , (2.1) x m x m = x + x − + x ∗ x , x ± = x ± x , x = x + i x , (2.2) z M = zu M , u M u M = 1 , z = ( z M z M ) / ≡ e φ . (2.3)As discussed in [11, 12], starting with the action of [1] in the above coordinates and fixingthe κ -symmetry light-cone gauge Γ + ϑ I = 0 on the two 10-d Majorana-Weyl GS spinors ϑ I , one may also choose the following analog of the conformal gauge √− g g αβ = diag( − z , z − ) . (2.4)4ince the resulting action contains x − only in the √− gg αβ ∂ α x + ∂ β x − term it admits asimple solution x + = p + τ , (2.5)which thus can be consistently imposed as a constraint additional to (2.4) to completelyfix the two-dimensional diffeomorphism invariance. With this choice x − decouples fromthe action (it may be determined from the equations of motion for g αβ or the analog ofthe Virasoro constraints where it appears only linearly). The resulting AdS light-cone gauge action may be written as S = T Z dτ Z πℓ dσ L , T = R πα ′ = √ λ π , (2.6) L = ˙ x ∗ ˙ x + (cid:0) ˙ z M + i p + z z N η i ( ρ MN ) ij η j (cid:1) + i p + ( θ i ˙ θ i + η i ˙ η i + θ i ˙ θ i + η i ˙ η i ) − ( p + ) z ( η i η i ) − z ( x ′∗ x ′ + z ′ M z ′ M ) − h p + z z M η i ( ρ M ) ij (cid:0) θ ′ j − i z η j x ′ (cid:1) + p + z z M η i ( ρ † M ) ij (cid:0) θ ′ j + i z η j x ′∗ (cid:1)i . (2.7)Here the θ i = ( θ i ) † , η i = ( η i ) † ( i = 1 , , ,
4) transform in the fundamental representationof SU (4) and parametrize the physical fermionic degrees of freedom (the remaining partsof the two ten dimensional Majorana-Weyl spinors in the original GS action). Thematrices ρ Mij are the off-diagonal blocks of the Dirac matrices in six dimensions in chiralrepresentation and ρ MN = ρ [ M ρ † N ] are the SO (6) generators (see Appendix A).The action has manifest SO (6) or SU (4) symmetry. It is quartic in the η -fermions andquadratic in the θ -fermions. As in the flat space light-cone gauge GS action, the factorsof p + can be absorbed by rescaling the fermions θ i and η i ( p + will still appear in theexpressions for conserved charges). For generality we introduced the parameter ℓ in therange of σ . For example, if we consider closed string case with the world-sheet topology of acylinder, before fixing any gauge we can always set ℓ = 1 by a coordinate transformation. In general, as in flat space case, the knowledge of x − is still required to construct the charges of thesymmetry algebra and vertex operators. Here, however, we will consider an observable that is determinedjust by the light-cone gauge action that does not contain x − . Here † stands for hermitian conjugation on the Grassmann algebra, i.e. fermions are complex. The above action is related to the one in (1.4),(1.5) in [12] by τ → ( p + ) − τ . It is also related to theaction in (5.29) in [13] by σ → T − σ . Another choice is to rescale τ and σ to set x + = τ and ℓ = p + as in the discussion of string interactionsin flat space, see also [12]. x m → kx m , z M → kz M . In thegauge-fixed action (2.7) this symmetry becomes x → kx, z M → kz M , σ → k − σ, ℓ → k − ℓ, so we can still set ℓ = 1 by such a rescaling. We can also consider the open stringcase defined on a strip or a half-plane (in the latter case ℓ = ∞ ).In the next two sections we shall consider the string path integral with the two-dimensional euclidean version of the action (2.7), i.e. with e i S = e − S E . The euclideanaction can be formally obtained from (2.7) by replacing σ → i σ (and assuming that ℓ = ∞ ). Setting p + = 1 leads to the action S E = T Z dτ Z ∞ dσ L E , (2.8) L E = ˙ x ∗ ˙ x + (cid:0) ˙ z M + i z z N η i ( ρ MN ) ij η j (cid:1) + i (cid:0) θ i ˙ θ i + η i ˙ η i + θ i ˙ θ i + η i ˙ η i (cid:1) − z ( η i η i ) + 1 z ( x ′∗ x ′ + z ′ M z ′ M )+ 2i h z z M η i ( ρ M ) ij (cid:0) θ ′ j − i z η j x ′ (cid:1) + 1 z z M η i ( ρ † M ) ij (cid:0) θ ′ j + i z η j x ′∗ (cid:1)i . (2.9)Dropping all σ -derivatives in (2.7) gives the light-cone Lagrangian for the AdS × S superparticle. When quantized [33, 12, 18], it reproduces the spectrum of IIB supergravityon AdS × S .The action (2.6),(2.7) has a natural “ground state” – the classical solution z = a = const , (2.10) x + = p + τ , x − = 0 , x , x , θ, η = 0 . (2.11)This solution – which is the direct counterpart of the point-like limit of the superstring inflat space – describes a massless geodesic parallel to the boundary of the Poincar´e patchrunning at a distance a from it. It reaches the boundary at spatial infinity ( x = ∞ ).The case of a = ∞ corresponds to the massless geodesic passing through the horizon orthe center of AdS in global coordinates. In global coordinates this massless geodesic isan arc that reaches the boundary of AdS (and then reflects back).To describe fluctuations near the solution (2.10) we may set z M = z M + ˜ z M , z M = e φ u M = a (0 , , , , , , (2.12)and then the quadratic fluctuation term in (2.7) will take the form (we rescaled thefermions by p + ) L = ˙ x ∗ ˙ x + ˙˜ z M ˙˜ z M − a − ( x ′∗ x ′ + ˜ z ′ M ˜ z ′ M )6 h i( θ i ˙ θ i + η i ˙ η i ) − a − η i ( ρ ) ij θ ′ j + h . c . i , (2.13)where ρ plays the role of the charge conjugation matrix (see [12]). This Lagrangiandescribes a collection of 8+8 massless excitations, i.e. it is exactly the same action that onefinds from flat space GS action when using a similar parametrization of the 16 fermioniccoordinates; the only difference is the presence of the “velocity of light” factor c = a − . Since the fluctuation spectrum contains 8 massless 2d bosons and 8 massless 2d fermionsthe 1-loop string partition function or 1-loop correction to the 2-d energy vanishes [36],in agreement with the fact that the massless geodesic should represent a BPS state.Including quartic interaction terms in (2.13) one may check that the string partitionfunction remains trivial also at the 2-loop order.Let us comment on the values of conserved charges on the solution (2.10). The expres-sions for the superconformal charges that correspond to the light-cone gauge action (2.7)were given in [11, 12] and for (2.10) we find that the only non-zero charge densities are P + = p + , K + = − a p + . (2.14)Here K + represents a component of the special conformal generator K m of SO (2 , a = 0.In general, using the global embedding coordinates ( η AB X A X B = − X + X + X + X + X − X = −
1) a massless geodesic in
AdS is described by (see, e.g., [36]): X A = N A + M A τ , η AB M A M B = η AB N A M B = 0 , η AB N A N B = − SO (2 ,
4) angular momentum is S AB = N A M B − N B M A . In particular, thechoice when the motion is along the third spatial direction and z = a = 1 correspondsto N A = (0 , , , , , , M A = ( p, , , p, , S = S = p . The relation between S AB generators and standard basis of conformal group generators on R , is as follows: S m = ( K m − P m ) , S m = ( K m + P m ) , S = D, L mn = S mn ( m, n = 0 , , , E = S = ( K + P ). In the present case then K m = P m ( m = 0 , P = − P = p and thus, up to a trivial rescaling, this is the same as in(2.14). Thus the global AdS energy here is same as the Poincar´e patch one, E = P = P ,but since K m is non-zero this does not represent a conformal primary state. It can be absorbed by rescaling σ → a − σ and then will appear in front of the action together withstring tension T as T a − and also will rescale the length of the cylinder: ℓ → a ℓ . We use that here x − = 0. The charge densities are constant so when integrated they will have aprefactor 2 πT ℓ = √ λ ℓ which we omit here. Expansion near null cusp background
