Beta, dipole and noncommutative deformations of M-theory backgrounds with one or more parameters

We construct new M-theory solutions starting from those that contain five U(1) isometries. We do this by reduction along one of the 5-torus directions, and then T-dualizing via the action of an O(4, 4) matrix and lifting back to 11-dimensions. The particular T-duality transformation is a sequence of O(2, 2) transformations embedded in O(4, 4), where the action of each O(2, 2) gives a Lunin-Maldacena deformation in 10-dimensions. We find general formulas for the metric and 4-form field of single and multiparameter deformed solutions, when the 4-form of the initial 11-dimensional background has at most one leg along the 5-torus. All the deformation terms in the new solutions are given in terms of subdeterminants of a 5 × 5 matrix, which represents the metric on the 5-torus. We apply these results to several M-theory backgrounds of the type AdS_r × X^11-r. By appropriate choices of the T-duality and reduction directions, we obtain analogues of beta, dipole and noncommutative deformations. We also provide formulas for backgrounds with only three or four U(1) isometries and study a case, for which our assumption for the 4-form field is violated.