Stringy Membranes in AdS/CFT

We study membrane configurations in AdS_{7/4}xS^{4/7}. The membranes are wrapped around the compact manifold S^{4/7} and are dynamically equivalent to bosonic strings in AdS_5. We thus conveniently identify them as"Stringy Membranes". For the case of AdS_7xS^4, their construction is carried out by embedding the Polyakov action for classical bosonic strings in AdS_5, into the corresponding membrane action. Therefore, every string configuration in AdS_5 can be realized by an appropriately chosen stringy membrane in AdS_7xS^4. We discuss the possibility of this being also the case for stringy membranes in AdS_4xS^7/Z^k (k>1 or k = 1). By performing a stability analysis to the constructed solutions, we find that the (membrane) fluctuations along their transverse directions are organized in multiple Lam\'{e} stability bands and gaps in the space of parameters of the configurations. In this membrane picture, strings exhibit a single band/gap structure.


Introduction.
Extended objects such as strings and membranes have played an important role in our understanding of fundamental interactions. Membranes first appeared in the early attempt of Dirac [1] to model an electron by a charged closed membrane and underlay the development of hadronic bag models [2,3] (for a review see [4]). The emergence of Yang-Mills theory as the conceptual foundation of the standard model of strong and electroweak interactions, brought about the still unresolved puzzle of color confinement in QCD. That strings model successfully the almost linear Regge trajectories, does not belittle the virtues of bosonic membranes as effective bags of the QCD vacuum, which also recover successfully the string limit. At this point we should also mention the striking equivalence between regularized spherical bosonic membranes and SU(∞) classical Yang-Mills theory, first observed by Goldstone and Hoppe [5]. 1 Moreover, let us also stress the analogies between the topological structure of Yang-Mills theory (self-duality) and closed bosonic membranes [10,11].
Besides offering a model for the description of elementary particles, quantum relativistic membranes became widely known as 2-dimensional generalizations of strings [12]. In contrast to strings however and due to their lacking a coupling constant, membranes are notoriously non-perturbative objects. As a consequence, systematic perturbative methods have not been developed for membranes, making them less attractive than strings as fundamental building blocks of matter. Indeed, the first superstring revolution underlined the prime role of superstrings as fundamental constituents. Despite that, the M-theory revolution [13,14] and later the "Matrix Theory Conjecture" [15] (see [16] for a review), paved the way for a more "democratic" framework [17] into which strings, membranes and p-branes of various dimensionalities coexist [18]. Indeed, the web of 10-dimensional string theories finds a unifying platform in a single 11-dimensional M-theory whose long-wavelength limit is just 11-dimensional supergravity.
In all of these developments, flat Minkowski spacetimes provided the majority of backgrounds for the study of membranes [19][20][21][22]. The spacetimes AdS 4/5/7 ×S 7/5/4 were known as (maximally supersymmetric) vacua of ten and eleven dimensional supergravity [23,24] but they were only rarely used as membrane backgrounds [25], despite furnishing them with exceptional features such as massless excitations and a discrete spectrum. 2 Moreover, the "membrane at the end of the universe" [26][27][28][29], as the membrane at the boundary of AdS 4 × S 7 spacetime came to be known, seemed to give rise to an OSp (8|4) superconformal field theory. The 1997 AdS/CFT correspondence of Maldacena [30][31][32] (see [33,34] for reviews) grouped many of the deep ideas that were present in 't Hooft's large-N expansion [35], the holographic principle [36,37] or the geometrization of RG flow (see e.g. [38,39]) into a tractable model of gauge/gravity duality.
According to the standard dictionary of the AdS/CFT correspondence, the energy of a state in the bulk of anti-de Sitter space, equals the scaling dimension of its dual CFT operator. In 2002, Gubser, Klebanov and Polyakov (GKP) [41] introduced a method for the calculation of the anomalous dimensions of certain local, gauge-invariant operators of N = 4 super Yang-Mills theory at strong 't Hooft coupling, a regime which is classically inaccessible by ordinary perturbation techniques. Their method consists in studying closed strings that spin inside AdS 5 × S 5 and in calculating their energy in terms of their other conserved charges, such as their spin or angular momentum, semiclassically. To every string state is then assigned an operator of N = 4, SYM, the bare dimension of which is some function of its spin and SU (4) quantum numbers. 3 As the dimensions of the operators typically receive quantum corrections, their renormalized values (anomalous dimensions) at strong coupling are expected to coincide with the corresponding energies of their dual bulk states, as obtained by the semiclassical treatment of GKP.
