Generalized type IIB supergravity equations and non-Abelian classical r-matrices

TL;DR

This paper investigates Yang-Baxter deformations of the $AdS_5 imes S^5$ superstring using non-Abelian classical r-matrices, deriving backgrounds that satisfy generalized supergravity equations and exploring conditions under which they reduce to standard type IIB supergravity solutions.

Contribution

It provides explicit examples of non-Abelian r-matrix deformations and analyzes their associated supergravity backgrounds, including T-dualized solutions that satisfy standard supergravity equations.

Findings

All backgrounds satisfy generalized supergravity equations.

Some T-dualized backgrounds satisfy standard type IIB supergravity.

Certain deformations are locally equivalent to undeformed $AdS_5 imes S^5$ after coordinate transformations.

Abstract

We study Yang-Baxter deformations of the $AdS_5 \times S^5$ superstring with non-Abelian classical $r$-matrices which satisfy the homogeneous classical Yang-Baxter equation (CYBE). By performing a supercoset construction, we can get deformed $AdS_5 \times S^5$ backgrounds. While this is a new area of research, the current understanding is that Abelian classical $r$-matrices give rise to solutions of type IIB supergravity, while non-Abelian classical $r$-matrices lead to solutions of the generalized supergravity equations. We examine here some examples of non-Abelian classical r-matrices and derive the associated backgrounds explicitly. All of the resulting backgrounds satisfy the generalized equations. For some of them, we derive "T-dualized" backgrounds by adding a linear coordinate dependence to the dilaton and show that these satisfy the usual type IIB supergravity equations. Remarkably, some of the "T-dualized" backgrounds are locally identical to undeformed $AdS_5 \times S^5$ after an appropriate coordinate transformation, but this seems not to be generally the case.