Ben Hoare

On deformations of

A bstractWe study the deformed AdS5 × S5 supercoset model of arXiv:1309.5850 which depends on one parameter κ and has classical quantum group symmetry. We confirm the conjecture that in the “maximal” deformation limit, κ → ∞, this model is T-dual to “flipped” double Wick rotation of the target space AdS5 × S5, i.e. dS5 × H5 space supported by an imaginary 5-form flux. In the imaginary deformation limit, κ → i, the corresponding target space metric is of a pp-wave type and thus the resulting light-cone gauge S-matrix becomes relativistically invariant. Omitting non-unitary contributions of imaginary WZ terms, we find that this tree-level S-matrix is equivalent to that of the generalized sine-Gordon model representing the Pohlmeyer reduction of the undeformed AdS5 × S5 superstring model. We also study in some detail similar deformations of the AdS3 × S3 and AdS2 × S2 supercosets. The bosonic part of the deformed AdS3 × S3 model happens to be equivalent to the symmetric case of the sum of the Fateev integrable deformation of the SL(2) and SU(2) principal chiral models, while in the AdS2 × S2 case the role of the Fateev model is played by the 2d “sausage” model. The κ = i limits are again directly related to the Pohlmeyer reductions of the corresponding AdSn × Sn supercosets: (2,2) super sine-Gordon model and its complex sine-Gordon analog. We also discuss possible deformations of AdS3 × S3 with more than one parameter.

AdS_n

Abstract We consider two integrable deformations of 2d sigma models on supercosets associated with AdS n × S n . The first, the “η-deformation” (based on the Yang–Baxter sigma model), is a one-parameter generalization of the standard superstring action on AdS n × S n , while the second, the “λ-deformation” (based on the deformed gauged WZW model), is a generalization of the non-abelian T-dual of the AdS n × S n superstring. We show that the η-deformed model may be obtained from the λ-deformed one by a special scaling limit and analytic continuation in coordinates combined with a particular identification of the parameters of the two models. The relation between the couplings and deformation parameters is consistent with the interpretation of the first model as a real quantum deformation and the second as a root of unity quantum deformation. For the AdS 2 × S 2 case we then explore the effect of this limit on the supergravity background associated with the λ-deformed model. We also suggest that the two models may form a dual Poisson–Lie pair and provide direct evidence for this in the case of the integrable deformations of the coset associated with S 2 .

x

We consider superstring theory on AdS3 ×S 3 ×T 4 supported by a combination of RR and NSNS 3-form fluxes (with parameter of the NSNS 3-form q). This theory interpolates between the pure RR flux model (q = 0) whose spectrum is expected to be described by a (thermodynamic) Bethe ansatz and the pure NSNS flux model (q = 1) which is described by the supersymmetric extension of the SL(2,R) × SU(2) WZW model. As a first step towards the solution of this integrable theory for generic value of q we compute the corresponding tree-level S-matrix for massive BMN-type excitations. We find that this S-matrix has a surprisingly simple dependence on q: the diagonal amplitudes have exactly the same structure as in the q = 0 case but with the BMN dispersion relation e 2 = p 2 +1 replaced by the one with shifted momentum and mass, e 2 = (p±q) 2 +1−q 2 . The off-diagonal amplitudes are then determined from the classical Yang-Baxter equation. We also construct the Pohlmeyer reduced model corresponding to this superstring theory and find that it depends on q only through the rescaled mass parameter, � → p 1 − q 2 �, implying that its relativistic S-matrix is q-independent.

S^n

A bstractWe study the deformed AdS5 × S5 supercoset model of arXiv:1309.5850 which depends on one parameter κ and has classical quantum group symmetry. We confirm the conjecture that in the “maximal” deformation limit, κ → ∞, this model is T-dual to “flipped” double Wick rotation of the target space AdS5 × S5, i.e. dS5 × H5 space supported by an imaginary 5-form flux. In the imaginary deformation limit, κ → i, the corresponding target space metric is of a pp-wave type and thus the resulting light-cone gauge S-matrix becomes relativistically invariant. Omitting non-unitary contributions of imaginary WZ terms, we find that this tree-level S-matrix is equivalent to that of the generalized sine-Gordon model representing the Pohlmeyer reduction of the undeformed AdS5 × S5 superstring model. We also study in some detail similar deformations of the AdS3 × S3 and AdS2 × S2 supercosets. The bosonic part of the deformed AdS3 × S3 model happens to be equivalent to the symmetric case of the sum of the Fateev integrable deformation of the SL(2) and SU(2) principal chiral models, while in the AdS2 × S2 case the role of the Fateev model is played by the 2d “sausage” model. The κ = i limits are again directly related to the Pohlmeyer reductions of the corresponding AdSn × Sn supercosets: (2,2) super sine-Gordon model and its complex sine-Gordon analog. We also discuss possible deformations of AdS3 × S3 with more than one parameter.

supercosets

Abstract The type IIB supergravity AdS 3 × S 3 × T 4 background with mixed RR and NSNS 3-form fluxes is a near-horizon limit of a non-threshold bound state of D5–D1 and NS5–NS1 branes. The corresponding superstring world-sheet theory is expected to be integrable, opening the possibility of computing its exact spectrum for any values of the coefficient q of the NSNS flux and the string tension. In arXiv:1303.1447 we have found the tree-level S-matrix for the massive BMN excitations in this theory, which turned out to have a simple dependence on q . Here, by analyzing the constraints of symmetry and integrability, we propose an exact massive-sector dispersion relation and the exact S-matrix for this world-sheet theory. The S-matrix generalizes its recent construction in the q = 0 case in arXiv:1303.5995 .

