Geometries with twisted spheres and non-abelian T-dualities

A bstract Spectral flow in two-dimensional superconformal field theories is known to correspond to a geometrical mixing between two circles in the gravity dual. We generalize this operation to the geometries which have SO (k +1) ×SO (k +1) isometries with k > 1 and perform various non-abelian T-dualities of the resulting twisted backgrounds. Combination of non-abelian twists and dualities leads to a new solution generating technique in supergravity, and we apply it to the geometries dual to supersymmetric states in N = 4 super-Yang-Mills theory.