The GKP string serves as a benchmark of the Maldacena conjecture because it compares the spectra on both sides of the correspondence [33,34], beyond BPS or nearly BPS (BMN) states. The proposal provoked a flurry of research activity (see e.g. [42]). In addition to the AdS 5 /CFT 4 proposal, the role of M-theory in AdS/CFT has been investigated in AdS 7 ×S 4 [43][44][45] and AdS 4 × S 7 /Z k backgrounds [46][47][48]. However, the existence of precise formulations of boundary theories for the AdS 4 class of backgrounds has no match with the AdS 7 ones. In the latter case potential interest arises through the work of Witten [49] who showed that a model for large-N QCD 4 can be obtained by toroidally compactifying the CFT dual of M-theory on AdS 7 ×S 4 . In this sense, the investigation of membrane solutions in these backgrounds gains in importance [51][52][53][54][55][56][57][58][59][60][61][62].
In our present work we construct membrane configurations in stringy disguise, which we conveniently call "stringy membranes". Their essential property is that they are partially wrapped around a compact dimension and reproduce the action, equations of motion and conserved charges of a string. 4 The bonus of this is twofold: firstly, at the level of classical quadratic fluctuations around stringy membrane solutions, we reveal the existence of an infinite set of purely membrane modes, in addition to the expected purely stringy ones. Secondly, just as the AdS 5 /CFT 4 parameter matching affords to strings in the bulk of AdS an effective string tension √ λ, our stringy membranes are similarly endowed with an effective tension √ λ = R √ λ/g s s . 5 In order to construct configurations with the above properties, we embed the bosonic Polyakov action for strings in AdS 5 into the AdS 7 × S 4 membrane action. We demonstrate that every AdS 5 string solution corresponds to a properly constructed membrane of AdS 7 × S 4 and every AdS 4 ⊂ AdS 5 string solution can be written as a membrane of AdS 4 × S 7 /Z k . An advantage of this construction can be seen through the quadratic fluctuation analysis around our specific stringy membrane solutions, which we perform in detail. We find that an independent subset of fluctuations, which is transverse to the direction of the stringy membrane, admits a Lamé multi-band/multi-gap structure, which is characteristic of their membrane nature. In our fluctuation analysis, string excitations are represented by single-band/single-gap configurations, suggesting that our AdS 7 membranes are collective excitations of their AdS 5 stringy counterparts.
Our paper is organized as follows. We begin in section 2 with a brief reminder of the equations that determine the motion of a bosonic membrane in AdS 7 × S 4 . In section 3 we demonstrate how some simple AdS 7 × S 4 membrane ansätze reproduce the action and equations of motion of the following two spinning string configurations of [41]: (I) the AdS 3 closed & folded string and (II) the string that pulsates in AdS 5 . This is not a mere coincidence and we then proceed to prove (in conformity with [64]) that all bosonic string ansätze in AdS 5 that are consistent with the conformal gauge, can be generated by appropriate membrane ansätze in AdS 7 × S 4 . The extension of these considerations to AdS 4 × S 7 /Z k is discussed in sections 3.1 and 3.2. The stability of our solitons is examined in section 4. We discuss our results in section 5. In appendix A we revisit the GKP string configurations (I) and (II) that we use in our paper and in appendix B we briefly discuss Lamé's equation. 4 An interesting, yet questionable by many (since nonlinear sigma models in more than two dimensions are not renormalizable by power counting), application of wrapping is the semiclassical quantization of an 11-dimensional supermembrane that is wrapped around a torus [63]. 5 R is the wrapping radius, gs the string coupling constant and s the fundamental string length.
