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# 3-Algebras in String Theory

Written By

Matsuo Sato

Submitted: November 13th, 2011 Published: July 11th, 2012

DOI: 10.5772/46480

From the Edited Volume

## Linear Algebra

Edited by Hassan Abid Yasser

Chapter metrics overview

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## 1. Introduction

In this chapter, we review 3-algebras that appear as fundamental properties of string theory. 3-algebra is a generalization of Lie algebra; it is defined by a tri-linear bracket instead of by a bi-linear bracket, and satisfies fundamental identity, which is a generalization of Jacobi identity , , . We consider 3-algebras equipped with invariant metrics in order to apply them to physics.

It has been expected that there exists M-theory, which unifies string theories. In M-theory, some structures of 3-algebras were found recently. First, it was found that by using $u\left(N\right)\oplus u\left(N\right)$Hermitian 3-algebra, we can describe a low energy effective action of N coincident supermembranes , , , , , which are fundamental objects in M-theory.

With this as motivation, 3-algebras with invariant metrics were classified , , , , , , , , , , , , , . Lie 3-algebras are defined in real vector spaces and tri-linear brackets of them are totally anti-symmetric in all the three entries. Lie 3-algebras with invariant metrics are classified into ${𝒜}_{4}$algebra, and Lorentzian Lie 3-algebras, which have metrics with indefinite signatures. On the other hand, Hermitian 3-algebras are defined in Hermitian vector spaces and their tri-linear brackets are complex linear and anti-symmetric in the first two entries, whereas complex anti-linear in the third entry. Hermitian 3-algebras with invariant metrics are classified into $u\left(N\right)\oplus u\left(M\right)$and $sp\left(2N\right)\oplus u\left(1\right)$Hermitian 3-algebras.

Moreover, recent studies have indicated that there also exist structures of 3-algebras in the Green-Schwartz supermembrane action, which defines full perturbative dynamics of a supermembrane. It had not been clear whether the total supermembrane action including fermions has structures of 3-algebras, whereas the bosonic part of the action can be described by using a tri-linear bracket, called Nambu bracket , , which is a generalization of Poisson bracket. If we fix to a light-cone gauge, the total action can be described by using Poisson bracket, that is, only structures of Lie algebra are left in this gauge . However, it was shown under an approximation that the total action can be described by Nambu bracket if we fix to a semi-light-cone gauge . In this gauge, the eleven dimensional space-time of M-theory is manifest in the supermembrane action, whereas only ten dimensional part is manifest in the light-cone gauge.

The BFSS matrix theory is conjectured to describe an infinite momentum frame (IMF) limit of M-theory  and many evidences were found. The action of the BFSS matrix theory can be obtained by replacing Poisson bracket with a finite dimensional Lie algebra's bracket in the supermembrane action in the light-cone gauge. Because of this structure, only variables that represent the ten dimensional part of the eleven-dimensional space-time are manifest in the BFSS matrix theory. Recently, 3-algebra models of M-theory were proposed , , , by replacing Nambu bracket with finite dimensional 3-algebras' brackets in an action that is shown, by using an approximation, to be equivalent to the semi-light-cone supermembrane action. All the variables that represent the eleven dimensional space-time are manifest in these models. It was shown that if the DLCQ limit of the 3-algebra models of M-theory is taken, they reduce to the BFSS matrix theory , , as they should , , , , , .

## 2. Definition and classification of metric Hermitian 3-algebra

In this section, we will define and classify the Hermitian 3-algebras equipped with invariant metrics.

### 2.1. General structure of metric Hermitian 3-algebra

The metric Hermitian 3-algebra is a map $V×V×V\to V$defined by $\left(x,y,z\right)↦\left[x,y;z\right]$, where the 3-bracket is complex linear in the first two entries, whereas complex anti-linear in the last entry, equipped with a metric $$, satisfying the following properties:

the fundamental identity

$\left[\left[x,y;z\right],v;w\right]=\left[\left[x,v;w\right],y;z\right]+\left[x,\left[y,v;w\right];z\right]-\left[x,y;\left[z,w;v\right]\right]$uid2

the metric invariance

$<\left[x,v;w\right],y>-=0$uid3

and the anti-symmetry

$\left[x,y;z\right]=-\left[y,x;z\right]$uid4

for

$x,y,z,v,w\in V$uid5

The Hermitian 3-algebra generates a symmetry, whose generators $D\left(x,y\right)$are defined by

$D\left(x,y\right)z:=\left[z,x;y\right]$uid6

From (), one can show that $D\left(x,y\right)$form a Lie algebra,

$\left[D\left(x,y\right),D\left(v,w\right)\right]=D\left(D\left(x,y\right)v,w\right)-D\left(v,D\left(y,x\right)w\right)$uid7

