Three-parameter integrable deformation of ℤ4 permutation supercosets

Abstract

A three-parameter integrable deformation of Z 4 permutation supercosets is constructed.These supercosets are of the form F /F 0 where F 0 is the bosonic diagonal subgroup of the product supergroup F = G × G. They include the AdS 3 × S 3 and AdS 3 × S 3 × S 3 supercosets. This deformation encompasses both the bi-Yang-Baxter deformation of the semi-symmetric space σ-model on Z 4 permutation supercosets and the mixed flux model.Truncating the action at the bosonic level, we show that one recovers the bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term. in [6] rely on the deformation being governed by a solution of the modified classical Yang-Baxter equation. The two-parameter bi-Yang-Baxter deformation of the principal chiral model was introduced in [5] and shown to be integrable in [7,8]. As demonstrated in [9], the bi-Yang-Baxter deformation of the SU (2) principal chiral model is equivalent to the two-parameter deformation of S 3 found in [10]. In [6] the Yang-Baxter deformation of the symmetric space σ-model was constructed and shown to be integrable. For symmetric spaces that take the form of Z 2 permutation cosets (1.2), the deformed model is equivalent to a particular one-parameter model contained within the bi-Yang-Baxter deformation of the principal chiral model on F 0 .Adding the standard topological Wess-Zumino term [11][12][13] to the principal chiral model is well-known to preserve its classical integrability. Up to the expected quantization of the level, the resulting model interpolates between the principal chiral and conformal Wess-Zumino-Witten models. Generalising the SU (2) construction of [14,15], the Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term for general Lie groups was derived in [16]. In order to construct this deformation it was assumed that the solution of the modified classical Yang-Baxter deformation governing the deformation cubes to its negative. This assumption is natural since it is satisfied by the standard Drinfel'd-Jimbo R-matrix for a non-split real form of a semi-simple Lie algebra [17][18][19]. In this article we will continue to focus on deformations governed by such R-matrices.By allowing for an antisymmetric term in the action, the two-parameter deformation of [10] was generalised to an integrable four-parameter deformation of S 3 in [2]. Observing that this model also contains the TsT transformation of the SU (2) Wess-Zumino-Witten model [20, 21], it was proposed in [9] that the four-parameter model should be understood as the combined bi-Yang-Baxter deformation and TsT transformation of the σ-model on S 3 plus Wess-Zumino term. The bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term was constructed in [1]. Applying a TsT transformation in the directions of the Cartan subalgebra, which is associated with adding a compatible abelian solution of the classical Yang-Baxter equation [22-29] to the Drinfel'd-Jimbo R-matrix, and considering the SU (2) case it was shown that this model indeed genera...

Dressing cosets and multi-parametric integrable deformations

“…Finally, knowing that (∂ ± ll −1 ) 2 = 0 and (∂ ± ll −1 ) 1 = A ± permits to rewrite Eq. (5.5) as 14) or, equivalently, as…”

S matrix for a three-parameter integrable deformation of AdS3 × S3 strings

“…Additionally, one can also accommodate the Wess-Zumino (WZ) term in the deformation [37], which allows to incorporate all AdS 3 × S 3 mixed-flux backgrounds within a three-parameter family of deformations, as it was done in ref. [38]. It is worth stressing that it is presently unknown whether this three-parameter deformation yields a superstring JHEP11(2020)022 background.…”

“…Our main goal concerns the first step in this roadmap, namely the study of the S matrix for the most general three-parameter deformation of ref. [38]. In particular, below we will construct its bosonic tree-level S matrix, which will provide an important input for the comprehensive study of this background.…”

“…In section 2 we review the geometry found in ref. [38]. In section 3 we present the computation of the S matrix.…”

Two-parameter integrable deformations of the AdS3× S3× T4 superstring

“…Last but not least, for supercosets with superisometry algebra g ⊕ g a Wess-Zumino-Witten term can be added to the action, corresponding to the introduction of NS-NS flux [4]. Combining the WZW term with the two-parameter deformation, one can construct an integrable three-parameter deformation of the semi-symmetric space sigma model [70]. It would be interesting to obtain a closed formula for the NS-NS and R-R fluxes and see if unimodular R-matrices also give rise to Weyl-invariant theories in this more general setting.…”