On integrable deformations of superstring sigma models related to AdSn×Sn supercosets

We address the question of the exact form of the dispersion relation for light-cone string excitations in string theory in AdS3×S3×T4 with mixed R–R and NS–NS 3-form fluxes. The analogy with string theory in AdS5×S5 suggests that in addition to the data provided by the perturbative near-BMN expansion and symmetry algebra considerations there is another source of information for the dispersion relation – the semiclassical giant magnon solution. In earlier work in arXiv:1303.1037 and arXiv:1304.4099 we found that the symmetry algebra constraints, which are consistent with a perturbative expansion, do not completely determine the form of the dispersion relation. The aim of the present paper is to fix the dispersion relation by constructing a generalisation of the known dyonic giant magnon soliton on S3 to the presence of a non-zero NS–NS flux described by a WZ term in the string action (with coefficient q). We find that the angular momentum of this soliton gets shifted by a term linear in world-sheet momentum p. We also discuss the symmetry algebra of the string light-cone S-matrix and show that the exact dispersion relation, which should have the correct perturbative BMN and semiclassical giant magnon limits, should also contain such a linear momentum term. The simplicity of the resulting bound-state picture provides a strong argument in favour of this dispersion relation.

On string theory on AdS3×S3×T4 with mixed 3-form flux: Tree-level S-matrix

Abstract We construct a two-parameter deformation of the Metsaev–Tseytlin action for supercosets with isometry group of the form G ˆ × G ˆ . The resulting action is classically integrable and is Poisson–Lie symmetric suggesting that the symmetry of the model is q -deformed, U q L ( G ˆ ) × U q R ( G ˆ ) . Focusing on the cases relevant for strings moving in AdS 3 × S 3 × T 4 and AdS 3 × S 3 × S 3 × S 1 , we analyze the corresponding deformations of the AdS 3 and S 3 metrics. We also construct a two-parameter q -deformation of the u ( 1 ) psu ( 1 | 1 ) 2 ⋉ u ( 1 ) ⋉ R 3 -invariant R-matrix and closure condition, which underlie the light-cone gauge S-matrix and dispersion relation of the aforementioned string theories. With the appropriate identification of parameters, the near-BMN limit of the dispersion relation is shown to agree with that found from the deformed supercoset sigma model.

On deformations of AdSn × Sn supercosets

A bstractUsing unitarity methods, we compute, for several massive two-dimensional models, the cut-constructible part of the one-loop 2 → 2 scattering S-matrices from the tree-level amplitudes. We apply our method to various integrable theories, finding evidence that for supersymmetric models the one-loop S-matrix is cut-constructible, while for models without supersymmetry (but with integrability) the missing rational terms are proportional to the tree-level S-matrix and therefore can be interpreted as a shift in the coupling. Finally, applying our procedure to the world-sheet theory for the light-cone gauge-fixed AdS5 × S5 superstring we reproduce, at one-loop in the near-BMN expansion, the S-matrix known from integrability techniques.

Massive S-matrix of AdS 3 × S 3 × T 4 superstring theory with mixed 3-form flux

A bstractWe find an exact type IIB supergravity solution that represents a one-parameter deformation of the T-dual of the AdS5 × S5 background (with T-duality applied in all 6 abelian bosonic isometric directions). The non-trivial fields are the metric, dilaton and RR 5-form only. The latter has remarkably simple “undeformed” form when written in terms of a “deformation-rotated” vielbein basis. An unusual feature of this solution is that the dilaton contains a linear dependence on the isometric coordinates of the metric precluding a straightforward reversal of T-duality. If we still formally dualize back, we find exactly the metric, B-field and product of dilaton with RR field strengths as recently extracted from the η-deformed AdS5 × S5 superstring action in arXiv:1507.04239. We also discuss similar solutions for deformed AdSn × Sn backgrounds with n = 2, 3. In the η → i limit we demonstrate that all these backgrounds can be interpreted as special limits of gauged WZW models and are also related to (a limit of) the Pohlmeyer-reduced models of the AdSn × Sn superstrings.

Giant magnon solution and dispersion relation in string theory in AdS3×S3×T4 with mixed flux

A bstractContinuing the program initiated in arXiv:1304.1798 we investigate unitarity methods applied to two-dimensional integrable field theories. The one-loop computation is generalized to encompass theories with different masses in the asymptotic spectrum and external leg corrections. Additionally, the prescription for working with potentially singular cuts is modified to cope with an ambiguity that was not encountered before. The resulting methods are then applied to three light-cone gauge string theories; i) AdS3 × S3 × T4 supported by RR flux, ii) AdS3 ×S3 ×S3 ×S1 supported by RR flux and iii) AdS3 ×S3 ×T4 supported by a mix of RR and NSNS fluxes. In the first case we find agreement with the exact result following from symmetry considerations and in the second case with one-loop semiclassical computations. This agreement crucially includes the rational terms and hence supports the conjecture that S-matrices of integrable field theories are cut-constructible, up to a possible shift in the coupling. In the final case, under the assumption that our methods continue to give all rational terms, we give a conjecture for the one-loop phases.