On integrability of strings on symmetric spaces

In the absence of NSNS three-form flux the bosonic string on a symmetric space is described by a symmetric space coset sigma-model. Such models are known to be classically integrable. We show that the integrability extends also to cases with non-zero NSNS flux (respecting the isometries) provided that the flux satisfies a condition of the form H_{abc}H^{cde}~R_{ab}^{de}. We then turn our attention to the type II Green-Schwarz superstring on a symmetric space. We prove that if the space preserves some supersymmetry there exists a truncation of the full superspace to a supercoset space and derive the general form of the superisometry algebra. In the case of vanishing NSNS flux the corresponding supercoset sigma-model for the string is known to be integrable. We prove that the integrability extends to the full string by augmenting the supercoset Lax connection with terms involving the fermions which are not captured by the supercoset model. The construction is carried out to quadratic order in these fermions. This proves the integrability of strings on symmetric spaces supported by RR flux which preserve any non-zero amount of supersymmetry. Finally we also construct Lax connections for some supercoset models with non-zero NSNS flux describing strings in AdS(2,3) x S(2,3) x S(2,3) x T(2,3,4) backgrounds preserving eight supersymmetries.Comment: 34 pages; v3: Minor clarifications, matches published version; v4: Missing local Lorentz transformation added in eqs. (2.14), (2.15) and (4.64

Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly

The equations that follow from kappa symmetry of the type II Green-Schwarz string are a certain deformation, by a Killing vector field $K$, of the type II supergravity equations. We analyze under what conditions solutions of these `generalized' supergravity equations are trivial in the sense that they solve also the standard supergravity equations. We argue that for this to happen $K$ must be null and satisfy $dK=i_KH$ with $H=dB$ the NSNS three-form field strength. Non-trivial examples are provided by symmetric pp-wave solutions. We then analyze the consequences for non-abelian T-duality and the closely related homogenous Yang-Baxter sigma models. When one performs non-abelian T-duality of a string sigma model on a non-unimodular (sub)algebra one generates a non-vanishing $K$ proportional to the trace of the structure constants. This is expected to lead to an anomaly but we show that when $K$ satisfies the same conditions the anomaly in fact goes away leading to more possibilities for non-anomalous non-abelian T-duality.Comment: 11 pages; v2: Important clarifications in sec 5 and projections fixed there. References added. v3: Typos fixed and eq. (5.13) with the general form of the anomaly in terms of K and X adde

Classifying integrable symmetric space strings via factorized scattering

All symmetric space $AdS_n$ solutions of type II supergravity have recently been found for $n>2$. For the supersymmetric solutions (and their T-duals) it is known that the Green-Schwarz string is classically integrable. We complete the classification by ruling out integrability for the remaining non-supersymmetric solutions. This is achieved by showing that tree-level scattering on the worldsheet of a GKP or BMN string fails to factorize for these cases.Comment: 17 pages; v2: Improvements to sec 1, results now summarized in Tab 1. Matches published versio

Constraining integrable AdS/CFT with factorized scattering

We consider (warped) AdS string backgrounds which allow for a GKP spinning string/null cusp solution. Integrability implies that the worldsheet S-matrix should factorize, which in turn constrains the form of the warp factor as a function of the coordinates of the internal space. This constraint is argued to rule out integrability for all supersymmetric AdS7 and AdS6 backgrounds as well as AdS5 Gaiotto-Maldacena backgrounds and a few highly supersymmetric AdS4 and AdS3 backgrounds.Comment: 19 pages; v2: Further clarifications in sec 4 and 5. Published versio

Worldsheet scattering in AdS(3)/CFT(2)

We confront the recently proposed exact S-matrices for AdS(3)/CFT(2) with direct worldsheet calculations. Utilizing the BMN and Near Flat Space (NFS) expansions for strings on AdS(3) x S(3) x S(3) x S(1) and AdS(3) x S(3) x T(4) we compute both tree-level and one-loop scattering amplitudes. Up to some minor issues we find nice agreement in the tree-level sector. At the one-loop level however we find that certain non-zero tree-level processes, which are not visible in the exact solution, contribute, via the optical theorem, and give an apparent mismatch for certain amplitudes. Furthermore we find that a proposed one-loop modification of the dressing phase correctly reproduces the worldsheet calculation while the standard Hernandez-Lopez phase does not. We also compute several massless to massless processes.Comment: 20 pages, 1 figure; v2: Some clarifications in comparison to literatur

The AdS(n) x S(n) x T(10-2n) BMN string at two loops

We calculate the two-loop correction to the dispersion relation for worldsheet modes of the BMN string in AdS(n) x S(n) x T(10-2n) for n=2,3,5. For the massive modes the result agrees with the exact dispersion relation derived from symmetry considerations with no correction to the interpolating function h. For the massless modes in AdS(3) x S(3) x T(4) however our result does not match what one expects from the corresponding symmetry based analysis. We also derive the S-matrix for massless modes up to the one-loop order. The scattering phase is given by the massless limit of the Hernandez-Lopez phase. In addition we compute a certain massless S-matrix element at two loops and show that it vanishes suggesting that the two-loop phase in the massless sector is zero.Comment: 30 pages, 6 figures; v2: References and comment on type IIB added, acknowledgements updated; v3: Comparison to proposed exact massless S-matrix in sec 5.3 corrected. Only non-trivial phase appears at one loop. Additional minor clarification