Gravitational analog of the Non-Abelian T-duality

Non-abelian T duality (NATD) is a symmetry of the worldsheet action that allows one to generate new solutions of string theory by performing algebraic transformations of known geometries. Applications of such transformations to spheres have been especially fruitful in the past, but the results did not reduce to the abelian transformation in the decompactification limit unless the dimension of sphere was equal to three. We propose a unique counterpart of NATD in classical gravity that reproduces the correct decompactification limit for all spheres $S^n$ but generates an n-form field strength and therefore cannot be naturally embedded in a worldsheet theory of NS-NS fields unless n=3. We also propose a non-abelian version of the TsT transformation which produces solutions of type II supergravity describing continuous deformations of geometries with $S^n\times S^n$ and $S^n\times T^n$ factors.