The bosonic part of the (Howe-Tucker-) Polyakov action [65] for a membrane in D spacetime dimensions, in the presence of a Wess-Zumino flux term is: where P is the Planck length of D-dimensional spacetime, X m the spacetime coordinates, and σ a = {τ, σ, δ} are the membrane/world-volume coordinates (σ, δ ∈ [0, 2π)). On the other hand A mnp (X) is an antisymmetric 3-form tensor field, g mn (X) is the spacetime metric, γ ab the membrane/world-volume (auxiliary) metric and h ab is its induced metric on the membrane world-volume (pull-back): where h ab = γ ab is the equation of motion that is obtained by varying action 2.1 w.r.t. the auxiliary metric γ ab . An especially convenient gauge choice is the following: The Polyakov action 2.1 then becomes: where the Poisson bracket, { , } is defined as: The constraints that follow from fixing the gauge 2.3 are: Let us now briefly consider the general motion of a classical, uncharged (no WZ term) bosonic membrane in AdS 7 × S 4 , as described in the global coordinate system of AdS 7 × S 4 (for AdS 7 × S 4 , it's = 2R. Setting R = 1 implies that = 2): 6 where Y µ and X i are the embedding coordinates of AdS 7 × S 4 (see 4.1-4.2) and ρ ≥ 0, t ∈ [0, 2π) , 7 θ 1 , θ 1 ∈ [0, π], and θ 2 , is: (2.9) Action 2.4 becomes: G AdS mn (y)G AdS pq (y){y m , y p }{y n , y q }− (2.10) while the constraints that follow from fixing the gauge 2.6, 2.7 are (i, j = 1, 2): Action 2.10, and its constraints 2.11-2.12, are invariant under the global isometry SO(6, 2)× SO(5) of AdS 7 × S 4 . The following 28+10 Noether charges are conserved on-shell: (2.14) The charges that correspond to the cyclic coordinates of the action 2.10, t, φ 1 , φ 2 , φ 3 , φ 1 , φ 2 , are simpler in form and can be directly read off from 2.9-2.10: where L stands for the Lagrangian of the system that is defined as S = L dτ .

Spinning Membranes and Spinning Strings.
We shall now show that the folded closed string of [41], rotating in AdS 3 ⊂ AdS 5 , has the same action and equations of motion as a specific membrane soliton that spins in AdS 3 ⊂ AdS 7 × S 4 . This result will later be generalized to any string soliton that lives in pure 8 AdS 5 , for which an equivalent AdS 7 × S 4 membrane soliton will be found. Let us start from the following ansatz for a membrane that rotates in AdS 3 × S 1 ⊂ AdS 7 × S 4 : It reads, in embedding coordinates (R = 1, = 2), The Polyakov action 2.10 and the constraint equation 2.11 become: 9 Action 3.4 and constraint 3.5 are identical to the on-shell string Polyakov action (written in the conformal gauge) and the Virasoro constraint of the AdS 3 folded closed string configuration of [41]. Were it not for a factor of cos 2 θ 1 φ 2 1 , the off-shell action 3.3 would also coincide with the corresponding off-shell stringy action. However-in action 3.3-it is only ρ that has a nonzero equation of motion and that equation of motion is identical to the stringy one, A.5: The conserved charges of the membrane action 3.3 are also identical to the ones obtained 9 In D = 11 dimensions, a simple relation between the Planck length 11, the string coupling constant gs and the string fundamental length can be deduced by dimensionally reducing 11-dimensional supergravity to 10 dimensions, Rc −→ gs = for strings, A.11-A.12 (for ω 2 > 1): Therefore the two systems are dynamically equivalent. Another (string) solution of [41] consists of a closed string that oscillates around the center of AdS 5 . It can also be written in terms of a pulsating, AdS 7 × S 4 membrane as follows: In embedding coordinates, the ansatz reads: and has the following membrane/string Polyakov action and constraint equation: The same comments that were made in the previous case can be repeated here as well. Our stringy membrane is dynamically equivalent to the pulsating string of [41] with identical equations of motion, A.18, A.19 (with w = 1): t cosh 2 ρ + 2ṫρ cosh ρ sinh ρ = 0 (3.14) ρ + sinh ρ cosh ρ ṫ 2 + 1 = 0. (3.15) Now, all of the previous results can be generalized to any 10 string soliton that rotates in AdS 5 and has no rotating counterpart in S 5 (dubbed "pure" solitons for convenience). We thus prove: 3.0.1. Every pure AdS 5 string soliton has an equivalent AdS 7 × S 4 membrane soliton (and not vice versa).
Proof: Start with 2.10 and 2.11-2.12, the membrane Polyakov action in AdS 7 × S 4 (in the gauge, γ 00 = − det h ij , γ 0i = 0, γ ij = h ij ) and its constraint equations: where y m ≡ (t, ρ, θ 1 , θ 2 , φ 1 , φ 2 , φ 3 ) and x m ≡ θ 1 , θ 2 , φ 1 , φ 2 . G mn (y, x) are the components of the metric 2.9. Taking σ as the string world-sheet coordinate, immediately gives: 10 A word of caution is due here. Not all ansätze are consistent with the conformal gauge. The statements herein presented concern string solitons that are compatible with the choice of the conformal gauge in Polyakov action. It would be interesting to be able to generalize them to the case of the Nambu-Goto action as well, i.e. independently of the gauge choice.