There is an one-to-one correspondence between the metric Hermitian 3-algebra and a class of metric complex super Lie algebras . Such a class satisfies the following conditions among complex super Lie algebras $S={S}_{0}\oplus {S}_{1}$, where ${S}_{0}$and ${S}_{1}$are even and odd parts, respectively. ${S}_{1}$is decomposed as ${S}_{1}=V\oplus \overline{V}$, where $V$is an unitary representation of ${S}_{0}$: for $a\in {S}_{0}$, $u,v\in V$,

$\left[a,u\right]\in V$uid8

and

$<\left[a,u\right],v>+=0$uid9

$\overline{v}\in \overline{V}$is defined by

$\overline{v}=<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},v>$uid10

The super Lie bracket satisfies

$\begin{array}{c}\hfill \left[V,V\right]=0,\phantom{\rule{1.em}{0ex}}\left[\overline{V},\overline{V}\right]=0\end{array}$uid11

From the metric Hermitian 3-algebra, we obtain the class of the metric complex super Lie algebra in the following way. The elements in ${S}_{0}$, $V$, and $\overline{V}$are defined by (), (), and (), respectively. The algebra is defined by () and

$\begin{array}{ccc}& & \left[D\left(x,y\right),z\right]:=D\left(x,y\right)z=\left[z,x;y\right]\hfill \\ & & \left[D\left(x,y\right),\overline{z}\right]:=-\overline{D\left(y,x\right)z}=-\overline{\left[z,y;x\right]}\hfill \\ & & \left[x,\overline{y}\right]:=D\left(x,y\right)\hfill \\ & & \left[x,y\right]:=0\hfill \\ & & \left[\overline{x},\overline{y}\right]:=0\hfill \end{array}$uid12

One can show that this algebra satisfies the super Jacobi identity and ()-() as in .

Inversely, from the class of the metric complex super Lie algebra, we obtain the metric Hermitian 3-algebra by

$\left[x,y;z\right]:=\alpha \left[\left[y,\overline{z}\right],x\right]$uid13

where $\alpha$is an arbitrary constant. One can also show that this algebra satisfies ()-() for () as in .

### 2.2. Classification of metric Hermitian 3-algebra

The classical Lie super algebras satisfying ()-() are $A\left(m-1,n-1\right)$and $C\left(n+1\right)$. The even parts of $A\left(m-1,n-1\right)$and $C\left(n+1\right)$are $u\left(m\right)\oplus u\left(n\right)$and $sp\left(2n\right)\oplus u\left(1\right)$, respectively. Because the metric Hermitian 3-algebra one-to-one corresponds to this class of the super Lie algebra, the metric Hermitian 3-algebras are classified into $u\left(m\right)\oplus u\left(n\right)$and $sp\left(2n\right)\oplus u\left(1\right)$Hermitian 3-algebras.

First, we will construct the $u\left(m\right)\oplus u\left(n\right)$Hermitian 3-algebra from $A\left(m-1,n-1\right)$, according to the relation in the previous subsection. $A\left(m-1,n-1\right)$is simple and is obtained by dividing $sl\left(m,n\right)$by its ideal. That is, $A\left(m-1,n-1\right)=sl\left(m,n\right)$when $m\ne n$and $A\left(n-1,n-1\right)=sl\left(n,n\right)/\lambda {1}_{2n}$.

Real $sl\left(m,n\right)$is defined by

$\left(\begin{array}{cc}{h}_{1}& c\\ i{c}^{†}& {h}_{2}\end{array}\right)$uid15

where ${h}_{1}$and ${h}_{2}$are $m×m$and $n×n$anti-Hermite matrices and $c$is an $n×m$arbitrary complex matrix. Complex $sl\left(m,n\right)$is a complexification of real $sl\left(m,n\right)$, given by

$\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$uid16

where $\alpha$, $\beta$, $\gamma$, and $\delta$are $m×m$, $n×m$, $m×n$, and $n×n$complex matrices that satisfy

$\text{tr}\alpha =\text{tr}\delta$uid17

Complex $A\left(m-1,n-1\right)$is decomposed as $A\left(m-1,n-1\right)={S}_{0}\oplus V\oplus \overline{V}$, where

$\begin{array}{c}\hfill \left(\begin{array}{cc}\alpha & 0\\ 0& \delta \end{array}\right)\in {S}_{0}\\ \hfill \left(\begin{array}{cc}0& \beta \\ 0& 0\end{array}\right)\in V\\ \hfill \left(\begin{array}{cc}0& 0\\ \gamma & 0\end{array}\right)\in \overline{V}\end{array}$uid18

