The AdS 5 × S 5 mirror model as a string
aa r X i v : . [ h e p - t h ] J a n HU-EP-14/21, HU-MATH-14/12ITP-UU-14/18, SPIN-14/16
The AdS × S mirror model as a string sigma model Gleb Arutyunov ∗ Institute for Theoretical Physics and Spinoza Institute,Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
Stijn J. van Tongeren † Institut f¨ur Mathematik und Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin,IRIS Geb¨aude, Zum Grossen Windkanal 6, 12489 Berlin
Doing a double Wick rotation in the worldsheet theory of the light cone AdS × S superstringresults in an inequivalent, so-called mirror theory that plays a central role in the field of integrabilityin AdS/CFT. We show that this mirror theory can be interpreted as the light cone theory of a freestring on a different background. This background is related to dS × H by a double T duality, andhas hidden supersymmetry. The geometry can also be extracted from an integrable deformation ofthe AdS × S sigma model, and we prove the observed mirror duality of these deformed models atthe bosonic level as a byproduct. While we focus on AdS × S , our results apply more generally. PACS numbers: 11.25.Tq
Integrability has and continues to be of central impor-tance in furthering our understanding of the AdS/CFTcorrespondence [1], giving important insights into finitecoupling quantum field theory. Through fruitful inter-play between results on both sides of the correspondence,remarkable progress has been made in the spectral prob-lem in particular [2]. Namely, we can describe scalingdimensions in planar N = 4 supersymmetric Yang-Millstheory (SYM) at finite ’t Hooft coupling through the cor-responding energy levels of a string on AdS × S , andthese energy levels can be computed exactly thanks tothe integrability of the string [3]. More precisely, underthe assumption that integrability persists at the quantumlevel, the spectrum of the AdS × S superstring can bedetermined by means of the thermodynamic Bethe ansatzapplied to a doubly Wick rotated version of its worldsheettheory [4, 5], as put forward in [6] and worked out in [7–11]. Since the light cone gauge fixed AdS × S string isnot Lorentz invariant however, this double Wick rotationresults in an inequivalent quantum field theory, the so-called mirror theory [7]. The associated mirror transfor-mation also appears extensively in the exact descriptionof polygonal Wilson loops or equivalently planar scatter-ing amplitudes [12–14]. Given the central importance ofthe mirror theory, we would like to elevate it beyond thestatus of a technical tool. This raises a question thathas gone unanswered since the model’s introduction in2007, namely whether the mirror theory itself can arisedirectly by light cone gauge fixing a free string on somebackground. Here we show that this is the case, giv-ing an interesting relation between strings on differentbackgrounds different from other well known dualities.Our results extend to the integrable deformation of theAdS × S superstring of [15], show an intriguing link tode Sitter space, and indicate new backgrounds with hid-den supersymmetry in the associated string sigma model.Our construction is based on a simple observation re- garding the bosonic light cone string action that directlyproduces the desired ‘mirror’ metric for a fairly genericclass of backgrounds. The doubly Wick-rotated lightcone action is obtained by a formal exchange and in-version of the metric components of the two directionsmaking up the light cone coordinates, and a sign flip onthe B field. It is not obvious that the resulting met-ric is part of a string background, but we demonstrateexplicitly that the mirror version of AdS × S is a solu-tion of type IIB supergravity with nontrivial dilaton andRamond-Ramond five-form. The double Wick rotationof the corresponding Green-Schwarz fermions is compat-ible with our considerations, as we have explicitly verifiedat quadratic level and will report on elsewhere [16].Interestingly, the mirror space we obtain from AdS × S is formally related to dS × H by a double T duality,H being the five-dimensional hyperboloid. The spacealso has a curvature