Let us now turn to another simple but less trivial solution of the (euclidean) superstringaction for which the fluctuation spectrum is massive and the full fluctuation Lagrangianhas constant coefficients – the null cusp background [22, 19]. Starting with (2.9) onefinds z = r τσ , x = x = 0 . (3.1)In addition we have (we set p + = 1) x + = τ , x − = − σ , x + x − = − z . (3.2)This solution is describing a euclidean world surface of an open string ending on the AdSboundary (we assume that τ and σ change from 0 to ∞ ). Since x + x − = 0 at z = 0 thissurface ends on a null cusp.Our aim will be to compute the expectation value of the corresponding Wilson looprepresented [37, 38] by the euclidean AdS × S string path integral with the null cuspboundary conditions hW cusp i = Z string = Z [ dxdzdθdη ] e − S E . (3.3)The semiclassical computation of this path integral is based on expanding near the so-lution (3.1). An important feature of this expansion is that it is possible to choose the As was already mentioned above, the null cusp solution is related [19] by an analytic continuationand a global conformal transformation to the infinite spin limit of the folded string solution [5] which isindeed a “homogeneous” solution [35]. We thank Martin Kruczenski for informing us about this form of the null cusp solution in the AdSlight-cone gauge. Here the analog of the Virasoro constraint gives ( ˙ x + x ′− + ˙ x − x ′ + ) + ˙ zz ′ = 0. Let us mentionthat there is an even simpler solution – a (euclidean) surface ending on a straight line at the boundary, x + = τ, x − = b τ, z = bσ , σ ∈ (0 , ∞ ). In this case the string partition function should be trivial,i.e. equal to 1 [38, 12]. While simple at first sight, this solution is however not homogeneous – i.e. thecoefficients of the action of small fluctuations are functions of worldsheet coordinates. Thus, carrying outhigher order perturbative calculations seems quite involved. Here it is assumed that x + and x − are already integrated out using the light-cone gauge conditions.Starting, say, with the Nambu version of superstring action and imposing the orthogonal-gauge conditionson the induced metric (2.4) as well as (2.5) as δ -function conditions in the path integral one may thenget rid of the integrals over x + and x − . τ, σ ). Namely, let us define the stringcoordinate fluctuations by z = r τσ ˜ z , ˜ z = e ˜ φ = 1 + ˜ φ + . . . , z M = r τσ ˜ z M , ˜ z M = e ˜ φ ˜ u M (3.4)˜ u a = y a y , ˜ u = 1 − y y , y ≡ X a =1 ( y a ) , a = 1 , ..., , (3.5) x = r τσ ˜ x , θ = 1 √ σ ˜ θ , η = 1 √ σ ˜ η . (3.6)A further redefinition of the worldsheet coordinates ( τ, σ ) → ( t, s ) (we will denote by( p , p ) the corresponding two-dimensional momenta, i.e. ( p , p ) = − i( ∂ t , ∂ s )) t = ln τ , s = ln σ , dtds = dτ dστ σ , τ ∂ τ = ∂ t , σ∂ σ = ∂ s . (3.7)leads then to the following euclidean action (2.8), (2.9): S E = T Z dt Z ∞−∞ ds L , (3.8) L = (cid:12)(cid:12) ∂ t ˜ x + ˜ x (cid:12)(cid:12) + 1˜ z (cid:12)(cid:12) ∂ s ˜ x − ˜ x (cid:12)(cid:12) (3.9)+ (cid:0) ∂ t ˜ z M + ˜ z M + i˜ z ˜ η i ( ρ MN ) ij ˜ η j ˜ z N (cid:1) + 1˜ z (cid:0) ∂ s ˜ z M − ˜ z M (cid:1) + i(˜ θ i ∂ t ˜ θ i + ˜ η i ∂ t ˜ η i + ˜ θ i ∂ t ˜ θ i + ˜ η i ∂ t ˜ η i ) − z (˜ η ) + 2i h z ˜ η i ( ρ M ) ij ˜ z M ( ∂ s ˜ θ j − ˜ θ j − i˜ z ˜ η j ( ∂ s x − x ))+ 1˜ z ˜ η i ( ρ † M ) ij ˜ z M ( ∂ s ˜ θ j − ˜ θ j + i˜ z ˜ η j ( ∂ s x ∗ − x ∗ )) i . (3.10)Given that the coefficients in the fluctuation action are constant, we should find for thepartition function in (3.3) Z string = e − W , W = W + W + W + ... = f ( λ ) V , (3.11) V = V , V ≡ Z dt Z ds , (3.12)where W = S E is the value of the classical action on the solution and W , W , ... arequantum corrections. The cusp anomaly function f ( λ ) has thus the following inversestring tension expansion f ( λ ) = √ λπ h a √ λ + a ( √ λ ) + a ( √ λ ) + · · · i . (3.13) The presence of extra in the volume factor is due to our choice of unit of scale, see below.