Choosing x 3 = φ 1 = δ for the coordinate of S 4 with metric component G S 33 = cos 2 θ 1 , The result 3.0.1 follows upon setting x m =3 = 0, y m>5 = 0, and performing the δ-integration: i.e. a pure AdS 5 string soliton. For comparison, we juxtapose the corresponding AdS 5 × S 5 string Polyakov action in the conformal gauge (γ ab = η ab ) and the corresponding Virasoro constraints: The equations of motion that correspond to θ 1 and φ 1 in 3.26 are trivially satisfied and the remaining equations of motion of 3.26 will be identical to the ones that are obtained by varying the string action 3.29. Thus the two systems are dynamically equivalent.
The not vice versa part in 3.0.1 follows from the fact that we may construct many inequivalent membrane actions with dependence on both σ and δ.

Stringy Membranes in AdS
Going over to the AdS 4 × S 7 case, construction 3.0.1 has to be modified in the following way. Assuming complete dependence of the string's spacetime coordinates on the worldsheet coordinates {τ , σ}, it is only a subset of all possible AdS 5 string solitons that can be obtained from an appropriate membrane ansatz on AdS 4 × S 7 -namely all string solitons that live in AdS 4 ⊂ AdS 5 . For example, both stringy anti-de Sitter solitons encountered in this paper (3.1, 3.9) are of this genre, living in AdS 3 ⊂ AdS 4 ⊂ AdS 5 . Thus, they can be reproduced by an AdS 4 × S 7 membrane: 3.1.1. Every pure string soliton of AdS 4 ⊂ AdS 5 11 has an equivalent AdS 4 × S 7 membrane soliton (and not vice versa).
Dropping the condition of full dependence on the world-sheet coordinates 3.32, it should be possible to apply this method and find, (i) AdS 4,7 × S 7,4 membrane equivalents to special string configurations that live in AdS 5 × S 5 and (ii) AdS 4 × S 7 membranes that are equivalent to strings that live in AdS 5 .

Stringy Membranes in
We can also consider stringy membranes in more general backgrounds, such as AdS 4 × S 7 /Z k . For k = 1, this is just AdS 4 × S 7 . On the other hand, AdS 4 × S 7 /Z k geometries provide the gravitational backgrounds of the ABJM correspondence [47]: For k = 1, the dual gauge theory reduces to N = 8 SCFT. In the case of the SU (2)×SU (2) gauge group, it becomes the N = 8 Bagger-Lambert-Gustavsson (BLG) theory, [46,67]. Now the question has been posed, whether a logarithmic type behavior for the anomalous dimensions of either theory's states is possible within this correspondence as well. Based on what has been said above, the answer is affirmative from the point of view of membranes. 11 AdS4 ⊂ AdS5 means that one of the two azimuthal angles of S 3 of AdS5 is set to zero. 12 Super Chern-Simons theory.
To see this, consider the metric of AdS 4 × S 7 /Z k [28]: It is easy to obtain solutions 3.1 and 3.9 from this metric. All that is needed is to supplement the AdS ansätze with y = kδ (R = 1, = 1/2) and set the six remaining angles of S 7 equal to zero. In fact, one could formulate the following proposition: 3.2.1. Every pure string soliton of AdS 4 ⊂ AdS 5 has an equivalent AdS 4 × S 7 /Z k membrane soliton (and not vice versa).
Of course, more general statements than 3.2.1 exist, since type IIA string theory action on AdS 4 × CP 3 is obtainable from the supermembrane action on AdS 4 × S 7 by double dimensional reduction [68][69][70].
This concludes our presentation of anti-de Sitter space stringy membranes. In the following section we shall examine their stability properties.

Membrane Fluctuations.