() is rewritten as $V\to \overline{V}$defined by

$B=\left(\begin{array}{cc}0& \beta \\ 0& 0\end{array}\right)↦{B}^{†}=\left(\begin{array}{cc}0& 0\\ {\beta }^{†}& 0\end{array}\right)$uid19

where $B\in V$and ${B}^{†}\in \overline{V}$. () is rewritten as

$\left[X,Y;Z\right]=\alpha \left[\left[Y,{Z}^{†}\right],X\right]=\alpha \left(\begin{array}{cc}0& y{z}^{†}x-x{z}^{†}y\\ 0& 0\end{array}\right)$uid20

for

$\begin{array}{c}\hfill X=\left(\begin{array}{cc}0& x\\ 0& 0\end{array}\right)\in V\\ \hfill Y=\left(\begin{array}{cc}0& y\\ 0& 0\end{array}\right)\in V\\ \hfill Z=\left(\begin{array}{cc}0& z\\ 0& 0\end{array}\right)\in V\end{array}$uid21

As a result, we obtain the $u\left(m\right)\oplus u\left(n\right)$Hermitian 3-algebra,

$\begin{array}{c}\hfill \left[x,y;z\right]=\alpha \left(y{z}^{†}x-x{z}^{†}y\right)\end{array}$uid22

where $x$, $y$, and $z$are arbitrary $n×m$complex matrices. This algebra was originally constructed in .

Inversely, from (), we can construct complex $A\left(m-1,n-1\right)$. () is rewritten as

$D\left(x,y\right)=\left(x{y}^{†},{y}^{†}x\right)\in {S}_{0}$uid23

() and () are rewritten as

$\begin{array}{ccc}& & \left[\left(x{y}^{†},{y}^{†}x\right),\left({x}^{\text{'}}{y}^{\text{'}†},{y}^{\text{'}†}{x}^{\text{'}}\right)\right]=\left(\left[x{y}^{†},{x}^{\text{'}}{y}^{\text{'}†}\right],\left[{y}^{†}x,{y}^{\text{'}†}{x}^{\text{'}}\right]\right)\hfill \\ & & \left[\left(x{y}^{†},{y}^{†}x\right),z\right]=x{y}^{†}z-z{y}^{†}x\hfill \\ & & \left[\left(x{y}^{†},{y}^{†}x\right),{w}^{†}\right]={y}^{†}x{w}^{†}-{w}^{†}x{y}^{†}\hfill \\ & & \left[x,{y}^{†}\right]=\left(x{y}^{†},{y}^{†}x\right)\hfill \\ & & \left[x,y\right]=0\hfill \\ & & \left[{x}^{†},{y}^{†}\right]=0\hfill \end{array}$uid24

This algebra is summarized as

$\left[\left(\begin{array}{cc}x{y}^{†}& z\\ {w}^{†}& {y}^{†}x\end{array}\right),\left(\begin{array}{cc}{x}^{\text{'}}{y}^{\text{'}†}& {z}^{\text{'}}\\ {w}^{\text{'}†}& {y}^{\text{'}†}{x}^{\text{'}}\end{array}\right)\right]$uid25

which forms complex $A\left(m-1,n-1\right)$.

Next, we will construct the $sp\left(2n\right)\oplus u\left(1\right)$Hermitian 3-algebra from $C\left(n+1\right)$. Complex $C\left(n+1\right)$is decomposed as $C\left(n+1\right)={S}_{0}\oplus V\oplus \overline{V}$. The elements are given by

$\begin{array}{c}\hfill \left(\begin{array}{cccc}\alpha & 0& 0& 0\\ 0& -\alpha & 0& 0\\ 0& 0& a& b\\ 0& 0& c& -{a}^{T}\end{array}\right)\in {S}_{0}\\ \hfill \left(\begin{array}{cccc}0& 0& {x}_{1}& {x}_{2}\\ 0& 0& 0& 0\\ 0& {x}_{2}^{T}& 0& 0\\ 0& -{x}_{1}^{T}& 0& 0\end{array}\right)\in V\\ \hfill \left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& {y}_{1}& {y}_{2}\\ {y}_{2}^{T}& 0& 0& 0\\ -{y}_{1}^{T}& 0& 0& 0\end{array}\right)\in \overline{V}\end{array}$uid26

where $\alpha$is a complex number, $a$is an arbitrary $n×n$complex matrix, $b$and $c$are $n×n$complex symmetric matrices, and ${x}_{1}$, ${x}_{2}$, ${y}_{1}$and ${y}_{2}$are $n×1$complex matrices. () is rewritten as $V\to \overline{V}$defined by $B↦\overline{B}=U{B}^{*}{U}^{-1}$, where $B\in V$, $\overline{B}\in \overline{V}$and

$U=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)$uid27

Explicitly,

$B=\left(\begin{array}{cccc}0& 0& {x}_{1}& {x}_{2}\\ 0& 0& 0& 0\\ 0& {x}_{2}^{T}& 0& 0\\ 0& -{x}_{1}^{T}& 0& 0\end{array}\right)↦\overline{B}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& {x}_{2}^{*}& -{x}_{1}^{*}\\ -{x}_{1}^{†}& 0& 0& 0\\ -{x}_{2}^{†}& 0& 0& 0\end{array}\right)$uid28

