On Supersymmetry of the Covariant 3-Algebra Model for M-Theory

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)

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 supersymmetry 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 11LM 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].

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 MM10Mi 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  . TT10,0i ,,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 XXXMN,  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, although 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 MM where X = XTi and = iT . There is no ghost  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) 221i , 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 2222 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].

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 =,trpapa2, paxI (7) next section, when we derive the BFSS matrix theory and 4g 2   the IIB matrix model, X i iUN1, ,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. g  4 We consider backgrounds 1122 iij pa,,x  xx (9)   24 X ==diag,,,,pppp12 N (1) gg  i I pa ,,,  xi  X =0 (2) 22 1 X M =, M (3) where ij,,,9 d . In the large N limit, this action is 01g 0 equivalent to

2 ==0, (4) 1111di 2 ij2 0  St=d2  rFDxxx   , g   42 4 where  =0,1,,dd 1 10 and Id ,,10. ii i i 1  pp01,,, pd  1 ( i=1, ,N) represent N points  i Dx i ,,  randomly distributed in a d-dimensional space. There are 22 infinitely many such configurations. X M represents an 0 1 eleven-dimensional constant vector. By using SO(1,10) where  is redefined to  . This fact is proved symmetry, we can choose (3) as a background w itho ut g loss of generality. g will be identified with a coupling perturbatively and non-perturbatively in the large N limit M constant. g  corresponds to X 0 =0, which leads as in the case of the large N reduced model [41-44]. 6This kind of mechanism and interpretation was originally found in Under the conditions (1)-(6), the super transforma- [35]. tions (8) and (10) reduces to

 ai 3-algebra model for M-theory that is obtained by a matrix regularization of a supermembrane action. As a result,  xiII=  there are two possibilities for 3-algebra models for M- theory. One is a covariant 3-algebra model for M-theory i =,papa  xxij ,, ij that possesses more than 32 supersymmetries as in this   2g paper. Another is a   1 supersymmetric 3-algebra by which (9) is invariant. Moreover, (9) and (11) reduces model for M-theory that is o btained by a matrix regu- 7 to larization of a non-covariant supermembrane action .  X M =0 0 (10) 5. Acknowledgements  =0, 0 We would like to thank T. Asakawa, K. Hashimoto, N. because the action (5) reduces to the action (9) and Kamiya, H. Kunitomo, T. Matsuo, S. Moriyama, K. Mu- M 0 S =0. This is consistent with the fact that X 0 and rakami, J. Nishimura, S. Sasa, F. Sugino, T. Tada, S. 0 are fixed. Terashima, S. Watamura, K. Yoshida, and especially H. Therefore, if we choose the backgr ounds with d = 1 , Kawai and A. Tsuchiya for valuable discussions. we obtain the BFSS matrix theory in the large N limit,

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