Are stringy membranes stable? Intuitively, one would expect that the δ-component of a stringy membrane, that is wound around a great circle of S 4/7 , would be unstable towards a lower energy configuration that is obtained by its collapsing to a point on either pole. This would indeed be the case for the simplest string extending along a great circle of a sphere and has no other dynamical parts [71]. Besides, since stringy membranes share a common Lagrangian and equations of motion with their equivalent strings, they are expected to inherit many of their stabilities/instabilities. Now, unstable strings may be stabilized in a multitude of ways, e.g. by adding more angular momenta [71,72], stable AdS components [73,74], pulsation [75], by orientifold projections [76], or even flux terms [77,78]. Surprisingly enough, even those stringy configurations that are known to possess unstable modes, have been studied and have been proven very useful in the context of AdS/CFT [71,74], as their instabilities are sometimes unseen in the dual gauge sector [80,81]. One possible explanation for this state of affairs is that these solutions can be easily extended to more stable configurations, while preserving their wanted dual gauge theory properties. Generalized, rigorous results (even numerical) concerning stability are however missing at the moment, mainly due to the difficulties that the corresponding analysis presents [82].
Stringy membranes are on the other hand membranes, not strings. We believe that this property may sometimes enhance the stability of the resulting system. For example, a single membrane component that is wound around a sphere has zero surface tension and is thus expected to be stable, in contrast to the similarly wound string that we saw above. Since we are actually proposing a model that attempts to reproduce the behavior of classical strings in AdS 5 , it would be interesting to be able to make concrete statements about its advantages/disadvantages in the domain of stability. Membrane fluctuations in various backgrounds have been studied in [83][84][85][86][87][88][89][90][91].
Interestingly, we shall find that our systems are governed by the Lamé equation. Lamé equations arise when one separates variables in Laplace's equation using an ellipsoidal coordinate system [92]. They belong to the class of the so-called quasi-exactly solvable (QES) systems [93,94], because their solutions may be determined algebraically in some cases [95][96][97][98][99]. Owing to the fact that their stabilities and instabilities are organized in bands and gaps, Lamé systems enjoy a wide range of physical applications: (a) they provide an alternative to the Kronig-Penney model for the motion of electrons in one-dimensional crystals [95,100]; (b) they govern explosive particle production (preheating) due to parametric resonance in post-inflationary cosmology [101][102][103]; (c) they arise in the study of sphaleron fluctuations in the φ 4 [104,105] and 1+1 dimensional abelian Higgs model [106,107]; (d) they are closely related to the spectral curve of SU (2) BPS monopoles [108,109]; (e) they come up in many occasions in supersymmetric quantum mechanics [110][111][112][113], etc. [114][115][116][117][118]. They have also appeared in string fluctuations in anti-de Sitter space [75,119,120]. The examination of stringy membranes in the present work, is suggestive of a much richer Lamé band/gap structure for their fluctuations. We will have more to say about the stabilities and instabilities of stringy membranes at the end of this section.
where Y 0 , X 0 , Λ 0 , Λ 0 is a classical solution that satisfies the equations of motion and constraints 4.4-4.8. The (quadratic) action for the fluctuations is: To lowest order, these fluctuations obey the following equations: and the following constraints (note that our fluctuations live in tangent space): In order to pass from the general case of an M2-brane in AdS p+2 × S q to the general case of a stringy membrane in AdS p+2 × S q (i.e. before considering any particular ansatz) we plug, into the equations of the solutions 4.4-4.8 and those of the fluctuations 4.12-4.15, setting also R = 1. This leads to the following equations of motion, fluctuation equations, and constraints: We could not come up with any coordinate transformation that eliminates this dependence on δ. It seems therefore that the equivalence between stringy membranes and strings cannot be extended beyond leading order.
In order to facilitate further analysis, we shall only study fluctuations along the directions that are transverse to the membrane, i.e. directions for which Y µ 0 = X i 0 = 0. From equations 4.22-4.25, we see that these fluctuations decouple from the ones that take place parallel to the stringy membrane. Having said this, the corresponding equations become:
where ρ (σ) 2 is the σ-periodic and even function 16 (displayed for various ω, in figure 1), The fluctuation equations for the transverse directions Y µ = X i = 0 (4. 29-4.30) can be transformed to the Jacobi form of Lamé's equation [121][122][123][124][125]: so long as we set, 15 Due care should be taken in this subsection, in order to distinguish the elliptic modulus k, from the parameter κ of ansatz 4.31 and the complete elliptic integral of the first kind K. 16 For large enough ω, it turns out that we may approximate ρ 2 = κ 2 · cd 2 κωσ 1/ω 2 ∼ κ 2 cos 2 σ.