9o compute the 1-loop coefficient a let us consider the quadratic part of the fluctuationLagrangian which identifies the spectrum of excitations L = ( ∂ t ˜ φ ) + ( ∂ s ˜ φ ) + ˜ φ + | ∂ t ˜ x | + | ∂ s ˜ x | + | ˜ x | + ( ∂ t y a ) + ( ∂ s y a ) + 2i (˜ θ i ∂ t ˜ θ i + ˜ η i ∂ t ˜ η i ) + 2i ˜ η i ( ρ ) ij ( ∂ s ˜ θ j − ˜ θ j ) + 2i ˜ η i ( ρ † ) ij ( ∂ s ˜ θ j − ˜ θ j ) . (3.14)We thus find the same mass spectrum as in conformal gauge [35, 19], up to normalizationof the mass scale. The bosonic modes are: one field ( ˜ φ ) with m = 1; two fields (˜ x, ˜ x ∗ )with m = ; five fields ( y a ) with m = 0. As in (2.13) (or as in flat space), the fermionsparametrized by θ i and η i have an off-diagonal kinetic operator but now with non-zeromass terms L F = iΘ K F Θ T , Θ = ( θ i , θ i , η i , η i ) ≡ ( θ, θ † , η, η † ) (3.15) K F = p − (i p + ) ρ p − (i p + ) ρ † +(i p − ) ρ p p − ) ρ † i p . (3.16)The matrices ρ (carrying lower indices) and ρ † (carrying upper indices) are related asin Appendix A. The determinant of the fermionic kinetic operator is det K F = ( p + ) implying that all 8 physical fermionic degrees of freedom have m = . The equality ofmasses of all the fermionic modes is required by the SO (6) symmetry of the null cuspbackground [31].Having the same mass spectrum implies the same (UV finite) result for the 1-loop Here we used that θ i = θ † i , η i = η † i and ignored a total derivative term. Here the classical solution (3.1) is z = e ( t − s ) which differs by a rescaling of s and t from the formused in [19]. The mass scales in the light-cone and the conformal gauges are related as m . c . = m . . It is interesting to note that the analogs of the first three modes in the closed string picture (i.e. forfluctuations near the long folded spinning string [5] in
AdS part of AdS ) are the angular AdS mode“transverse” to the profile of the string and the two AdS modes “transverse” to the AdS subspace ofthe solution [6, 35]. Whenever indices on fermions are not written explicitly we will implicitly assume that θ and η carryupper indices while θ † and η † carry lower indices. W = − ln Z = V Z d p (2 π ) h ln( p + 1) + 2 ln( p + ) + 5 ln p − p + ) i = − π V , (3.17)i.e. we get a = − a = − K , K = ∞ X k =0 ( − k (2 k + 1) = 0 . ... . (3.18) An important feature of the light-cone gauge action expanded near the cusp solutionis that the bosonic propagator is diagonal. This is a useful simplification for higherloop calculations as we shall now demonstrate by the explicit computation of the 2-loopcoefficient (3.18) in the cusp anomaly. Finding a amounts to computing all connectedvacuum Feynman diagrams in the background of the null cusp (3.1). We will thus needto expand the light-cone gauge Lagrangian (3.9) to the quartic order. We begin by analyzing the one-particle irreducible contributions to the partition function.At 2-loops they correspond to the sunset and double-bubble diagrams, see fig. 1. The Note that with the choice of normalization we use here the light-cone and the conformal gauge volumefactors are related by V = 4 V conf . . Let us note that in [20] the coefficient a was calculated in the conformal gauge by using a T-dualversion of the AdS × S action in the Poincar´e patch coordinates. This approach is convenient because theT-dual action is only quadratic in the fermions [39]. To get such an action one must fix the κ -symmetryby choosing the so called S-gauge [11]. One may wonder whether it is possible to combine the virtues ofthe bosonic light cone gauge x + = τ with the simplicity of the T-dual Green-Schwarz action. It turns outthat the choice of the S-gauge is not compatible with the bosonic light-cone gauge. Indeed, the equationof motion for x − would be 0 = d ∗ dx + + ( δ IJ d ∗ + s IJ d ) ¯ ϑ I Γ + dϑ J = ( δ IJ d ∗ + s IJ d ) ¯ ϑ I Γ + dϑ J , which is incontradiction with the fact that in the S-gauge ( δ IJ d ∗ + s IJ d ) ¯ ϑ I Γ + dϑ J = 0. As usual, s IJ = diag(1 , − W = − ln Z string are obtained from W = h S int i − h S i c + · · · , (4.1)where S int is the interacting part of the action (3.8),(3.9) containing cubic and quarticterms. As usual, the Wick contractions are made by inserting the appropriate propagators,and the subscript c indicates that only connected diagrams are to be included. At the2-loop level, the first term in (4.1) gives the double-bubble diagram while the second termgives the sunset diagram as well as the connected graph with two tadpoles which will bediscussed in the next subsection.For the bosonic sunset diagrams we need the following cubic terms from the action, S (3)˜ φ ˜ x ˜ x ∗ = − Z dtds ˜ φ | ∂ s ˜ x − ˜ x | S (3)˜ φ = Z dtds ˜ φ [( ∂ t ˜ φ ) − ( ∂ s ˜ φ ) ] S (3)˜ φy = Z dtds ˜ φ [( ∂ t y a ) − ( ∂ s y a ) ] . (4.2)The fact that the bosonic propagator is diagonal implies that the sunset graph is simplygiven by W . sunset = − h S (3)˜ φ ˜ x ˜ x S (3)˜ φ ˜ x ˜ x + S (3)˜ φ S (3)˜ φ + S (3)˜ φy S (3)˜ φy i . (4.3)All the terms in the above expression can be readily computed. For instance, the firstterm yields, in momentum space, − Z d p d q d r δ (2) ( p + q + r ) G ˜ φ ˜ φ ( p ) (cid:0) q + (cid:1) G ˜ x ˜ x ( q ) (cid:0) r + (cid:1) G ˜ x ˜ x ( r )= − Z d p d q d r δ (2) ( p + q + r ) (1 + 4 q )(1 + 4 r )( p + 1)( q + )( r + ) , (4.4) Here we temporarily set the string tension T to one. In the following, we will sometimes ignore alsothe obvious 2-d volume factor V . The dependence on T and V is easily reinserted at the end of thecalculation. G ˜ x ˜ x ∗ ( p ) = 2 p + , G ˜ φ ˜ φ ( p ) = 1 p + 1 , G y a y b ( p ) = δ ab p . (4.5)To evaluate the momentum integrals, we employ the same regularization scheme used in[20, 21]. Manipulation of tensor structures in the numerators are performed in d = 2, andthe resulting scalar integrals are computed in an analytic (e.g., dimensional) regularizationscheme in which power divergent contributions are set to zero. Namely, we will set Z d p (2 π ) ( p ) n = 0 , n ≥ . (4.6)Introducing the notations [20, 21]I[ m ] = Z d p (2 π ) p + m ) (4.7)I[ m , m , m ] = Z d p d q d r (2 π ) δ ( p + q + r )( p + m )( q + m )( r + m ) , (4.8)we obtain for the above integral Z d p d q d r δ (2) ( p + q + r ) (1 + 4 q )(1 + 4 r )( p + 1)( q + )( r + ) = − I[1 , , ] . (4.9)This integral is proportional to the Catalan’s constant in (3.18) since in generalI[2 m , m , m ] = K8 π m . (4.10)The computation of the remaining contributions is analogous.The second term in (4.3) gives a result proportional to I[1] , while the last term turnsout to vanish. When everything is put together, we obtain the following simple answerfor the bosonic sunset diagram W . sunset = I[1 , , ] + I[1] . (4.11)Note that for non-zero masses the integral I[ m , m , m ] is finite, while I[ m ] is logarith-mically UV divergent. When any of the masses vanishes, both types of integrals exhibitIR singularities. The direct cancellation of these divergences amounts to carefully accounting for the contribution ofthe path integral measure and was shown to occur in the conformal gauge [21]. W . bubble = h S (4) i , (4.12)where S (4) includes the following quartic vertices, S (4)˜ φ ˜ x ˜ x ∗ = 4 Z dtds ˜ φ | ∂ s ˜ x − ˜ x | (4.13) S (4)˜ φ = Z dtds ˜ φ [( ∂ t ˜ φ ) + ( ∂ s ˜ φ ) + 16 ˜ φ ] (4.14) S (4)˜ φ y = Z dtds ˜ φ [( ∂ t y a ) + ( ∂ s y a ) ] (4.15) S (4) y = − Z dtds y a y a ( ∂ t y b ∂ t y b + ∂ s y b ∂ s y b ) . (4.16)It turns out that the only non-vanishing contribution comes from the ˜ φ -interaction, andthe final result is W . bubble = − I[1] . (4.17)Next, let us consider the vertices coming from the fermionic part of the Green-Schwarzaction. For the sunset diagram we need