Schr\"odinger geometries arising from Yang-Baxter deformations

TL;DR

This paper explores new Schr"odinger geometries derived from Yang-Baxter deformations involving combined $rak{so}(2,4)$ and $rak{so}(6)$ generators, expanding the known correspondence between supergravity solutions and classical r-matrices.

Contribution

It introduces examples of r-matrices with both $rak{so}(2,4)$ and $rak{so}(6)$ components, leading to new Schr"odinger spacetimes and gravity duals of dipole theories.

Findings

Constructed explicit r-matrices with both $rak{so}(2,4)$ and $rak{so}(6)$ components.

Derived Schr"odinger geometries from Yang-Baxter sigma-models.

Reproduced metrics and B-fields via Yang-Baxter deformations.

Abstract

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 $\mathfrak{so}(2,4)$ or $\mathfrak{so}(6)$. 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.