In figure 2 we have plotted the potential 4.39 for various values of ρ 0 . The corresponding Lagrange multipliers are given by, and make the following substitutions in 4.34: u = τ · √ cosh 2ρ 0 + K k 2 and z = y µ (τ ) , h = n 2 cosh 2ρ 0 + 2k 2 2 m 2 + 1 , ν (ν + 1) = 4m 2 + 2 , k = sinh ρ 0 √ cosh 2ρ 0 Let us now summarize and discuss our results: we have analyzed the fluctuations of the AdS 7 × S 4 stringy membranes 3.1, 3.9 along their transverse directions Y µ 0 = X i 0 = 0 and have found that they fall under Lamé's equation 4.34. For ν (ν + 1) ∈ R and 0 < k < 1, equation 4.34 always has an infinite set of real eigenvalues a s ν k 2 and b s ν k 2 that correspond to periodic eigenfunctions. 17 These eigenvalues can be classified into four groups, according to the parity (even or odd) and period (equal to 2K or 4K) of their corresponding eigenfunctions (see appendix B). For a generic eigenvalue h (not necessarily of a periodic eigenfunction), Lamé's equation 4.34 is stable iff all corresponding eigenfunctions z (u, h) are bounded, otherwise it is unstable. It turns out that the intervals of stability are determined by the eigenvalues of periodic solutions:  Solutions of Lamé's equation are stable within the above intervals and unstable outside them. The contractions imply that the relative order of the corresponding endpoints is not a priori known and may thus be reversed, for different values of ν ∈ R, s = 0 , 1 , 2 , . . . and k ∈ (0 , 1). Another interesting property of Lamé eigenvalues is known as "coexistence". In short, coexistence implies that ν ∈ N iff Lamé's equation has exactly ν + 1 intervals of stability (bands), following exactly ν + 1 intervals of instability (gaps). See figure 3 for plots of the Lamé bands (colored) and gaps (white) for ν = 1 and ν = 5.
We thus see that the stability of Lamé solutions is organized in ( 5 Discussion. In this paper, we have studied the stringy properties of uncharged bosonic membranes in AdS 7 × S 4 and AdS 4 × S 7 /Z k . We have examined the conditions under which the string sigma model in AdS 5 ×S 5 may be embedded in the membrane sigma model in AdS 4,7 ×S 7,4 . Specifically, we have found that all string configurations of AdS 5 may be reproduced by membranes living in AdS 7 ×S 4 . Moreover, all string solitons that live in AdS 4 ⊂ AdS 5 may be reproduced by membranes of AdS 4 × S 7 . We have also shown how logarithmic scaling violations (i.e. E − S ∼ ln S) for membranes living in AdS 4 × S 7 /Z k may be obtained, generalizing the work of Hartnoll and Nuñez [54].
There's absolutely no magic in obtaining stringy behavior from membranes on AdS m × S n . The corresponding setups are essentially one-dimensional in each of the two product spaces, having no dynamics in one of them (the n-sphere). Viewed together as an ensemble, they have two independent dimensions. Hence their membrane nature. Our treatment is very similar to that of Duff-Howe-Inami-Stelle [64], albeit with a different motivation [54,56,126]. Compared to [64], and apart from considering only bosonic membranes in an AdS m × S n background (i.e. a product of two manifolds), we haven't actually performed a double dimensional reduction (as e.g. in [56]), although it may have seemed so. In this work, we have been primarily interested in the applications of the GKP method. A posteriori, analogous string-membrane reductions could be found in [127][128][129].
Secondly, we have analyzed the stability of stringy membranes in the linearized approximation. We have demonstrated that the similarities between stringy membranes and strings cannot be extended beyond leading order, since the perturbation equations depend on the second world-volume coordinate δ, which cannot be eliminated from the equations themselves. By studying the stability of stringy membranes along their transverse directions we have found that they are governed by Lamé's equation. Therefore, they typically exhibit the standard stability/instability pattern of bands and gaps. Interestingly (for m = 0 in table 1), our analysis recovers the single-band/single-gap structure of the AdS 3 string case [119,120]. At this point, important issues of interpretation arise for both strings and membranes. Firstly, does the Lamé band/gap structure that anti-de Sitter strings and membranes possess, admit a particle interpretation? Moreover, what is the holographic dual of the Lamé instability phenomenon in question? In what follows, we conclude our work with a detailed exposition of our results as well as some prospects for further work on open issues that emerge from them.