the following cubic interactions, S (3)˜ φ ˜ ηθ = − Z dtds ˜ η i ( ρ ) ij ( ∂ s ˜ θ j − ˜ θ j ) ˜ φ − h . c .S (3) y ˜ η ˜ θ = +i Z dtds ˜ η i ( ρ a ) ij ( ∂ s ˜ θ j − ˜ θ j ) y a − h . c .S (3)˜ x ˜ η ˜ η = + Z dtds ˜ η i ( ρ ) ij ˜ η j ( ∂ s ˜ x − ˜ x ) − h . c .S (3) y ˜ η ˜ η = +i Z dtds ˜ η i ( ρ a ) ij ˜ η j ∂ t y a . (4.18)The fact that the bosonic propagator is diagonal leads to a dramatic reduction of thenumber of possible terms. The fermionic contribution to the sunset diagram is W . sunset = − h S (3)˜ φ ˜ η ˜ θ S (3)˜ φ ˜ η ˜ θ + S (3)˜ x ˜ η ˜ η S (3)˜ x ˜ η ˜ η + S (3) y ˜ η ˜ η S (3) y ˜ η ˜ η + S (3) y ˜ η ˜ θ S (3) y ˜ η ˜ θ + 2 S (3) y ˜ η ˜ η S (3) y ˜ η ˜ θ i . (4.19)As an example of a typical calculation let us detail the analysis of h S (3)˜ φ ˜ η ˜ θ S (3)˜ φ ˜ η ˜ θ i ; Wickcontractions yield the following expression − h S (3)˜ φ ˜ η ˜ θ S (3)˜ φ ˜ η ˜ θ i = − (2i) G ˜ φ ˜ φ ( r ) (cid:2) (i p − )(i q + ) A − ( q + ) B (cid:3) (4.20)where A = Tr h ρ † G ˜ θ † ˜ η † ( p ) ρ † G ˜ θ † ˜ η † ( − q ) + ρ G ˜ θ ˜ η ( p ) ρ G ˜ θ ˜ η ( − q ) i B = Tr h ρ G ˜ η ˜ η † ( p ) ρ † G ˜ θ † ˜ θ ( − q ) + ρ † G ˜ η † ˜ η ( p ) ρ G ˜ θ ˜ θ † ( − q ) i . (4.21)14he fermion propagators appearing in this expression are (proportional to) the relevantentries of the inverse of the fermionic kinetic operator (3.16), and are given by G ˜ θ i ˜ η j ( p ) = − p − i2 p + ρ † , G ˜ θ i ˜ η j ( p ) = − p − i2 p + ρ ,G ˜ θ i ˜ θ j ( p ) = G ˜ η i ˜ η j ( p ) = − p p + . (4.22)After collecting all contributions in (4.19) and reducing them to scalar integrals, the finalresult for the fermionic sunset diagram turns out to be W . sunset = − I[ , , ] + 2 I[ ] + 2 I[ ]I[1] −
52 I[ ]I[0] . (4.23)Finally, we have to include the fermionic contributions to the double-bubble topology. Itis easy to see that the diagram with two fermion bubbles, which, in principle, arises dueto the ˜ η interaction, vanishes, and so do all diagrams with an ˜ η ˜ η -loop. Then the onlynon-trivial contributions come from the following boson-fermion 4-vertices S (4) yy ˜ η ˜ θ = − i2 Z dtds y a y a ˜ η i ( ρ ) ij ( ∂ s ˜ θ j − ˜ θ j ) − h . c . (4.24) S (4)˜ φ ˜ φ ˜ η ˜ θ = +2i Z dtds ˜ φ ˜ η i ( ρ ) ij ( ∂ s ˜ θ j − ˜ θ j ) − h . c . (4.25)After reduction to scalar integrals we obtain the following result W . bubble = − ]I[1] + 52 I[ ]I[0] . (4.26)Thus, the bosonic and fermionic one-particle irreducible contributions, (4.11), (4.12),(4.23) and (4.26), sum up to a divergent 2-loop correction to the partition function. ThisUV divergence is of log -type and thus should be canceled by additional non-1PI connecteddiagram contributions to restore the expected 2-loop finiteness of the superstring theory.This is indeed what happens as we shall see below. So far we have considered only the one-particle irreducible contributions; however, ln Z string receives contributions from all connected graphs. In particular, at two loops we mighthave non-vanishing tadpole diagrams of the topology given in fig. 2. We will see thatsuch tadpole diagrams play an important role for reproducing the 2-loop result for thecusp anomaly found previously in the conformal gauge.15igure 2: The 2-loop tadpole topology. The only non-vanishing contri-bution corresponds to fermionic ˜ θ ˜ η bubbles, represented by dashed lines,connected by the ˜ φ propagator.The (spontaneously broken) symmetries of the theory forbid single-fermion terms fromappearing in the effective action. Thus, only the bosonic fields may exhibit nontrivial1-point functions. Let us begin by discussing the contributions of bosonic loops. Giventhe structure of the bosonic 3-vertices (4.2) and the fact that the bosonic propagator isdiagonal, it is easy to see that all 2-loop graphs with bosonic tadpoles must have ˜ φ as theinner leg, which is at zero momentum by momentum