Linus Wulff

Superisometries and integrability of superstrings

A bstractFor type II supergravity backgrounds with superisometries the corresponding transformations and conserved currents for the superstring are constructed up to fourth order in Θ. It is then shown how, for certain backgrounds related to near horizon geometries of intersecting branes, the components of the superisometry current can be used to construct a Lax connection demonstrating the classical integrability of the string in these backgrounds. This includes examples of AdS2 and AdS3 backgrounds with a D(2, 1; α) isometry group which have not previously been studied from an integrability point of view. The construction of the Lax connection is carried out up to second order in Θ.

Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations

A bstractWe determine the constraints imposed on the 10d target superspace geometry by the requirement of classical kappa-symmetry of the Green-Schwarz superstring. In the type I case we find that the background must satisfy a generalization of type I supergravity equations. These equations depend on an arbitrary vector Xa and imply the one-loop scale invariance of the GS sigma model. In the special case when Xa is the gradient of a scalar ϕ (dilaton) one recovers the standard type I equations equivalent to the 2d Weyl invariance conditions of the superstring sigma model. In the type II case we find a generalized version of the 10d supergravity equations the bosonic part of which was introduced in arXiv:1511.05795. These equations depend on two vectors Xa and Ka subject to 1st order differential relations (with the equations in the NS-NS sector depending only on the combination Xa = Xa + Ka). In the special case of Ka = 0 one finds that Xa = ∂aϕ and thus obtains the standard type II supergravity equations. New generalized solutions are found if Ka is chosen to be a Killing vector (and thus they exist only if the metric admits an isometry). Non-trivial solutions of the generalized equations describe K-isometric backgrounds that can be mapped by T-duality to type II supergravity solutions with dilaton containing a linear isometry-breaking term. Examples of such backgrounds appeared recently in the context of integrable η-deformations of AdSn × Sn sigma models. The classical kappa-symmetry thus does not, in general, imply the 2d Weyl invariance conditions for the GS sigma model (equivalent to type II supergravity equations) but only weaker scale invariance type conditions.

Target space supergeometry of η and λ-deformed strings

A bstractWe study the integrable η and λ-deformations of supercoset string sigma models, the basic example being the deformation of the AdS5 × S5 superstring. We prove that the kappa symmetry variations for these models are of the standard Green-Schwarz form, and we determine the target space supergeometry by computing the superspace torsion. We check that the λ-deformation gives rise to a standard (generically type II*) supergravity background; for the η-model the requirement that the target space is a supergravity solution translates into a simple condition on the R-matrix which enters the definition of the deformation. We further construct all such non-abelian R-matrices of rank four which solve the homogeneous classical Yang-Baxter equation for the algebra so

The one-loop worldsheet S-matrix for the AdSn × Sn × T10−2n superstring

\mathfrak{so}

Integrable Deformations of

(2, 4). We argue that most of the corresponding backgrounds are equivalent to sequences of non-commuting TsT-transformations, and verify this explicitly for some of the examples.

T

Starting with the Fˆ/G supercoset model corresponding to the AdSn×Sn superstring one can define the λ-model of arXiv:1409.1538 either as a deformation of the Fˆ/Fˆ gauged WZW model or as an integrable one-parameter generalisation of the non-abelian T-dual of the AdSn×Sn superstring sigma model with respect to the whole supergroup Fˆ. Here we consider the case of n=2 and find the explicit form of the 4d target space background for the λ-model for the PSU(1,1|2)/SO(1,1)×SO(2) supercoset. We show that this background represents a solution of type IIB 10d supergravity compactified on a 6-torus with only metric, dilaton Φ and the RR 5-form (represented by a 2-form F in 4d) being non-trivial. This implies that the λ-model is Weyl invariant at the quantum level and thus defines a consistent superstring sigma model. The supergravity solution we find is different from the one in arXiv:1410.1886 which should correspond to a version of the λ-model where only the bosonic subgroup of Fˆ is gauged. Still, the two solutions have equivalent scaling limit of arXiv:1504.07213 leading to the isometric background for the metric and eΦF which is related to the η-deformed AdS2×S2 sigma model of arXiv:1309.5850. Similar results are expected in the AdS3×S3 and AdS5×S5 cases.

-Dual

A bstractWe compute the massive-sector worldsheet S-matrix for superstring theories in AdSn × Sn × T10−2n (with n = 2, 3, 5) in the near BMN expansion up to one-loop order in inverse string tension. We show that, after taking into account the wave function renormalization, the one-loop S-matrix is UV finite. In an appropriate regularization scheme the S-matrix is consistent with the underlying symmetries of the superstring theory, i.e. for the n=3,5casesitcoincideswiththeoneimpliedbythelight-conegaugesymmetrieswiththe dressing phases determined from the crossing equations. For the n = 2, 3 cases we observe that the massless modes decouple from the one-loop calculation of massive mode scattering, i.e. the 2n-dimensional supercoset sigma model and the full 10-dimensional superstring happen to have the same massive one-loop S-matrix.

\sigma

A bstractIn 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 HabcHcde ∼ Rabde. 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 AdS2,3 × S2,3 × S2,3 × T2,3,4 backgrounds preserving eight supersymmetries.