singularity, but this is not necessar-ily problematic for the string sigma model. In fact, themirror sigma model inherits the symmetries and in par-ticular the integrability of the light cone AdS × S sigmamodel, hinting at well defined behavior. The bosonic su (2) ⊕ ⊂ psu (2 | ⊕ ⊕ H symmetry of the AdS × S light cone string matches the so (4) ⊕ symmetry of ourbackground ( H is a central element corresponding tothe worldsheet Hamiltonian). The supersymmetry ofthe mirror model is not realized through superisome-tries however, as the mirror background admits no Killingspinors. This is natural because the central element ofthe symmetry algebra of the mirror theory is nonlinearlyrelated to its Hamiltonian ( C ∼ sinh ˜ H ) [7], so that thefull superalgebra should be nonlinearly realized on thefermions of the mirror background.The mirror sigma model’s integrability is not obviousfrom its geometry. Interestingly however, we can obtain(the bosonic part of) this sigma model as a limit of the in-tegrable deformation of the AdS × S coset sigma modelconstructed in [15]. The way to take this limit naturallyfollows when we use the main observation of this letterto prove the mirror duality [17] of these deformed models(at the bosonic level). Provided technical complicationsin extracting the fermions in these deformed models canbe overcome, this relation would manifest classically in-tegrability and κ symmetry of the mirror background. DOUBLE WICK ROTATIONS OF LIGHT CONEGAUGE FIXED STRINGS
We want to understand whether the double Wick ro-tated light cone worldsheet theory of a string on a givenbackground can be realized by light cone gauge fixing astring on another background. At the bosonic level thisworks quite elegantly. We will consider d -dimensionalbackgrounds with coordinates { t, φ, x µ } and metric ds ≡ g mn dx m dx n = − g tt dt + g φφ dφ + g µν dx µ dx ν , the components depending only on the transverse coor-dinates x µ , and B fields that are nonzero only in thetransverse directions. The bosonic string action is givenby S = − T Z d τ d σ ( g mn dx m dx n − B mn dx m ∧ dx n ) , where T is the string tension. To fix a light cone gaugewe introduce light cone coordinates [18] x + = t, x − = φ − t, and fix (in the first order formalism) x + = τ, p + = 1 . The action then takes the form (see e.g. [17]) S = T Z d τ d σ (cid:16) − √ Y + B µν ˙ x µ x ′ ν (cid:17) , where Y = ( ˙ x µ x ′ µ ) − ( ˙ x µ ˙ x µ − g tt )( x ′ ν x ′ ν + 1 /g φφ ) , and dots and primes refer to temporal and spatial deriva-tives on the worldsheet, having rescaled σ → T σ . Wenow observe that a double Wick rotation of the world-sheet coordinates τ → i ˜ σ, σ → − i ˜ τ . gives an action of the same form, with g tt interchangedfor 1 /g φφ and B for − B . Thus we can obtain the doubleWick rotated worldsheet theory also directly by gaugefixing a string on a background with a metric with g tt and 1 /g φφ interchanged and a B field with opposite sign. Note that we generically denote quantities in the doubleWick rotated theory (the mirror theory) by tildes.Taking this construction and feeding it the metric ofAdS × S in global coordinates [19] ds = − (1 + ρ ) dt + dρ ρ + ρ d Ω + (1 − r ) dφ + dr − r + r d Ω , we directly obtain ds = 11 − r ( − dt + dr ) + r d Ω + 11 + ρ (cid:0) dφ + dρ (cid:1) + ρ d Ω , (1)the metric that would result in the (bosonic) mirror the-ory. The transverse directions should not be affected bythis transformation, but for the light cone directions thisis more subtle and the φ direction need not keep its range.For the mirror version of AdS × S we will take φ non-compact in fact, upon considering the relation of our mir-ror space to the spaces appearing in the deformed sigmamodels of [15]. DEFORMED AdS × S COSET SIGMA MODELS
The family of deformed sigma models of [15] can belabeled by a parameter κ ∈ [0 , ∞ ), the undeformedAdS × S string sigma model sitting at κ = 0. Thecorresponding metric is given by [20] ds = − f + ( ρ ) f − ( κ ρ ) dt + 1 f + ( ρ ) f − ( κ ρ ) dρ + ρ d Θ ρ + f − ( r ) f + ( κ r ) dφ + 1 f − ( r ) f + ( κ r ) dr + r d Θ r , where f ± ( x ) = 1 ± x and d Θ is a deformation of thethree-sphere metric in Hopf coordinates d Θ ρ ≡