conservation. Moreover, diagramswith a ˜ φ or a y bubble vanish identically by t ↔ s symmetry due to the form of thecorresponding vertices. Thus, the only potentially non-trivial terms come from two ˜ x ˜ x ∗ loops and the S (3)˜ φ ˜ x ˜ x ∗ interaction. Each bubble contributes a factor − Z d p (2 π ) ( p + ) G ˜ x ˜ x ∗ ( p ) = − Z d p (2 π ) p + p + = − Z d p (2 π ) , (4.27)which is zero in our regularization scheme, as explained above. Therefore, we concludethat all bosonic tadpoles vanish.Since all bosonic non-1PI diagrams vanish identically, the total bosonic contributioncomes from summing up the expressions (4.11) and (4.12), i.e. is given by W = 14 2 π √ λ V I[1 , , ] , (4.28)where we have reinstated the explicit dependence on the inverse of the string tension T − = π √ λ and the overall two-dimensional volume factor V .Let us now turn to the analysis of the fermionic tadpoles. Since the ˜ η ˜ η ∗ two-point func-tion G ˜ η ˜ η ∗ ( p ) = − p p + 14 is parity-odd, the bubble integral containing only this propagatorvanishes identically. Thus, the potentially non-trivial tadpoles may come only from thevertices in the first two lines of (4.18). The S (3) y ˜ η ˜ θ vertex, however, leads to a vanishingresult since each bubble is proportional to Tr (cid:2) ρ a G ˜ η ˜ θ ( p ) (cid:3) ∝ Tr ρ a = 0. The remaining16ermionic tadpole with a ˜ φ internal leg is, on the other hand, non-trivial and gives − h S (3)˜ φ ˜ η ˜ θ S (3)˜ φ ˜ η ˜ θ i non-1PI = − (2i) G ˜ φ ˜ φ (0) (cid:16) Z d p (2 π ) (i p − )Tr[ − ( ρ † ) G ˜ θ † ˜ η † ( p ) − ( ρ ) G ˜ θ ˜ η ( p )] (cid:17) , (4.29)which after reduction to scalar integrals yields W . tadpole = − ] . (4.30)This log divergent term is precisely what we need to cancel a similar divergent term in W . sunset , see (4.23). The presence of the tadpole is therefore necessary to guaranteea finite answer for the cusp anomaly.Let us mention that it is the “vacuum-vacuum” transition amplitude or the backgroundpartition function (3.3) that is a physical observable that should be UV finite. As for theeffective action Γ given by the sum of 1PI graphs evaluated in a non-trivial background,it is, in general, UV finite only after a field renormalization. The presence of the tadpolefor ˜ φ means that here one would need such a renormalization to make Γ finite. We do notneed to worry about this renormalization if our interest is to compute the full partitionfunction in (3.3).Combining all the partial results we find the answer for W at 2-loops in light-conegauge, W = W . sunset + W . bubble + W . sunset + W . bubble + W . tadpole = 2 π √ λ V (cid:16) I[1 , , ] − I[ , , ] (cid:17) = −
14 2 π √ λ V I[1 , , ] = − K8 π √ λ V . (4.31)The result is manifestly finite and reproduces the value of a in (3.18). We observe that,as in the conformal gauge calculation of [20, 21], the net effect of the fermions is to changethe sign of the bosonic result for W .We conclude that the AdS light-cone gauge result is thus in perfect agreement with thestring theory computation in the conformal gauge [20, 21] and with the strong-couplingprediction of the Bethe ansatz [25]. 17 cknowledgments We are grateful to M. Kruczenski, T. McLoughlin, R. Metsaev and D. Volin for usefuldiscussions. This work was supported in part by the US National Science Foundationunder DMS-0244464 (S.G.), PHY-0608114 and PHY-0855356 (R.Ro.) and PHY-0643150(C.V.), the US Department of Energy under contracts DE-FG02-201390ER40577 (OJI)(R.Ro.) and DE-FG02-91ER40688 (C.V.), the Fundamental Laws Initiative Fund atHarvard University (S.G.) and the A. P. Sloan Foundation (R.Ro.). It was also supportedby the EPSRC (R.Ri.) and by the PPARC (A.T.). S.G. and R.Ri. would like to thank theSimons Center for Geometry and Physics for hospitality during the 7th Simons workshopon Physics and Mathematics.