() is rewritten as

$\begin{array}{ccc}\hfill \left[X,Y;Z\right]& :=& \alpha \left[\left[Y,\overline{Z}\right],X\right]\hfill \\ & =& \alpha \left[\left[\left(\begin{array}{cccc}0& 0& {y}_{1}& {y}_{2}\\ 0& 0& 0& 0\\ 0& {y}_{2}^{T}& 0& 0\\ 0& -{y}_{1}^{T}& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& {z}_{2}^{*}& -{z}_{1}^{*}\\ -{z}_{1}^{†}& 0& 0& 0\\ -{z}_{2}^{†}& 0& 0& 0\end{array}\right)\right],\left(\begin{array}{cccc}0& 0& {x}_{1}& {x}_{2}\\ 0& 0& 0& 0\\ 0& {x}_{2}^{T}& 0& 0\\ 0& -{x}_{1}^{T}& 0& 0\end{array}\right)\right]\hfill \\ & =& \alpha \left(\begin{array}{cccc}0& 0& {w}_{1}& {w}_{2}\\ 0& 0& 0& 0\\ 0& {w}_{2}^{T}& 0& 0\\ 0& -{w}_{1}^{T}& 0& 0\end{array}\right)\hfill \end{array}$uid29

for

$\begin{array}{c}\hfill X=\left(\begin{array}{cccc}0& 0& {x}_{1}& {x}_{2}\\ 0& 0& 0& 0\\ 0& {x}_{2}^{T}& 0& 0\\ 0& -{x}_{1}^{T}& 0& 0\end{array}\right)\in V\\ \hfill Y=\left(\begin{array}{cccc}0& 0& {y}_{1}& {y}_{2}\\ 0& 0& 0& 0\\ 0& {y}_{2}^{T}& 0& 0\\ 0& -{y}_{1}^{T}& 0& 0\end{array}\right)\in V\\ \hfill Z=\left(\begin{array}{cccc}0& 0& {z}_{1}& {z}_{2}\\ 0& 0& 0& 0\\ 0& {z}_{2}^{T}& 0& 0\\ 0& -{z}_{1}^{T}& 0& 0\end{array}\right)\in V\end{array}$uid30

where ${w}_{1}$and ${w}_{2}$are given by

$\left({w}_{1},{w}_{2}\right)=-\left({y}_{1}{z}_{1}^{†}+{y}_{2}{z}_{2}^{†}\right)\left({x}_{1},{x}_{2}\right)+\left({x}_{1}{z}_{1}^{†}+{x}_{2}{z}_{2}^{†}\right)\left({y}_{1},{y}_{2}\right)+\left({x}_{2}{y}_{1}^{T}-{x}_{1}{y}_{2}^{T}\right)\left({z}_{2}^{*},-{z}_{1}^{*}\right)$uid31

As a result, we obtain the $sp\left(2n\right)\oplus u\left(1\right)$Hermitian 3-algebra,

$\left[x,y;z\right]=\alpha \left(\left(y\odot \stackrel{˜}{z}\right)x+\left(\stackrel{˜}{z}\odot x\right)y-\left(x\odot y\right)\stackrel{˜}{z}\right)$uid32

for $x=\left({x}_{1},{x}_{2}\right)$, $y=\left({y}_{1},{y}_{2}\right)$, $z=\left({z}_{1},{z}_{2}\right)$, where ${x}_{1}$, ${x}_{2}$, ${y}_{1}$, ${y}_{2}$, ${z}_{1}$, and ${z}_{2}$are n-vectors and

$\begin{array}{ccc}\hfill \stackrel{˜}{z}& =& \left({z}_{2}^{*},-{z}_{1}^{*}\right)\hfill \\ \hfill a\odot b& =& {a}_{1}·{b}_{2}-{a}_{2}·{b}_{1}\hfill \end{array}$uid33

## 3. 3-algebra model of M-theory

In this section, we review the fact that the supermembrane 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. We also review that the models reduce to the BFSS matrix theory in the DLCQ limit.

### 3.1. Supermembrane and 3-algebra model of M-theory

The fundamental degrees of freedom in M-theory are supermembranes. The action of the covariant supermembrane action in M-theory  is given by

$\begin{array}{ccc}\hfill {S}_{M2}& =& \int {d}^{3}\sigma \left(\sqrt{-G}+\frac{i}{4}{ϵ}^{\alpha \beta \gamma }\overline{\Psi }{\Gamma }_{MN}{\partial }_{\alpha }\Psi \left({\Pi }_{\beta }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}M}{\Pi }_{\gamma }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}N}+\frac{i}{2}{\Pi }_{\beta }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}M}\overline{\Psi }{\Gamma }^{N}{\partial }_{\gamma }\Psi \hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}-\frac{1}{12}\overline{\Psi }{\Gamma }^{M}{\partial }_{\beta }\Psi \overline{\Psi }{\Gamma }^{N}{\partial }_{\gamma }\Psi \right)\right)\hfill \end{array}$uid35

where $M,N=0,\cdots ,10$, $\alpha ,\beta ,\gamma =0,1,2$, ${G}_{\alpha \beta }={\Pi }_{\alpha }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}M}{\Pi }_{\beta M}$and ${\Pi }_{\alpha }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}M}={\partial }_{\alpha }{X}^{M}-\frac{i}{2}\overline{\Psi }{\Gamma }^{M}{\partial }_{\alpha }\Psi$. $\Psi$is a $SO\left(1,10\right)$Majorana fermion.

