Journal of Modern Physics, 2012, 3, 1813-1818 http://dx.doi.org/10.4236/jmp.2012.311226 Published Online November 2012 (http://www.SciRP.org/journal/jmp) On Supersymmetry of the Covariant 3-Algebra Model for M-Theory Matsuo Sato Department of Natural Science, Faculty of Education, Hirosaki University, Hirosaki, Japan Email: [email protected] Received September 5, 2012; revised October 4, 2012; accepted October 12, 2012 ABSTRACT We examine a natural supersymmetric extension of the bosonic covariant 3-algebra model for M-theory proposed in [1]. It possesses manifest SO(1,10) symmetry and is constructed based on the Lorentzian Lie 3-algebra associated with the U(N) Lie algebra. There is no ghost related to the Lorentzian signature in this model. It is invariant under 64 super- symmetry transformations although the supersymmetry algebra does not close. From the model, we derive the BFSS matrix theory and the IIB matrix model in a large N limit by taking appropriate vacua. Keywords: M-Theory; 3-Algebra; Matrix Model; String Theory 1. Introduction 3-algebra. That is, it can be written in the 3-algebra manifest form as The BFSS matrix theory is conjectured to describe infi- 2 nite momentum frame (IMF) limit of M-theory in [2] and 3 11LM N ST d,, g XX X many evidences were found. However, because of the M 2 12 g limit, SO(1,10) symmetry is not manifest in these models; it includes only time and nine matrices corresponding to where ,, denotes Nambu-Poisson bracket [15,16]. nine spatial coordinates. As a result, it is very difficult to Therefore, a bosonic covariant 3-algebra model for M- derive full dynamics of M-theory. For example, we do theory was proposed in [1]. not know the manner to describe longitudinal momentum In this paper, we examine a natural supersymmetric transfer of D0-branes. Therefore, we need a covariant extension of the bosonic covariant model in [1]3, matrix model for M-theory that possesses manifest SO 1 2 SXXX LM,, N (1,10) symmetry. 12 Recently, structures of 3-algebras [3-5] were found in (1) 1 1 MN the effective actions of the multiple M2-branes [6-14] MN XX,, . and 3-algebras have been intensively studied [15-31]. 4 One can expect that structures of 3-algebras play more The bosons X L and the Majorana fermions are fundamental roles in M-theory2 than the accidental struc- spanned by the elements of the Lorentzian Lie 3-algebra tures in the effective descriptions. associated with the U(N) Lie algebra. This action defines The BFSS matrix theory and the IIB matrix model [35] a zero-dimensional field theory and possesses manifest can be obtained by the matrix regularization of the Pois- SO(1,10) symmetry. By expanding fields around appro- son brackets of the light-cone membrane theory [36] and priate vacua, we derive the BFSS matrix theory and the of Green-Schwarz string theory in Schild gauge [35], IIB matrix model in a large N limit. respectively. Because the regularization replaces a two- dimensional integral over a world volume by a trace 2. A Supersymmetric Extension over matrices, the BFSS matrix theory and the IIB ma- We examine a following model, trix model are one-dimensional and zero-dimensional field theories, respectively. On the other hand, the bos- 1 LM N2 SXXX ,, onic part of the membrane action has a structures of a 12 (1) 1 1 ABJM theory can also be rewritten in a 3-algebra manifest form [14]. XXMN,, . 2A formulation of M-theory by a cubic matrix action was proposed by 4 MN Smolin [32-34]. 3This extension was originally proposed in Appendix of [1]. Copyright © 2012 SciRes. JMP 1814 M. SATO L where X with L 0,1, ,10 are vectors and where UN , 1 U 1 and the other fields are are Majorana spinors of SO(1,10). This action defines a not transformed. zero-dimensional field theory and possesses manifest SO Third, the action is invarian t under another part of su- (1,10) symmetry. There is no coupling constant. persymmetry transformation, so called the dynamical su- X M and are spanned by the elements of the Lor- persymmetry transformation, entzian Lie 3-algebra associated with the U(N) Lie alge- XiMM (8) bra, 22 M MM10Mi MM X XT XT XT, 20Xi 2 0 (9) 10i (2) 10i i LMN 10TT iT, XX, X (10) 202 LMN 2 where iN 1, 2, ,2 . The algebra is defined by 1 ab 20 =, 0S (11) TTT,, 0, 1 0 ij