• Scaling dimensions and stringy membranes.  [54] and in complete analogy with the stringy case [41], this membrane configuration is expected to correspond to twist-2 gauge theory operators, with anomalous scaling dimensions given, at strong coupling, by the equations of the corresponding leading Regge trajectories (writing S for the charge S 1 = S 12 in 2.16 and defining √ λ ≡ R 2 /g s 3 s ): At the classical level, it is rather easy to obtain the full "short" series whereas finding the "long" series presents more challenges. A method that potentially generates all of the subleading "long" terms was presented in [130], along with the proof of a formula that links the expressions for the anomalous dimensions, in the "short" and the "long" regimes (see also appendix A). What is more, the long series was found to satisfy the Moch-Vermaseren-Vogt (MVV) constraints that follow from a "reciprocity", aka "parity-preserving" relation.
Originally proposed by Gribov and Lipatov [131] in the context of deep inelastic scattering (DIS), the so-called "strong" reciprocity relation has been verified for twist-two operators, up to three loops in perturbative QCD [132] and up to four loops in weakly coupled, N = 4, SYM [133,134]. It was claimed in [130] that reciprocity is very likely satisfied by twist-two operators in string perturbation theory as well. As we have just seen, all of these statements naturally carry over to stringy membranes.
On the other hand it is known that the "cusp anomalous dimension" f (λ) receives quantum corrections that are calculated in superstring theory by evaluating the Lamé fluctuation determinants [119]. Since the quadratic supermembrane sigma model on AdS 7/4 × S 4/7 is completely different from the corresponding model of superstrings (we have seen an instance of this in the fluctuations of stringy membranes), we expect that the quantum corrections to the anomalous dimensions of twist-2 operators, as calculated from AdS 7/4 × S 4/7 supermembranes, will differ from the purely stringy ones.
The equations of motion and all constraint equations of stringy membranes 4.20-4.21 are identical to the corresponding equations of strings that rotate in anti-de Sitter space. As such, they may be shown to be equivalent to the generalized sinh-Gordon equation through a reduction of the Pohlmeyer type [135]. In complete analogy with AdS strings, stringy membranes in AdS (2,3,4) thus turn out to be equivalent to the Liouville, sinh-Gordon and B 2 -Toda model respectively (cf. [136][137][138]).
Another outcome of our analysis concerns the dual gauge theories. The generalization of the GKP method to theories which contain extended objects other than strings, offers a method to compare their dual CFTs by means of studying the integrable sectors that stand for terms that are built by permuting trace fields Z and impurities, W, Y, D±.
they probably share in the bulk. That is, useful insights about the behavior of one theory can be extracted by studying a similar sector of the other, despite the fact that the theories might significantly differ or even have different dimensionalities. In the present work, the following dualities that contain states/operators for which ∆−S ∼ ln S were examined: Gauge Theory dual Gravity Theory all possess common integrable sectors, and we would also like to guess that this "AdS family" could contain more members (e.g. QCD, N = 6, quiver Super Chern-Simons [47], N = 1, SYM [54,[139][140][141][142][143], etc.). In another-yet similar-direction, it has been shown in [144] that N = 0 , 1 , 2 , 4, SYM theories all possess a common universal one-loop dilatation operator.
We couldn't think of a more general argument showing that all (super-) string theories that can be formulated on AdS 5 and their dual gauge sectors, are included in an AdS 7 × S 4 (super-) membrane theory and its dual SCFT respectively. Moreover, double dimensional reduction [64], doesn't generally work in the case, membranes/AdS 4,7 × S 7,4 −→ strings/AdS 5 × S 5 , thus we have no a priori reason to expect that string theory on AdS 5 × S 5 is contained in M-theory on AdS 4,7 × S 7,4 . It would, nevertheless, be extremely interesting to investigate the extent up to which the results of Duff-Howe-Inami-Stelle [64] can be applied to the AdS 4,7 × S 7,4 case as well. That is find out which embeddings of the full Green-Schwarz action on AdS 5 × S 5 [145][146][147][148][149][150], into the full supermembrane action on AdS 4,7 × S 7,4 [151][152][153][154] are allowed, much along the (bosonic) lines of the present paper.