Appendix A: Notation
We mostly follow the notation of [11] but define x ± and x, x ∗ without √ factors. Four-dimensional indices (along the AdS boundary) are a, b = 0 , , , SO (6) indices are M, N = 1 , ..., SU (4) indices are i, j = 1 , , ,
4. For the fermionic variables we have θ † i = θ i , η † i = η i , θ ≡ θ i θ i , η ≡ η i η i . The matrices ρ M are off-diagonal blocks of the six-dimensional Dirac matrices in chiralrepresentation: ρ Mij = − ρ Mji , ( ρ M ) il ρ Nlj + ( ρ N ) il ρ Mlj = 2 δ MN δ ij , ( ρ M ) ij ≡ − ( ρ Mij ) ∗ (A.1) ρ MN ij = 12 [( ρ M ) il ρ Nlj − ( ρ N ) il ρ Mlj ] . (A.2)One can choose the following explicit representation for the ρ Mij matrices ρ ij = − − , ρ ij = − i 0 0 00 0 0 − i0 0 i 0 , ρ ij = − − ,ρ ij = − i0 0 i 00 − i 0 0i 0 0 0 , ρ ij = − i 0 0 00 − i 0 0 , ρ ij = − − , As usual, their explicit form is not needed to carry out the calculations described in thetext. We found it convenient however to use at times the representation described here.18 eferences [1] R.R. Metsaev and A.A. Tseytlin, “Type IIB superstring action in AdS(5) x S(5) back-ground,” Nucl. Phys. B , 109 (1998) [arXiv:hep-th/9805028].[2] M. B. Green and J. H. Schwarz, “Covariant Description Of Superstrings,” Phys. Lett. B , 367 (1984).[3] R. R. Metsaev, “Type IIB Green-Schwarz superstring in plane wave Ramond-Ramondbackground,” Nucl. Phys. B , 70 (2002) [arXiv:hep-th/0112044]. R. R. Metsaev andA. A. Tseytlin, “Exactly solvable model of superstring in plane wave Ramond-Ramondbackground,” Phys. Rev. D , 126004 (2002) [arXiv:hep-th/0202109].[4] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, “Strings in flat space and pp wavesfrom N = 4 super Yang Mills,” JHEP , 013 (2002) [arXiv:hep-th/0202021].[5] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A semi-classical limit of the gauge/stringcorrespondence,” Nucl. Phys. B , 99 (2002) [arXiv:hep-th/0204051].[6] S. Frolov and A. A. Tseytlin, “Semiclassical quantization of rotating superstring in AdS(5)x S(5),” JHEP , 007 (2002) [arXiv:hep-th/0204226].[7] C. G. Callan, H. K. Lee, T. McLoughlin, J. H. Schwarz, I. Swanson and X. Wu, “Quantizingstring theory in AdS(5) x S5: Beyond the pp-wave,” Nucl. Phys. B , 3 (2003) [arXiv:hep-th/0307032]. C. G. Callan, T. McLoughlin and I. Swanson, “Holography beyond the Penroselimit,” Nucl. Phys. B , 115 (2004) [arXiv:hep-th/0404007].[8] G. Arutyunov and S. Frolov, “Integrable Hamiltonian for classical strings on AdS(5) x S5,”JHEP , 059 (2005) [arXiv:hep-th/0411089]. G. Arutyunov, S. Frolov, J. Plefka andM. Zamaklar, “The off-shell symmetry algebra of the light-cone AdS(5) x S5 superstring,”J. Phys. A , 3583 (2007) [arXiv:hep-th/0609157].[9] N. Beisert, “The dilatation operator of N = 4 super Yang-Mills theory and integrability,”Phys. Rept. , 1 (2005) [arXiv:hep-th/0407277].[10] G. Arutyunov and S. Frolov, “Foundations of the AdS × S Superstring. Part I,” J. Phys.A , 254003 (2009) [arXiv:0901.4937 [hep-th]].[11] R.R. Metsaev and A.A. Tseytlin, “Superstring action in AdS × S : κ -symmetry lightcone gauge,” Phys. Rev. D , 046002 (2001) [arXiv:hep-th/0007036].[12] R.R. Metsaev, C.B. Thorn and A.A. Tseytlin, “Light-cone Superstring in AdS Space-time,”Nucl. Phys. B , 151 (2001) [arXiv:hep-th/0009171].[13] A.A. Tseytlin, “On limits of superstring in AdS(5) x S5,” Theor. Math. Phys. , 1376(2002) [Teor. Mat. Fiz. , 69 (2002)] [arXiv:hep-th/0201112].
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