This action is invariant under dynamical supertransformations,

$\begin{array}{ccc}\hfill \delta \Psi & =& ϵ\hfill \\ \hfill \delta {X}^{M}& =& -i\overline{\Psi }{\Gamma }^{M}ϵ\hfill \end{array}$uid36

These transformations form the $𝒩=1$supersymmetry algebra in eleven dimensions,

$\begin{array}{ccc}\hfill \left[{\delta }_{1},{\delta }_{2}\right]{X}^{M}& =& -2i{ϵ}_{1}{\Gamma }^{M}{ϵ}_{2}\hfill \end{array}$uid37
$\begin{array}{ccc}\hfill \left[{\delta }_{1},{\delta }_{2}\right]\Psi & =& 0\hfill \end{array}$uid38

The action is also invariant under the $\kappa$-symmetry transformations,

$\begin{array}{ccc}\hfill \delta \Psi & =& \left(1+\Gamma \right)\kappa \left(\sigma \right)\hfill \\ \hfill \delta {X}^{M}& =& i\overline{\Psi }{\Gamma }^{M}\left(1+\Gamma \right)\kappa \left(\sigma \right)\hfill \end{array}$uid39

where

$\Gamma =\frac{1}{3!\sqrt{-G}}{ϵ}^{\alpha \beta \gamma }{\Pi }_{\alpha }^{L}{\Pi }_{\beta }^{M}{\Pi }_{\gamma }^{N}{\Gamma }_{LMN}$uid40

If we fix the $\kappa$-symmetry () of the action by taking a semi-light-cone gauge Advantages of a semi-light-cone gauges against a light-cone gauge are shown in , , 

${\Gamma }^{012}\Psi =-\Psi$uid42

we obtain a semi-light-cone supermembrane action,

$\begin{array}{ccc}\hfill {S}_{M2}& =& \int {d}^{3}\sigma \left(\sqrt{-G}+\frac{i}{4}{ϵ}^{\alpha \beta \gamma }\left(\overline{\Psi }{\Gamma }_{\mu \nu }{\partial }_{\alpha }\Psi \left({\Pi }_{\beta }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mu }{\Pi }_{\gamma }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\nu }+\frac{i}{2}{\Pi }_{\beta }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mu }\overline{\Psi }{\Gamma }^{\nu }{\partial }_{\gamma }\Psi -\frac{1}{12}\overline{\Psi }{\Gamma }^{\mu }{\partial }_{\beta }\Psi \overline{\Psi }{\Gamma }^{\nu }{\partial }_{\gamma }\Psi \right)\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}+\overline{\Psi }{\Gamma }_{IJ}{\partial }_{\alpha }\Psi {\partial }_{\beta }{X}^{I}{\partial }_{\gamma }{X}^{J}\right)\right)\hfill \end{array}$uid43

where ${G}_{\alpha \beta }={h}_{\alpha \beta }+{\Pi }_{\alpha }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mu }{\Pi }_{\beta \mu }$, ${\Pi }_{\alpha }^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mu }={\partial }_{\alpha }{X}^{\mu }-\frac{i}{2}\overline{\Psi }{\Gamma }^{\mu }{\partial }_{\alpha }\Psi$, and ${h}_{\alpha \beta }={\partial }_{\alpha }{X}^{I}{\partial }_{\beta }{X}_{I}$.

In , it is shown under an approximation up to the quadratic order in ${\partial }_{\alpha }{X}^{\mu }$and ${\partial }_{\alpha }\Psi$but exactly in ${X}^{I}$, that this action is equivalent to the continuum action of the 3-algebra model of M-theory,

$\begin{array}{ccc}\hfill {S}_{cl}& =& \int {d}^{3}\sigma \sqrt{-g}\left(-\frac{1}{12}{\left\{{X}^{I},{X}^{J},{X}^{K}\right\}}^{2}-\frac{1}{2}{\left({A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},{X}^{I}\right\}\right)}^{2}\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}-\frac{1}{3}{E}^{\mu \nu \lambda }{A}_{\mu ab}{A}_{\nu cd}{A}_{\lambda ef}\left\{{\varphi }^{a},{\varphi }^{c},{\varphi }^{d}\right\}\left\{{\varphi }^{b},{\varphi }^{e},{\varphi }^{f}\right\}+\frac{1}{2}\Lambda \hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}-\frac{i}{2}\overline{\Psi }{\Gamma }^{\mu }{A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},\Psi \right\}+\frac{i}{4}\overline{\Psi }{\Gamma }_{IJ}\left\{{X}^{I},{X}^{J},\Psi \right\}\right)\hfill \end{array}$uid44