Going further, one could attempt to study the difference of the membrane and string Polyakov actions, S 2 − S 1 , in more complex setups. Similarly, one could prove that any membrane soliton may be obtained by going to higher-dimensional extended objects (e.g. a 3-or a 5-brane) living in more spacetime dimensions. In general one could claim that any p-brane soliton, living in pure AdS m , may be obtained from a (p + 1)-brane living in AdS m × S m+n+1−m or a (p + q)-brane living in an adequately generalized spacetime.
A Spinning Strings in AdS 5 × S 5 .
(ii). ω 2 < 1 : Two oppositely oriented rigidly rotating Wilson loops, with This system has two cyclic coordinates-namely t and θ-so that the conservation laws are the following:  The string essentially contains four segments extending between ρ = 0 and ρ = ρ 0 and this accounts for the factor 4 in front of the ρ-integrals. One also has to calculate the length of the string, where κ is a factor needed to fix ρ(σ = π/2) = ρ 0 .
For the case (i) of the closed and folded string with ω 2 > 1, it's ω · tanh ρ 0 = 1 so that the integrals A.7 -A.9 take simpler forms and can be expressed in terms of complete elliptic functions: 20 We have plotted ρ (σ) for various values of ω in figure 4. In figure 5 we have plotted the energy of the string as a function of its spin, E = E(S). 20 Our conventions for the elliptic integrals and elliptic functions follow Abramowitz-Stegun [155]. Following [130] we may also establish a kind of duality between short and long folded closed strings in AdS 3 . To begin, there's a known formula between the complete elliptic integrals of the first and second kinds, namely Legendre's relation (see e.g. [155]): where the arguments of the elliptic integrals k = 1/ω 2 and k = 1/ω 2 satisfy k + k = 1.
We thus see that large values of ω → ∞ ("short" strings) correspond to values of ω → 1 + near unity ("long" strings) and A.13 then provides a map between the corresponding energies and spins. Solving A.11 and A.12 for E(k) and K(k) and substituting in A.13, we get the following duality relation between classical folded short and long strings: where the value of the 't Hooft coupling, λ = 4 /α 2 has also been used. There's yet another useful expression of A.14 in terms of the anomalous dimension γ ≡ E − S, The second string configuration that we will examine consists of a closed string that pulsates at the equator of S 3 of AdS 5 : t = t (τ ) , ρ = ρ(τ ), θ = 0, φ 1 = wσ, φ 2 = 0 × θ 1 = θ 2 = φ 1 = φ 2 = φ 3 = 0 . (A.16) Figure 6. ρ = ρ (τ ) of the pulsating closed string A. 16. In AdS 5 × S 5 embedding coordinates the solution reads: The equations of motion and the Virasoro constraints become: This is an oscillatory time-periodic solution that we have plotted for various ρ 0 in figure 6.

B Lamé's Equation.
We saw in section 4 that the fluctuation equations of stringy membranes 3.1-3.9, can be reduced to the Jacobian form of Lamé's equation, namely, d 2 z du 2 + h − ν (ν + 1) k 2 sn 2 u|k 2 z = 0, (B.1) where ν (ν + 1) ∈ R and 0 < k < 1 [123,125]. The potential of Lamé's equation, sn 2 u|k 2 is a doubly periodic function with (primitive) real and imaginary periods equal to 2K k 2 and 2iK k 2 respectively. It is depicted in figure 7. where n = 0 , 1 , 2 , . . .. The eigenvalues have the following ordering properties [123,125]: a 0 ν < a 1 ν < a 2 ν < a 3 ν . . . , a n ν → ∞ as n → ∞ where the contractions have been included to signify that the relative order of the contracted terms is not generally known and can therefore be reversed, for given values of ν and k 2 . For ν ∈ R, the expression ν (ν + 1) ∈ R is symmetric under ν ↔ −ν − 1 so that, without loss of generality, we may take ν ≥ −1/2 and ν (ν + 1) ≥ −1/4. If further ν ∈ N, then the first 2ν + 1 Lamé functions are polynomials (Lamé polynomials), while the remaining, transcendental solutions coexist, i.e., a n ν = b n ν , for n, ν ∈ N and n ≥ ν + 1. Finally, we will say a few things about the Lamé functions of imaginary periods. We first observe that B.1 has the following symmetry [117,118,123,125]: so that, when z (u) has a real period of 2 p K (p = 1, 2) and satisfies B.1, z (u ) ≡ z (u) will have an imaginary period 2 i p K and will satisfy the transformed equation: It turns out that the duality B.5, interchanges the bands of stability with the gaps of instability, in B.2 [117,118].