Schr\"odinger geometries arising from Yang-Baxter deformations

We present further examples of the correspondence between solutions of type IIB supergravity and classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have been composed of generators of only one of either $\mathfrak{s}\mathfrak{o}(2,4)$ or $\mathfrak{s}\mathfrak{o}(6)$. In this paper, we consider some examples of $r$-matrices with both of them. The $r$-matrices of this kind contain (generalized) Schr"odinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schr"odinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.