Finite-gap equations for strings on AdS_3 x S^3 x T^4 with mixed 3-form flux

We study superstrings on AdS_3 x S^3 x T^4 supported by a combination of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three form fluxes, and construct a set of finite-gap equations that describe the classical string spectrum. Using the recently proposed all-loop S-matrix we write down the all-loop Bethe ansatz equations for the massive sector. In the thermodynamic limit the Bethe ansatz reproduces the finite-gap equations. As part of this derivation we propose expressions for the leading order dressing phases. These phases differ from the well-known Arutyunov-Frolov-Staudacher phase that appears in the pure Ramond-Ramond case. We also consider the one-loop quantization of the algebraic curve and determine the one-loop corrections to the dressing phases. Finally we consider some classical string solutions including finite size giant magnons and circular strings.Comment: 44 pages, 3 figures. v2: references and a discussion about perturbative results adde

Classical integrability and quantum aspects of the AdS(3) x S(3) x S(3) x S(1) superstring

In this paper we continue the investigation of aspects of integrability of the type IIA AdS(3) x S(3) x S(3) x S(1) and AdS(3) x S(3) x T(4) superstrings. By constructing a one parameter family of flat connections we prove that the Green-Schwarz string is classically integrable, at least to quadratic order in fermions, without fixing the kappa-symmetry. We then compare the quantum dispersion relation, fixed by integrability up to an unknown interpolating function h(lambda), to explicit one-loop calculations on the string worldsheet. For AdS(3) x S(3) x S(3) x S(1) the spectrum contains heavy, as well as light and massless modes, and we find that the one-loop contribution differs depending on how we treat these modes showing that similar regularization ambiguities as appeared in AdS(4)/CFT(3) occur also here.Comment: 29 pages; v2: updated references and acknowledgmen

The all-loop integrable spin-chain for strings on AdS3 × S 3 × T 4: The massive sector

We bootstrap the all-loop dynamic S-matrix for the homogeneous psu (1, 1|2)2 spin-chain believed to correspond to the discretization of the massive modes of string theory on AdS3 × S 3 × T 4. The S-matrix is the tensor product of two copies of the su (1|1)2 invariant S-matrix constructed recently for the d (2, 1; α)2 chain, and depends on two anti-symmetric dressing phases. We write down the crossing equations that these phases have to satisfy. Furthermore, we present the corresponding Bethe Ansatz, which differs from the one previously conjectured, and discuss how our construction matches several recent perturbative calculations

AdS3/CFT2, finite-gap equations and massless modes

It is known that string theory on AdS 3 × M 7 backgrounds, where M 7 = S 3 × S 3 × S 1 or S 3 × T 4, is classically integrable. This integrability has been previously used to write down a set of integral equations, known as the finite-gap equations. These equations can be solved for the closed string spectrum of the theory. However, it has been known for some time that the finite-gap equations on these AdS 3 × M 7 backgrounds do not capture the dynamics of the massless modes of the closed string theory. In this paper we re-examine the derivation of the AdS 3 × M 7 finite-gap system. We find that the conditions that had previously been imposed on these integral equations in order to implement the Virasoro constraints are too strict, and are in fact not required. We identify the correct implementation of the Virasoro constraints on finite-gap equations and show that this new, less restrictive condition captures the complete closed string spectrum on AdS 3 × M 7

Two-dimensional S-matrices from unitarity cuts

Using 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 × S 5 superstring we reproduce, at one-loop in the near-BMN expansion, the S-matrix known from integrability techniques

Protected string spectrum in AdS(3)/CFT2 from worldsheet integrability

We derive the protected closed-string spectra of AdS3/CFT2 dual pairs with 16 supercharges at arbitrary values of the string tension and of the three-form fluxes. These follow immediately from the all-loop Bethe equations for the spectra of the integrable worldsheet theories. Further, representing the underlying integrable systems as spin chains, we find that their dynamics involves length-changing interactions and that protected states correspond to gapless excitations above the Berenstein-Maldacena-Nastase vacuum. In the case of AdS3 × S3 × T4 the degeneracies of such operators precisely match those of the dual CFT2 and the supergravity spectrum. On the other hand, we find that for AdS3 × S3 × S3 × S1 there are fewer protected states than previous supergravity calculations had suggested. In particular, protected states have the same su(2) charge with respect to the two three-spheres

On the massless modes of the AdS3/CFT2 integrable systems

We make a proposal for incorporating massless modes into the spin-chain of the AdS3/CFT2 integrable system. We do this by considering the α → 0 limit of the alternating d(2,1;α)2 spinchain constructed in arXiv:1106.2558. In the process we encounter integrable spin-chains with nonirreducible representations at some of their sites. We investigate their properties and construct their R-matrices in terms of Yangians

The complete AdS3 ×S3 × T4 worldsheet S matrix

We derive the non-perturbative worldsheet S matrix for fundamental excitations of Type IIB superstring theory on AdS3 ×S3 × T4 with Ramond-Ramond flux. To this end, we study the off-shell symmetry algebra of the theory and its representations. We use these to determine the S matrix up to scalar factors and we derive the crossing equations that these scalar factors satisfy. Our treatment automatically includes fundamental massless excitations, removing a long-standing obstacle in using integrability to study the AdS3/CFT2 correspondence

Near BMN dynamics of the AdS(3) x S(3) x S(3) x S(1) superstring

We investigate the type IIA AdS(3) x S(3) x M(4) superstring with M(4)=S(3) x S(1) or M(4)=T(4). String theory in this background is interesting because of AdS3/CFT2 and its newly discovered integrable structures. We derive the kappa symmetry gauge-fixed Green-Schwarz string action to quadratic order in fermions and quartic order in fields utilizing a near BMN expansion. As a first consistency check of our results we show that the two point functions are one-loop finite in dimensional regularization. We then perform a Hamiltonian analysis where we compare the energy of string states with the predictions of a set of conjectured Bethe equations. While we find perfect agreement for single rank one sectors, we find that the product SU(2) x SU(2) sector does not match unless the Bethe equations decouple completely. We then calculate 2 to 2 bosonic tree-level scattering processes on the string worldsheet and show that the two-dimensional S-matrix is reflectionless. This might be important due to the presence of massless worldsheet excitations which are generally not described by the Bethe equations.Comment: 28 pages; v2: Fixed signs and eq. (B.1), results unchanged, one reference added; v3: Matches published versio