Supercoset construction of Yang-Baxter deformed AdS$_5\times$S$^5$ backgrounds

We proceed to study Yang-Baxter deformations of the AdS$_5\times$S$^5$ superstring with the classical Yang-Baxter equation. We make a general argument on the supercoset construction and present the master formula to describe the dilaton in terms of classical $r$-matrices. The supercoset construction is explicitly performed for some classical $r$-matrices and the full backgrounds including the Ramond-Ramond (R-R) sector and dilaton are derived. Within the class of abelian $r$-matrices, the perfect agreement is shown for well-known examples including gravity duals of non-commutative gauge theories, $\gamma$-deformations of S$^5$ and Schr\"odinger spacetimes. It would be remarkable that the supercoset construction works well, even if the resulting backgrounds are not maximally supersymmetric. In particular, three-parameter $\gamma$-deformations of S$^5$ and Schr\"odinger spacetimes do not preserve any supersymmetries. As for non-abelian $r$-matrices, we will focus upon a specific example. The resulting background does not satisfy the equation of motion of the Neveu-Schwarz-Neveu-Schwarz (NS-NS) two-form because the R-R three-form is not closed.