where $I,J,K=3,\cdots ,10$and $\left\{{\varphi }^{a},{\varphi }^{b},{\varphi }^{c}\right\}={ϵ}^{\alpha \beta \gamma }{\partial }_{\alpha }{\varphi }^{a}{\partial }_{\beta }{\varphi }^{b}{\partial }_{\gamma }{\varphi }^{c}$is the Nambu-Poisson bracket. An invariant symmetric bilinear form is defined by $\int {d}^{3}\sigma \sqrt{-g}{\varphi }^{a}{\varphi }^{b}$for complete basis ${\varphi }^{a}$in three dimensions. Thus, this action is manifestly VPD covariant even when the world-volume metric is flat. ${X}^{I}$is a scalar and $\Psi$is a $SO\left(1,2\right)×SO\left(8\right)$Majorana-Weyl fermion satisfying (). ${E}^{\mu \nu \lambda }$is a Levi-Civita symbol in three dimensions and $\Lambda$is a cosmological constant.

The continuum action of 3-algebra model of M-theory () is invariant under 16 dynamical supersymmetry transformations,

$\begin{array}{ccc}& & \delta {X}^{I}=i\overline{ϵ}{\Gamma }^{I}\Psi \hfill \\ & & \delta {A}_{\mu }\left(\sigma ,{\sigma }^{\text{'}}\right)=\frac{i}{2}\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\left({X}^{I}\left(\sigma \right)\Psi \left({\sigma }^{\text{'}}\right)-{X}^{I}\left({\sigma }^{\text{'}}\right)\Psi \left(\sigma \right)\right),\hfill \\ & & \delta \Psi =-{A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},{X}^{I}\right\}{\Gamma }^{\mu }{\Gamma }_{I}ϵ-\frac{1}{6}\left\{{X}^{I},{X}^{J},{X}^{K}\right\}{\Gamma }_{IJK}ϵ\hfill \end{array}$uid45

where ${\Gamma }_{012}ϵ=-ϵ$. These supersymmetries close into gauge transformations on-shell,

$\begin{array}{ccc}& & \left[{\delta }_{1},{\delta }_{2}\right]{X}^{I}={\Lambda }_{cd}\left\{{\varphi }^{c},{\varphi }^{d},{X}^{I}\right\}\hfill \\ & & \left[{\delta }_{1},{\delta }_{2}\right]{A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},\phantom{\rule{1.em}{0ex}}\right\}={\Lambda }_{ab}\left\{{\varphi }^{a},{\varphi }^{b},{A}_{\mu cd}\left\{{\varphi }^{c},{\varphi }^{d},\phantom{\rule{1.em}{0ex}}\right\}\right\}\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}-{A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},{\Lambda }_{cd}\left\{{\varphi }^{c},{\varphi }^{d},\phantom{\rule{1.em}{0ex}}\right\}\right\}+2i{\overline{ϵ}}_{2}{\Gamma }^{\nu }{ϵ}_{1}{O}_{\mu \nu }^{A}\hfill \\ & & \left[{\delta }_{1},{\delta }_{2}\right]\Psi ={\Lambda }_{cd}\left\{{\varphi }^{c},{\varphi }^{d},\Psi \right\}+\left(i{\overline{ϵ}}_{2}{\Gamma }^{\mu }{ϵ}_{1}{\Gamma }_{\mu }-\frac{i}{4}{\overline{ϵ}}_{2}{\Gamma }^{KL}{ϵ}_{1}{\Gamma }_{KL}\right){O}^{\Psi }\hfill \end{array}$uid46

where gauge parameters are given by ${\Lambda }_{ab}=2i{\overline{ϵ}}_{2}{\Gamma }^{\mu }{ϵ}_{1}{A}_{\mu ab}-i{\overline{ϵ}}_{2}{\Gamma }_{JK}{ϵ}_{1}{X}_{a}^{J}{X}_{b}^{K}$. ${O}_{\mu \nu }^{A}=0$and ${O}^{\Psi }=0$are equations of motions of ${A}_{\mu \nu }$and $\Psi$, respectively, where

$\begin{array}{ccc}\hfill {O}_{\mu \nu }^{A}& =& {A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},{A}_{\nu cd}\left\{{\varphi }^{c},{\varphi }^{d},\phantom{\rule{1.em}{0ex}}\right\}\right\}-{A}_{\nu ab}\left\{{\varphi }^{a},{\varphi }^{b},{A}_{\mu cd}\left\{{\varphi }^{c},{\varphi }^{d},\phantom{\rule{1.em}{0ex}}\right\}\right\}\hfill \\ & & +{E}_{\mu \nu \lambda }\left(-\left\{{X}^{I},{A}_{ab}^{\lambda }\left\{{\varphi }^{a},{\varphi }^{b},{X}_{I}\right\},\phantom{\rule{1.em}{0ex}}\right\}+\frac{i}{2}\left\{\overline{\Psi },{\Gamma }^{\lambda }\Psi ,\phantom{\rule{1.em}{0ex}}\right\}\right)\hfill \\ \hfill {O}^{\Psi }& =& -{\Gamma }^{\mu }{A}_{\mu ab}\left\{{\varphi }^{a},{\varphi }^{b},\Psi \right\}+\frac{1}{2}{\Gamma }_{IJ}\left\{{X}^{I},{X}^{J},\Psi \right\}\hfill \end{array}$uid47

