Journal of Modern Physics, 2013, 4, 32-37
http://dx.doi.org/10.4236/jmp.2013.44A006 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Zariski 3-Algebra Model of M-Theory
Matsuo Sato
Department of Natural Science, Faculty of Education, Hirosaki University, Hirosaki, Japan
Received January 15, 2013; revised February 20, 2013; accepted March 3, 2013
Copyright © 2013 Matsuo Sato. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We review on Zariski 3-algebra model of M-theory. The model is obtained by Zariski quantization of a semi-light-cone
supermembrane action. The model has manifest 1


supersymmetry in eleven dimensions and its relation to the
supermembrane action is clear.
Keywords: M-Theory; 3-Algebra; Matrix Model; String Theory
1. Introduction 3
2d4
1
212
MN
MMN
MNM N
i
SG
i


 






Recently, structures of 3-algebras [1-3] were found in the
effective actions of the multiple M2-branes [4-12] and
3-algebras have been intensively studied [13-29]. It had
been expected that structures of 3-algebras play more
fundamental roles in M-theory than the accidental struc-
tures in the effective descriptions, and 3-algebra models
of M-theory w e re pr op osed [30-34] .
In this paper, we review one of the models, called
Zariski 3-algebra model of M-theory. This model has
manifest 1

supersymmetry in eleven dimensions
and the relation to the supermembrane action is clear. We
start with the fact found in [32] that the supermembrane
action in a semi-light-cone gauge is a gauge theory based
on a 3-algebra that is generated by the Nambu-Poisson
bracket [13,14]. The gauge theory’s thirty-two super-
symmetries form the 1

supersymmetry algebra in
eleven dimensions. By performing the Zariski quantiza-
tion, the action is second quantized and we obtain Zariski
3-algebra model of M-theory.
2. Supermembrane Action in a
Semi-Light-Cone Gauge
In this section, we review the fact that the supermem-
brane action in a semi-light-cone gauge can be described
by Nambu bracket, where structures of 3-algebra are
manifest. The 3-algebra models of M-theory are defined
based on the semi-light-cone supermembrane action.
The fundamental degrees of freedom in M-theory are
supermembranes. The covariant supermembrane action
in M-theory [35] is given by


  

,0,,10,,,0,1,2, M
(1)


where
M
MN G
 
 
and
2
MM M
i
X
 
.

 


1,10SO Majorana fermion. is a

This action is invariant under dynamical supertrans-
formations,
,,
.
MM
Xi






1
(2)

These transformations form the

supersym-
metry algebra in eleven dimensions,



121 2
12
,2,
,0.
MM
Xi 

 



(3)
The action is also invariant under the -symmetry
transformations,



1,
1,
MM
Xi



 
  (4)
where 1
3!
LMN.
L
MN
G

 
 
 (5)
C
opyright © 2013 SciRes. JMP

M. SATO 33
If we fix the -symmetry (4) of the action by taking
a semi-light -cone gauge [32]

012 , (6)


we obtain a semi-light-cone supermembrane action,
3
21
d ,
IJ
MIJ
ii
SG XX
 
 

 
 

    
 
 


42 12


 (7)
where
,i Gh X
 
 
  ,
2
I
I

X
and
I
hX X
 
In [32], it is shown under an approximation up to the quadratic order in .

and but exactly in 


X
,
that this action is equivalent to
  
 
2
2
311 11
d,, ,,,,,,
1222 3
, ,,
24
,,
I
JKabIacdbe f
abab cdef
I J
ab IJ
SgXXXAXEAAA
ii
AXX




cl
ab




 



 



 (8)
where and ,,3, ,10IJK

,,
abcab c 



is the Nambu-Poisson bracket.
I
X
is a scalar and

is a




1,2 8SO SO
E Majorana-Weyl fermion satisfy- ing (6).


is a Levi-Civita symbol in three dimen-
sions and

is a cosmological constant.
(8) is invariant under 16 dynamical supersymmetry
transformations,

  


,, ,
21
,,, ,,
6
III I
I
ab IIJK
I IJK
i
Xi AXX
AX XXX


 


 
 


012


ab

 
(9)
where . These supersymmetries close into gauge transformations on-shell, 










 
12 12
21
122 121
,,,,,,,
,,,,,,,,2,
,,, ,
4
IcdI ab
cd ab
abcdab cdA
abcdab cd
cd KL
cd KL
XXA
AiO
i
iO


A




 
 
 




 



 
(10)
where gauge parameters are given by 0
A
O

0O
212 1
2
J
K
ababJKa b
iAi XX


  .
and

are equations of motions of

A


and

, respectively, where



 
  


,, ,, ,
2
1
,,,,.
2
ab I
abI J
ab IJ
EX A X
OA XX






 

 
{,,, ,,,, ,
,,
Aabcd abcd
ab cdabcd
Iab
OA AAA
i
 

  




2112 0
(11)
(10) implies that a commutation relation between the
dynamical supersymmetry transformations is

  (12)
up to the equations of motions and the gauge transforma-
tions.
This action is invariant under a translation,




,, ,
II
XA










 (13)
I

are constants. where

M. SATO 34
The action is also invariant under 16 kinematical su-
persymmetry transformations
,  


 
 
1
(14)
and the other fields are not transformed. is a constant
and satisfy 012 . and  should come from
sixteen components of thirty-two
  
supersymmetry consists of remaining 16 target-space
supersymmetries and transmuted 16 -symmetries in
the semi-light-cone gauge [32,36,37].

