Abstract
We further study integrable deformations of the AdS$_5\times$S$^5$ superstring by following the Yang-Baxter sigma model approach with classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). Deformed string backgrounds specified by $r$-matrices are considered as solutions of type IIB supergravity, and therefore the relation between gravitational solutions and $r$-matrices may be called the gravity/CYBE correspondence. In this paper, we present a family of string backgrounds associated with a classical $r$-matrices carrying two parameters and its three-parameter generalization. The two-parameter case leads to the metric and NS-NS two-form of a solution found by Hubeny-Rangamani-Ross [hep-th/0504034] and another solution in [arXiv:1402.6147]. For all of the backgrounds associated with the three-parameter case, the metric and NS-NS two-form are reproduced by performing TsT transformations and S-dualities for the undeformed AdS$_5\times$S$^5$ background. As a result, one can anticipate the R-R sector that should be reproduced via a supercoset construction.
Highlights
What kinds of string backgrounds would be concerned with integrability? It is wellknown that symmetric cosets in general lead to classical integrability
We further study integrable deformations of the AdS5×S5 superstring by following the Yang-Baxter sigma model approach with classical r-matrices satisfying the classical Yang-Baxter equation (CYBE)
Deformed string backgrounds specified by r-matrices are considered as solutions of type IIB supergravity, and the relation between gravitational solutions and r-matrices may be called the gravity/CYBE correspondence
Summary
We consider here a deformation of the AdS5 bosonic part of (2.1) based on the following classical r-matrix of Jordanian type, rJor = E24 ∧ (c1E22 − c2E44) ,. It is convenient to rewrite the metric part and NS-NS two-form coupled part of the Lagrangian (2.1) into the following form, LG. For the su(2, 2) generators pμ (μ = 0, 1, 2, 3) and γ5, see appendix A. The metric part and NS-NS two-form part of the Lagrangian (2.7) are given by LG = −γαβ. Note that the resulting Lagrangian becomes complex in general. It is necessary to argue the reality condition. Both LG and LB become real if and only if c1 and c2 are related by the complex conjugation, c1 = c∗2. The result given in subsection 2.2 of [62] can be reproduced by imposing that c1
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