11 + κ ρ sin ζ ( dζ + cos ζdψ ) + sin ζdψ ,d Θ r ≡
11 + κ r sin ξ ( dξ + cos ξdχ ) + sin ξdχ . The B field is given by B = κ (cid:16) ρ sin 2 ζ κ ρ sin ζ dψ ∧ dζ − r sin 2 ξ κ r sin ξ dχ ∧ dξ (cid:17) , vanishing at κ = 0. The tension can be convenientlyparameterized as T = g p κ . The range of ρ is restricted to [0 , / κ ) to preserve thetimelike nature of t , with a curvature singularity at ρ =1 / κ . At κ = 0 there is no singularity but rather theconformal boundary of anti-de Sitter space at ρ = ∞ .Now let us introduce rescaled coordinates˜ t = κ t, ˜ φ = κ φ, ˜ r = κ ρ, ˜ ρ = κ r, and relabeled coordinates˜ ξ = ζ, ˜ ζ = ξ, ˜ χ i = ψ i , ˜ ψ i = χ i . If we then also introduce ˜ κ = 1 / κ , the metric ds =˜ κ ˜ ds becomes˜ ds = − f + ( ˜ κ ˜ r ) f − (˜ r ) d ˜ t + 1 f + ( ˜ κ ˜ r ) f − (˜ r ) d ˜ r + ˜ r d Θ ˜ r + f − ( ˜ κ ˜ ρ ) f + (˜ ρ ) d ˜ φ + 1 f − ( ˜ κ ˜ ρ ) f + (˜ ρ ) d ˜ ρ + ˜ ρ d Θ ˜ ρ , where now the d Θ factors contain tildes on κ and theangles. Up to the tildes and the factor of ˜ κ this isnothing but the deformed metric we started with, with g tt and 1 /g φφ interchanged. Similarly, the B field preciselypicks up a sign in addition to tildes and a factor of ˜ κ .This factor can now be absorbed in the string tension.Identifying quantities with and without tildes puts ussquarely in the situation of the previous section, provingthat the deformed bosonic light cone theory at tension T ( g ) and deformation value κ is equal to the double Wickrotated theory at tension T ( ˜ κ g ) and deformation value˜ κ . This is precisely the mirror duality observed in [17],which we have now proven at the bosonic level.Just as the undeformed limit κ → × S , the maximally deformed limit ˜ κ → φ coordinate is naturally noncompact. Let usnow discuss the mirror space in more detail. THE AdS × S MIRROR BACKGROUND
We obtained the mirror space (1) directly from AdS × S , but it is also closely related to dS × H . Namely, (byconstruction) our mirror space turns into dS × H uponapplying a timelike T duality in t and a noncompact Tduality in φ . Indeed after a timelike T duality the firstline of eqn. (1) becomes − (1 − r ) dt + dr − r + r d Ω , which is dS in static coordinates. Similarly, T dualityin φ turns the second line into(1 + ρ ) dφ + dρ ρ + ρ d Ω , which is H in analogous coordinates. The timelike andnoncompact nature of these T dualities makes this arather formal relation however. So while our originalstring lived on the maximally symmetric space AdS × S ,its mirrored version does not quite live on the maximallysymmetric space dS × H , but rather its doubly T dualcousin (1).At this stage we should mention that based on simplermodels [21] it was conjectured that the maximal defor-mation limit of the models of the previous section shoulddirectly correspond to dS × H [15]. Indeed, this spacewas already extracted in a different κ → ∞ limit com-bined with two spacelike T dualities [22]. This limit re-quires taking one of the coordinates outside its naturalrange however, thereby changing the coordinate that istimelike, and does not appear to be smoothly connectedto the general deformed geometry. Like dS × H , the re-sulting geometry is formally supported by an imaginaryfive-form flux. Our limit instead yields a geometry thatis smoothly related to the generic case, and naturally re-sults in a real worldsheet theory. We consider our doublyT dual relation to dS × H , albeit part timelike and non-compact, in line with the conjecture of [15]. Our mirrorspace is clearly related to the one obtained in [22] by onetime and three spacelike T dualities.Our mirror space is a product of two five-dimensionalspaces, with curvature R = 4 1 − r − r − ρ ρ , showing a (naked) singularity at r = 1. This means thatin strong contrast to AdS × S , the mirror space is sin-gular. Sigma models on singular backgrounds are notnecessarily ill-defined however, and the integrability ofour string sigma model is actually a promising indicationof good behavior.A more pressing question is that of consistency of ourmirror space as a string background, and we would liketo show that