2112 0.
 
A commutation relation between the kinematical su-
persymmetry transformations is given by

supersym-
metry parameters in eleven dimensions, corresponding to
eigen values of , respectively. This
11
012





 


(15)
A commutator of dynamical supersymmetry transfor-
mations and kinematical ones acts as

  


21 1212021 121200
,, ,
2
III II
I
i
A XX
 
Xi

 


 
 

 
21 12,
(16)
where the commutator that acts on the other fields van-
ishes. Thus, the commutation relation is given by


  

(17)
where

is a translation.

If we change a basis of the supersymmetry transforma-
tions as

,,i

 

 
 (18)
We obtain
21 12
21 12
12
,
,
0.
21



 

 

  
 

 
 


,


(19)
These thirty-two supersymmetry transformations are
summarised as



 1
and (19) implies the


supersymmetry algebra in eleven dimensions,
21 12.


 


(20)
3. Zariski Quantization
In this section, we review the Zariski Quantization and
apply it for the semi-light-cone supermembrane action
(8). In [34], it is shown that the Zariski quantization is a
second quantization and the Zariski quantized action re-
duces to the supermembrane action if the fields are re-
stricted to one-body states.
First, we define elements of linear spaces by


rr
uu
YZ 


X0,
rr
r
ru 


 (21)
where the basis u
Z
are labeled by polynomials


,uuxx,
12
in the valuables 12
x
x
 
,uxx Z
with real or com-
plex coefficients. The summation is taken over all the
polynomials of two valuables

12 u satisfies
au u
Z
aZ

where a is a real (complex) number. The
coefficients


Yu

are functions over 3-dimensional
spaces. Summation is defined naturally as linear spaces.
The quantum Zariski product is defined as 





00000000
0000
00
0000
.
rrss
rs
rs
rs
uuv v
ru sv
uuv vuvuv
uvuv
YZ YZ
YZYZ YYZZ





 

 
 
 


 


 

XX
 


(22)
Any polynomial can be decomposed uniquely as
12
M

 


121 2, u vMhMMN
ZZabuuuuu u 

 

12MM N
vbuuu

uauu u
u, where is a real (complex) number
and are irreducible normalized polynomials.
a
i
uv
Z
Z is define d by
(23)
where

 
 
is defined by
12
121 21
:,
!N
N
MMMN S
uuuu uuuuu
N
 

 



 
(24)
where
N
S

1, 2,,N is the perm utation gr ou p of

is the Moyal product defined by


1122
112
2
,
rr
rr
r
ij ijij
riiij jj 0!
gfg
rx xxxxx


 
  

 
f
(25)

M. SATO 35
where and run from 1 to 2.
r
ir j

is define d b y
 
00
.
r
rr
ru
uZ







,ip
rr






 (26)
We define derivatives on by derivatives with
respect to as 1, 2,
i




0.
rr
r
rr
uu
ii
ru
YZ 








X (27)
One can show that the quantum Zariski product is
Abelian, associative and distributive, and the derivative
is commutative and satisfies the Leibniz rule [34].
We define the Zariski quantized Nambu-Poisson
bracket by

 
00 0000
000
000
,,
:
,
ijk ijk
ijk uv wuvw
ijk
uvw
YYYZZZ


 

  

 


 

X XXX
 



,, 1,2,3ijk

,, ,,,Y ABZ



 

,XX

 

,, 
  XX

(28)
where . By definition, the bracket is
skew-symmetric. By using the above properties, one can show that it satisfies the Leibniz rule and the fundamen- tal identity;
 
,,,,,,,,, ,,,AB XYZABXYZXABYZX
 



 

,, ,,ABXYZ 
(29)
for any . Thus, the Zariski quantized
Nambu-Poisson bracket has the same Nambu-Poisson structure as the original Nambu-Poison brac k et. We define a metric for by
 
 
00 00
00
00
00
00
,
00
,0
,d d
d,
r
r
pp
uv uv
uv
pr
uv w
uvr w
YY ZZ
YY Z

 


   






XXXXX X
 
where w Z is defined by
2
if, otherwise0,
ww
Zawaz Z
az
(30)
where is a real (complex) number and is a nor-
malized polynomial, whose monomial of the highest total
degree has coefficient 1.
This metric is invariant under a gauge transformation
generated by the Zariski quantized Nambu-Poisson
bracket [34] as
34121342
,,,, ,,0.

 


 


XXXXX XXX (31)
By performing the Zariski quantization of the super-
membrane action in a semi-light-cone gauge (8), we ob-
tain

2
2
3alg 11 1
,,,,,, ,,
12 23
IJKuabIuvwacdbef
Mabuuabcdefuvvuww
SE



 
 
 
 
XXXAXA A A

1

,, , ,.
24
ua b IJ
abu uIJ
ii




  
 
 
AX X



(32)
The Zariski quantization preserves the supersym-
metries of the semi-light-cone supermembrane theory,
because the quantum Zariski product is Abelian, associa-
tive and distributive, and admits a commutative deriva-
tive satisfying the Leibniz rule.
4. Conclusion
Zariski 3-algebra model of M-theory has manifest

supersymmetry in eleven dimensions because
Zariski quantization preserves the supersymmetry of the
supermembrane action in the semi-light-cone gauge. The
relation between the model and the supermembrane ac-
tion is clear: If the fields are restricted to one-body states,
the model reduces to the supermembrane action.
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