Ja n 20 09 ON Dielectric Membranes

By placing a collection of membranes in 11 dimensional background supergravity six form field and considering the non-linear action of multiple membranes we obtain the potential for transverse scalar fields of membranes and study some possible vacuum configurations. We find that the system has a stable vacuum in which is a formation of membranes into fuzzy three sphere. Ganjali@theory.ipm. ac.ir When a neutral material put in an external electric field the electric charges separate from each other and an electric dipole occurs. Similar phenomena exists in string theory where there are extended objects as Dp branes and higher rank antisymmetric form fields as charge of these objects. In particular, When a set of neutral D branes put in an external antisymmetric background field one observes the ”polarized” D branes which are expanded into a higher dimensional world volume theory [1]. One may naturally demand such phenomena for membranes in M-theory. There are some efforts to studying this in the context of M-theory [2] where it was used some idea such as considering massive ABJM theory [3] or Basu-Harvey equation [4] as boundary conditions of membranes. But recently, some wonderful works was done in understanding the dynamics of multiple M2 branes. Firstly, it was found an action [5] which was based on an strange algebra, Lie-3 algebra, which is expected to describe the low energy dynamics of multiple membranes. Based on this action, the authors of [6] found a non-linear action for M2 branes which in low energy limit reduces to that of [5]. Beside, the Lie-3 algebra valued fields allows one to write down an action for the coupling of supergravity background form fields C3 and C6 to world volume of membranes [7]. So, by using the nonlinear action of multiple membranes and the action of couplings of background form fields to world volume of membranes one can study the effects of placing multiple membranes in these external fields. The aim of this note is to study that in details. In particular, we will find that when a collection of membranes are placed in a constant background C6 fields the configuration of membranes has a trivial vacua in which the potential V (X) = 0 and all fields are represented by some commuting matrices. But, the theory still has a nontrivial vacua in which has lower energy and so is stable in comparision with commuting one. Such vacuum is a configuration of membranes which are formed in fuzzy three sphere S and are ended on five branes. On Triple Product: First of all, we briefly introduce triple product. All fields in BLG theory are valued in a Lie-3 algebra and can be expanded in terms of generators of the algebra as X = X aT a in which the generators T a construct the following product [T , T , T ] = fdT , (1) where fd are some structure constants and obey a fundamental identity f gf abg h = f abc gf gde h + f abd gf gec h + f abe gf cdg h, (2) Note that it can be shown that the main results of this paper can be obtained without the assumption of triple algebra on the theory. In this paper we use the indices as a, b, . = +,−, 1, 2, . , Dim G; μ, ν, . = 0, 1, 2; Î , Ĵ , . = 0, 1, . , 10 and I, J, . = 3, 4, . , 10. And similarly, one can expand the gauge fields and spinor fields in terms of the generators. 1 There has been a lot of efforts to finding solutions of (2), see for example [5, 7, 8, 9, 10, 11]. An interesting solution was presented in [8, 9, 10] in which by decomposing T a = {T−, T, T } one may demands [T−, T, T ] = 0, [T , T , T ] = fT [T, T , T ] = [T , T ] = fcT c (3) In this fashion, the T− mode completely decouples from the theory and one obtains a theory with ordinary Lie algebra generators T a which is lifted by T. Non-linear action for M2 branes: Proposed bosonic part of non-linear Lagrangian for multiple membranes for gauge group U(N) is given by LM2 = −T2STr (