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

Abstract

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 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 3 or 4 U (1) isometries and study a case, for which our assumption for the 4-form field is violated.

Tri-vector deformations in d = 11 supergravity

“…and finally an uplift back to d = 11. As expected, this reflects the fact that the corresponding deformation is simply a d = 11 extension of a TsT [43,44].…”

“…and a slightly more involved deformation prescription (3.16) instead of the open/closed map. When the Killing vectors form a U(1) 3 subgroup, this prescription reproduces an uplift of TsT to d = 11 [43,44]. For a single commuting ∂ * the tri-vector can be written as Ω = ∂ * ∧ β, and non-abelian YB deformations w.r.t.…”

Time-dependent AdS backgrounds from S-branes

“…From (9) it is clear that we should haveḊ = 0. However, a time independent metric part in the general S-brane solution is not allowed when it is charged [25] but it is possible when it is chargeless [36]. Therefore, when S-brane has a charge, there has to be some smearing for a non-dilatonic p-brane.…”

“…In [36] a simple formula was derived to obtain Lunin-Maldacena deformations [34] of solutions of d=11 supergravity which possess three U(1) isometries. The formula given in [36] is valid only when these U(1) directions do not mix with any other coordinate in the metric and when the 4-form field strength has at most one overlapping with them.…”

“…This generates the time-dependent version of the dyonic M2-brane, M2 ⊂ M5, of [35] which has AdS 7 × S 4 near-horizon too. This deformation is actually a particular Uduality transformation and a simple formula for the deformed background was given in [36] which we use here. We conclude in section 4 with some comments and possible future directions.…”