Killing spinors from classical r-matrices

Abstract

The Yang-Baxter (YB) deformation is a systematic way of performibg integrable deformations of two-dimensional symmetric non-linear sigma models. The deformations can be labeled by classical $r$-matrices satisfying the classical YB equation. This YB deformation is also applicable to type IIB superstring theory defined on $\mathrm{AdS}_5\times S^5$. In this case, a simple class of YB deformation is associated with TsT transformations of string backgrounds. The data of this transformation is encoded in an antisymmetric bi-vector $\Theta$ (which is often called $\beta$ field or non-commutativity parameter). In this article, we give a simple recipe for obtaining the explicit expression for the Killing spinors of the TsT transformed background starting from $\Theta$. We moreover discuss the M-theory equivalent of the TsT transformation, allowing us to also give Killing spinors of 11-dimensional backgrounds. We discuss examples of TsT transformed backgrounds starting from flat space, $\mathrm{AdS}_{5}\times S^5$ and $\mathrm{AdS}_{7}\times S^4$. We find that in this way we can relate the $\Omega$-deformation to YB deformations.