it is part of a solution of supergravity. Asour metric is related to the metric of dS × H by twoT dualities, it is natural to assume that our (type IIB)supergravity solution is supported by only a dilaton Φand a self-dual five-form flux F , just as dS × H formallyis [23]. We could in fact induce our solution from thedS × H one through the T dualities, but let us notassume familiarity with exotic supergravities. We thenhave to solve the equations of motion for the dilaton4 ∇ Φ − ∇ Φ) = R, the metric R µν = − ∇ µ ∇ ν Φ + 14 · e F µρλσδ F ρλσδν , and the five-form ∂ ν (cid:16) √− gF ρλσδν (cid:17) = 0 . The equation for the dilaton is solved byΦ = Φ −
12 log(1 − r )(1 + ρ ) , where Φ is a constant. Although the metric (1) cor-responds to a direct product of two manifolds, the five-form has mixed components, t matching up with the fourtransverse coordinates of the submanifold containing φ and vice versa [24]. Introducing would-be volume forms ω t and ω φ for these two sets of five coordinates containing t and φ respectively, the five-form is given by F = 4 e − Φ ( ω φ − ω t ) . With this solution we can start adding Green-Schwarzfermions to the worldsheet theory in a canonical fashion.As indicated in the introduction, we can match these withan appropriate analytic continuation of the AdS × S fermions at least at the quadratic level [16]. While theunboundedness of the dilaton raises questions regardingthe mirror background in interacting string theory, thisis no immediate problem for the sigma model of a freestring.This background has no conventional supersymmetryas it does not admit Killing spinors. This follows fromthe variation of the dilatino λδ ǫ λ = ∂ µ Φ Γ µ ǫ, which does not vanish for any nonzero chiral spinor ǫ withour dilaton. Nevertheless, the string sigma model on ourmirror background has a hidden form of supersymmetry[25], inherited from the AdS × S string. (SUPER)SYMMETRY OF THE MIRROR MODEL Since double Wick rotations preserve symmetries, ourmodel must have the manifest psu (2 | ⊕ symmetry ofthe light cone AdS × S string. This symmetry need notbe linearly realized however, and in fact the action of thesupercharges cannot be. We can understand this fromthe form of the relevant superalgebras.The on shell symmetry algebra of the light cone AdS × S string is psu (2 | ⊕ ⊕ H , where the central element H corresponds to the worldsheet Hamiltonian. Consideringone of the two copies of psu (2 | Q and Q † satisfy { Q aα , Q † βb } = δ ab R βα + δ βα L ab + 12 δ ab δ βα H , where L and R generate the two bosonic su (2)’s. Wesee that as usual the generators of the superisometriesof the light cone AdS × S string anticommute to thegenerators of isometries, in particular time translations[26]. For the mirror theory we instead have [7] { ˜ Q aα , ˜ Q † βb } = δ ab R βα + δ βα L ab + T δ ab δ βα sinh ˜ H , where the rest of the anticommutators vanish when weinterpret the mirror theory as an on shell string withzero worldsheet momentum. Now were these mirror su-percharges to correspond to linearly realized superisome-tries of a string background, they ought to anticommuteto the associated Hamiltonian, not its hyperbolic sine.Put differently, identifying the mirror Hamiltonian as thegenerator of time translations, we are simply no longerdealing with a Lie superalgebra. Moreover, in the de-formed models the symmetry algebra of the worldsheettheory is expected to be a quantum deformed version of psu (2 | ⊕ , as supported by the S-matrix computationsof [20]. This symmetry does not appear to be realizedgeometrically on the deformed background however, andthere is no reason to assume that it should be linearly re-alized on the worldsheet fermions, even in the maximallydeformed limit where the algebra is no longer deformed.Note that the bosonic symmetry is nonetheless realizedlinearly, cf. the two three-spheres in eqn. (1).We should also briefly address the on and off shell sym-metry algebras of the mirror theory. It is well known thatthe worldsheet symmetry algebra of the AdS × S stringpicks up a central extension C ∼ sin P when going offshell [27], allowing the exact S-matrix to be determined.In a string-based mirror theory, the situation should beas follows. We start with psu (2 | ⊕ ⊕ sinh ˜ H worldsheetsymmetry, where we can match sinh ˜ H with C throughanalytic continuation. Going off shell by relaxing thelevel matching condition, the algebra should pick up acentral extension matching the analytic continuation of H . In this way the off shell symmetry algebras of boththeories, and hence their exact S-matrices, would be re-lated by the expected analytic continuation. OUTLOOK