() implies that a commutation relation between the dynamical supersymmetry transformations is

${\delta }_{2}{\delta }_{1}-{\delta }_{1}{\delta }_{2}=0$uid48

up to the equations of motions and the gauge transformations.

This action is invariant under a translation,

$\delta {X}^{I}\left(\sigma \right)={\eta }^{I},\phantom{\rule{2.em}{0ex}}\delta {A}^{\mu }\left(\sigma ,{\sigma }^{\text{'}}\right)={\eta }^{\mu }\left(\sigma \right)-{\eta }^{\mu }\left({\sigma }^{\text{'}}\right)$uid49

where ${\eta }^{I}$are constants.

The action is also invariant under 16 kinematical supersymmetry transformations

$\stackrel{˜}{\delta }\Psi =\stackrel{˜}{ϵ}$uid50

and the other fields are not transformed. $\stackrel{˜}{ϵ}$is a constant and satisfy ${\Gamma }_{012}\stackrel{˜}{ϵ}=\stackrel{˜}{ϵ}$. $\stackrel{˜}{ϵ}$and $ϵ$should come from sixteen components of thirty-two $𝒩=1$supersymmetry parameters in eleven dimensions, corresponding to eigen values $±$1 of ${\Gamma }_{012}$, respectively. This $𝒩=1$supersymmetry consists of remaining 16 target-space supersymmetries and transmuted 16 $\kappa$-symmetries in the semi-light-cone gauge , , .

A commutation relation between the kinematical supersymmetry transformations is given by

${\stackrel{˜}{\delta }}_{2}{\stackrel{˜}{\delta }}_{1}-{\stackrel{˜}{\delta }}_{1}{\stackrel{˜}{\delta }}_{2}=0$uid51

A commutator of dynamical supersymmetry transformations and kinematical ones acts as

$\begin{array}{ccc}& & \left({\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}\right){X}^{I}\left(\sigma \right)=i{\overline{ϵ}}_{1}{\Gamma }^{I}{\stackrel{˜}{ϵ}}_{2}\equiv {\eta }_{0}^{I}\hfill \\ & & \left({\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}\right){A}^{\mu }\left(\sigma ,{\sigma }^{\text{'}}\right)=\frac{i}{2}{\overline{ϵ}}_{1}{\Gamma }^{\mu }{\Gamma }_{I}\left({X}^{I}\left(\sigma \right)-{X}^{I}\left({\sigma }^{\text{'}}\right)\right){\stackrel{˜}{ϵ}}_{2}\equiv {\eta }_{0}^{\mu }\left(\sigma \right)-{\eta }_{0}^{\mu }\left({\sigma }^{\text{'}}\right)\hfill \end{array}$uid52

where the commutator that acts on the other fields vanishes. Thus, the commutation relation is given by

${\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}={\delta }_{\eta }$uid53

where ${\delta }_{\eta }$is a translation.

If we change a basis of the supersymmetry transformations as

$\begin{array}{ccc}& & {\delta }^{\text{'}}=\delta +\stackrel{˜}{\delta }\hfill \\ & & {\stackrel{˜}{\delta }}^{\text{'}}=i\left(\delta -\stackrel{˜}{\delta }\right)\hfill \end{array}$uid54

we obtain

$\begin{array}{ccc}& & {\delta }_{2}^{\text{'}}{\delta }_{1}^{\text{'}}-{\delta }_{1}^{\text{'}}{\delta }_{2}^{\text{'}}={\delta }_{\eta }\hfill \\ & & {\stackrel{˜}{\delta }}_{2}^{\text{'}}{\stackrel{˜}{\delta }}_{1}^{\text{'}}-{\stackrel{˜}{\delta }}_{1}^{\text{'}}{\stackrel{˜}{\delta }}_{2}^{\text{'}}={\delta }_{\eta }\hfill \\ & & {\stackrel{˜}{\delta }}_{2}^{\text{'}}{\delta }_{1}^{\text{'}}-{\delta }_{1}^{\text{'}}{\stackrel{˜}{\delta }}_{2}^{\text{'}}=0\hfill \end{array}$uid55

These thirty-two supersymmetry transformations are summarised as $\Delta =\left({\delta }^{\text{'}},{\stackrel{˜}{\delta }}^{\text{'}}\right)$and () implies the $𝒩=1$supersymmetry algebra in eleven dimensions,

${\Delta }_{2}{\Delta }_{1}-{\Delta }_{1}{\Delta }_{2}={\delta }_{\eta }$uid56

### 3.2. Lie 3-algebra models of M-theory

In this and next subsection, we perform the second quantization on the continuum action of the 3-algebra model of M-theory: By replacing the Nambu-Poisson bracket in the action () with brackets of finite-dimensional 3-algebras, Lie and Hermitian 3-algebras, we obtain the Lie and Hermitian 3-algebra models of M-theory , , respectively. In this section, we review the Lie 3-algebra model.

