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 5 U(1) isometries. We do this by reducing along one of the 5-torus directions, 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 5x5 matrix, which represents the metric on the 5-torus. We apply these results to several M-theory backgrounds of the type AdS_r x 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 3 or 4 U(1) isometries and study a case, for which our assumption for the 4-form field is violated.