ij ijk MN TTT,, = T,T = fT , (3) where =,trMN X X and 0 S is the k 2 i j k ijk 1 variation of the action (5) under (8), (9) and (10) . TT,, T = f T, We should note that the above super transformation is 2 ijk ij lk where ab,1 ,0,1,2,, N and f fhl is to- slightly different with a 3-algebra manifest super trans- ij tally anti-symmetrized. TT, is a Lie bracket of the formation, which is a straightforward analogue to that of U(N) Lie algebra. The metric of the elements is defined the BLG theory for multiple M2-branes; by XiM M 11 10 (12) TT,0,,1,TT i LMN XXLM,,X N . TT10,0i ,,0,TT0 (4) 6 0 iijij If we decompose this transformation, (8), (9) and (10) TT,0,TT,. h are the same, but (11) is different. In the analogue case, 0. There is no such symmetry5 because By using these relations, the action is rewritten as 0 0 S 0 . 22 11LM2 In the Lorentzian case, the action does possess super- Str=, X00 XMN X XXXMN, 42 symmetry because 20 cancels 0 S . However, 1 20 is inconsistent with the 3-algebra symmetry. As a XXMN, (5) 2 0 MN result, the supersymmetry algebra does not close, al- though it closes in a X M sector as one can see below. 1 MN 0 MN XX,, The commutators among the supersymmetry transfor- 2 mations act on X M as M Mi i where X = XT and = T . There is no ghost MM i i X i' M 12 21 1 2 in the theory, because X 1 or 1 does not appear in the action4. M 11''X 11 0 Let us summarize symmetry of the action. First, gauge symmetry is the N 2 -dimensional translation and U(N) ''X M TTXTab,, M i , 22 22 ab i symmetry associated with the Lorentzian Lie 3-algebra L N [10]. where abi'22 LN X a X b . Second, there are two kinds of shift symmetry. First If we change a basis of the supe rsymmetry transforma- one is the eleven-dimensional translation symmetry ge- tions as nerated by 121 (13) M M X =, (6) 221i , Where X M UN, M U 1 and the other fields up to the gauge transformation, we obtain MM are not transformed. Second one is a part of supersym- 11 11 XX metry, so called the kinematical supersymmetry, gener- M M ated by 2222 X X (14) M 11 =, (7) 21 12 X 0, 4Ghost-free Lorentzian 3-algebra theories were studied in [37,38]. 5This fact was originally shown in [39]. Copyright © 2012 SciRes. JMP M. SATO 1815 where is a translation. to SO(1,10) symmetric vacua. These 64 supersymmetry transformations are summa- We assume all the backgrounds (1), (2), (3) and (4) as rised as 12, and (14) implies the =2 su- independent vacua and fix them in the large N limit [40]. M M persymmetry algebra in eleven dime nsions in the X Thus, we do not in tegrate X 0 , 0 or the diagonal sector, elements of a and we expand the field s around the M M backgrounds as, X X . (15) X = pa Because the low energy effective description o f M- Xx=. theory is given by the =1 eleven-dimensional su- II (5) pergravity, the 2 supersymmetry in this sector is =, necessarily broken into the 1 supersymmetry, where we impose a chirality condition spontaneously. In the next section, we will show that the model reduces to the BFSS matrix theory and the IIB 10 =. (6) matrix model in a large N limit if appropriate vacua are Under these conditions, the first term of the action (5) chosen. is rewritten as Because the commutators among th e supersymmetry M transformations of X result in the eleven-dimensional 1 L 2 2 M Str10=, X XXMN translation (6), eigen values of X UN should be 4 6 interpreted as eleven-dimensional space-time . In the 1 2 2 =,trpapa2, paxI (7) next section, when we derive the BFSS matrix theory and 4g 2 the IIB matrix model, X i iUN1, ,9 and 2 X i iU0, ,9 N are identified with matrices in IJ xx,. the BFSS matrix theory and the IIB matrix model respectively. Therefore, our interpretation is consistent The second term is with the space-time interpretation in these models. 1 M 2 StrXXX20=,MN 2 3. BFSS Matrix Theory and IIB Matrix (8) 1 22 Model from Covariant 3-Algebra Model =,tr p a x10 x 10 ,. xI for M-Theory 2g 2 The covariant 3-algebra model for M-theory possesses a As a result, the total action is independent of x10 as large moduli that includes simultaneously diagonalizable follows, configurations. By treating appropriate configurations as 11 2 backgrounds, we derive the BFSS matrix theory and the Strpapa=, 2 IIB matrix model in the large N limit.