We have introduced a transformation of backgroundmetrics that allows us to interpret mirror versions oflight cone strings as light cone strings on different back-grounds. We then applied this to AdS × S and gavethe supergravity background in which a free string has alight cone worldsheet theory identical to the AdS × S mirror theory. With this direct interpretation of the mir-ror model we can ask new types of questions, the mainone being whether we can give meaning to this geomet-ric mirror transformation in the context of AdS/CFT.The fact that a double Wick rotation on the worldsheethas an interpretation in terms of free string theory leadsus to wonder whether a ‘similar analytic continuation’can be implemented in planar N = 4 SYM. If possi-ble, the end result should have an interesting relation toour string. It might be fruitful to approach this throughthe deformed sigma models which continuously connectAdS × S and the mirror background, where it may bepossible to implement the deformation (perturbatively)in planar N = 4 SYM. The unbounded dilaton in ourmirror background does not bode well for attempts atextending this beyond the sigma model of a free string(planar gauge theory), but then a double Wick rotationloses its simple physical interpretation on a higher genusRiemann surface to begin with.Our considerations clearly apply to many spaces otherthan AdS × S , though they are of course mainly ofinterest when a mirror model comes into play [28]. Ourprocedure for example directly applies to AdS × S × T ,and AdS × S × M supported by pure Ramond-Ramondfluxes. More interesting are AdS × CP (see e.g. [29]),and AdS × S × M supported by mixed fluxes [30], sinceits metric and B field respectively do not fit the simpli-fying assumptions of this letter. We will relax these as-sumptions when discussing the worldsheet fermions [16].Returning to our mirror background, it would be inter-esting to investigate possible consequences of its singularnature, starting for example with an analysis of classicalstring motion in our mirror space. Also, extracting theexplicit fermionic couplings in the deformed coset sigmamodels is a complicated but relevant problem which maysimplify our limit. Finally, while we now see that thedeformed geometry interpolates between two solutions ofsupergravity, demonstrating that it is one at any κ re-mains an interesting open question.We would like to thank R. Borsato, S. Frolov, B. Hoare,G. Korchemsky, M. de Leeuw, J. Maldacena, T. Mat-sumoto, A. Tseytlin, S. Vandoren, and B. de Wit for dis-cussions. S.T. is supported by the Einstein FoundationBerlin in the framework of the research project ”Gravita-tion and High Energy Physics” and acknowledges furthersupport from the People Programme (Marie Curie Ac-tions) of the European Union’s Seventh Framework Pro-gramme FP7/2007-2013/ under REA Grant AgreementNo 317089. G.A. acknowledges support by the Nether-lands Organization for Scientific Research (NWO) underthe VICI grant 680-47-602. The work by G.A. is also apart of the ERC Advanced grant research programme No.246974, “Supersymmetry: a window to non-perturbativephysics” and of the D-ITP consortium, a program of theNWO that is funded by the Dutch Ministry of Education,Culture and Science (OCW). ∗ [email protected]; Correspondent fellow at SteklovMathematical Institute, Moscow. † [email protected][1] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998),hep-th/9711200.[2] For reviews see [31, 32].[3] I. Bena, J. Polchinski, and R. Roiban, Phys. Rev. D69 ,046002 (2004), hep-th/0305116.[4] A. B. Zamolodchikov, Nucl. Phys.
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