Poisson-Lie duals of the eta deformed symmetric space sigma model

Poisson-Lie dualising the eta deformation of the G/H symmetric space sigma model with respect to the simple Lie group G is conjectured to give an analytic continuation of the associated lambda deformed model. In this paper we investigate when the eta deformed model can be dualised with respect to a subgroup G_0 of G. Starting from the first-order action on the complexified group and integrating out the degrees of freedom associated to different subalgebras, we find it is possible to dualise when G_0 is associated to a sub-Dynkin diagram. Additional U_1 factors built from the remaining Cartan generators can also be included. The resulting construction unifies both the Poisson-Lie dual with respect to G and the complete abelian dual of the eta deformation in a single framework, with the integrated algebras unimodular in both cases. We speculate that extending these results to the path integral formalism may provide an explanation for why the eta deformed AdS_5 x S^5 superstring is not one-loop Weyl invariant, that is the couplings do not solve the equations of type IIB supergravity, yet its complete abelian dual and the lambda deformed model are.Comment: 37 pages, v2: minor change

Bi-η\eta and bi-λ\lambda deformations of Z4\mathbb{Z}_4 permutation supercosets

Integrable string sigma models on AdS$_3$ backgrounds with 16 supersymmetries have the distinguishing feature that their superisometry group is a direct product. As a result the deformation theory of these models is particularly rich since the two supergroups in the product can be deformed independently. We construct bi-$\eta$ and bi-$\lambda$ deformations of two classes of $\mathbb{Z}_4$ permutation supercoset sigma models, which describe sectors of the Green-Schwarz and pure-spinor string worldsheet theories on type II AdS$_3$ backgrounds with pure R-R flux. We discuss an important limit of these models when one supergroup is undeformed. The associated deformed supergravity background should preserve 8 supersymmetries and is expected to have better properties than the full bi-deformation. As a step towards investigating the quantum properties of these models, we study the two-loop RG flow of the bosonic truncation of the bi-$\lambda$ deformation.Comment: 35 pages, published versio

Elliptic deformations of the AdS 3 × S 3 × T 4 string

With the aim of investigating the existence of an integrable elliptic deformation of strings on AdS3 × S3 × T4, we compute the tree-level worldsheet S-matrix of the elliptically-deformed bosonic sigma model on AdS3 × S3 in uniform light-cone gauge. The resulting tree-level S-matrix is compatible with the integrability of the model and has interesting features, including a hidden U(1) symmetry not manifest in the Lagrangian. We find that it cannot be embedded in the known exact integrable R-matrices describing deformations of the undeformed AdS3 × S3 × T4 light-cone gauge S-matrix including fermions. Therefore, we construct embeddings of the deformed 6-d metric in type II supergravity with constant dilaton and homogeneous fluxes. The simplicity of these solutions suggests they are promising candidates to lead to an integrable string sigma model including fermions

Integrable supersymmetric deformations of AdS3 × S3 × T4

We construct a family of type IIB string backgrounds that are deformations of AdS3 × S 3 × T 4 with a “squashed” AdS3 × S 3 metric supported by a combination of NSNS and RR fluxes. They have global SU(1, 1) × SU(2) symmetry, regular curvature, constant dilaton and preserve 8 supercharges. Upon compactification to 4 dimensions they reduce to N = 2 supersymmetric AdS2 × S 2 solutions with electric and magnetic Maxwell fluxes. These type IIB supergravity solutions can be found from the undeformed AdS3 × S 3 × T 4 background by a combination of T-dualities and S-duality. In contrast to T-duality, S-duality transformations of a type IIB supergravity background do not generally preserve the classical integrability of the corresponding Green-Schwarz superstring sigma model. Nevertheless, we show that integrability is preserved in the present case. Indeed, we find that these backgrounds can be obtained, up to T-dualities, from an integrable inhomogeneous Yang-Baxter deformation (with unimodular Drinfel’d-Jimbo R-matrix) of the original AdS3 × S 3 supercoset model

The twisted story of worldsheet scattering in η\eta-deformed AdS5×S5AdS_5 \times S^5

We study the worldsheet scattering theory of the $\eta$ deformation of the AdS$_\mathsf{5} \times$S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \times$S$^\mathsf{5}$ superstring, with trigonometric quantum-deformed symmetry. We compute the two-body worldsheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding exact factorized S matrix, and show that its perturbative expansion matches our tree-level results, once we correctly identify the deformed light-cone symmetry algebra of the string. Finally, we briefly revisit the computation of the corresponding S matrix for the $\eta$ deformation based on the distinguished Dynkin diagram, finding a tree-level S matrix that factorizes and solves the classical Yang-Baxter equation, in contrast to previous results.Comment: 40 pages, published in JHEP, v4: minor clarification

S matrix for a three-parameter integrable deformation of AdS3×S3AdS_3\times S^3 strings

We consider the three-parameter integrable deformation of the $AdS_3\times S^3$ superstring background constructed in arXiv:1811.00453. Working on the string worldsheet in uniform lightcone gauge, we find the tree-level bosonic S matrix of the model and study some of its limits.Comment: 28 pages, one Mathematica notebook attached; v2: minor changes, references adde