If we replace the Nambu-Poisson bracket in the action () with a completely antisymmetric real 3-algebra's bracket , ,

$\begin{array}{ccc}& & \int {d}^{3}\sigma \sqrt{-g}\to ⟨\phantom{\rule{1.em}{0ex}}⟩\hfill \\ & & \left\{{\varphi }^{a},{\varphi }^{b},{\varphi }^{c}\right\}\to \left[{T}^{a},{T}^{b},{T}^{c}\right]\hfill \end{array}$uid58

we obtain the Lie 3-algebra model of M-theory , ,

$\begin{array}{ccc}\hfill {S}_{0}& =& <-\frac{1}{12}{\left[{X}^{I},{X}^{J},{X}^{K}\right]}^{2}-\frac{1}{2}{\left({A}_{\mu ab}\left[{T}^{a},{T}^{b},{X}^{I}\right]\right)}^{2}\hfill \\ & & \phantom{\rule{1.em}{0ex}}-\frac{1}{3}{E}^{\mu \nu \lambda }{A}_{\mu ab}{A}_{\nu cd}{A}_{\lambda ef}\left[{T}^{a},{T}^{c},{T}^{d}\right]\left[{T}^{b},{T}^{e},{T}^{f}\right]\hfill \\ & & \phantom{\rule{1.em}{0ex}}-\frac{i}{2}\overline{\Psi }{\Gamma }^{\mu }{A}_{\mu ab}\left[{T}^{a},{T}^{b},\Psi \right]+\frac{i}{4}\overline{\Psi }{\Gamma }_{IJ}\left[{X}^{I},{X}^{J},\Psi \right]>\hfill \end{array}$uid59

We have deleted the cosmological constant $\Lambda$, which corresponds to an operator ordering ambiguity, as usual as in the case of other matrix models , .

This model can be obtained formally by a dimensional reduction of the $𝒩=8$BLG model , , ,

$\begin{array}{ccc}\hfill {S}_{𝒩=8BLG}& =& \int {d}^{3}x<-\frac{1}{12}{\left[{X}^{I},{X}^{J},{X}^{K}\right]}^{2}-\frac{1}{2}{\left({D}_{\mu }{X}^{I}\right)}^{2}-{E}^{\mu \nu \lambda }\left(\frac{1}{2}{A}_{\mu ab}{\partial }_{\nu }{A}_{\lambda cd}{T}^{a}\left[{T}^{b},{T}^{c},{T}^{d}\right]\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}+\frac{1}{3}{A}_{\mu ab}{A}_{\nu cd}{A}_{\lambda ef}\left[{T}^{a},{T}^{c},{T}^{d}\right]\left[{T}^{b},{T}^{e},{T}^{f}\right]\right)\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}+\frac{i}{2}\overline{\Psi }{\Gamma }^{\mu }{D}_{\mu }\Psi +\frac{i}{4}\overline{\Psi }{\Gamma }_{IJ}\left[{X}^{I},{X}^{J},\Psi \right]>\hfill \end{array}$uid60

The formal relations between the Lie (Hermitian) 3-algebra models of M-theory and the $𝒩=8$($𝒩=6$) BLG models are analogous to the relation among the $𝒩=4$super Yang-Mills in four dimensions, the BFSS matrix theory , and the IIB matrix model . They are completely different theories although they are related to each others by dimensional reductions. In the same way, the 3-algebra models of M-theory and the BLG models are completely different theories.

The fields in the action () are spanned by the Lie 3-algebra ${T}^{a}$as ${X}^{I}={X}_{a}^{I}{T}^{a}$, $\Psi ={\Psi }_{a}{T}^{a}$and ${A}^{\mu }={A}_{ab}^{\mu }{T}^{a}\otimes {T}^{b}$, where $I=3,\cdots ,10$and $\mu =0,1,2$. $<>$represents a metric for the 3-algebra. $\Psi$is a Majorana spinor of SO(1,10) that satisfies ${\Gamma }_{012}\Psi =\Psi$. ${E}^{\mu \nu \lambda }$is a Levi-Civita symbol in three-dimensions.

Finite dimensional Lie 3-algebras with an invariant metric is classified into four-dimensional Euclidean ${𝒜}_{4}$algebra and the Lie 3-algebras with indefinite metrics in , , , , . We do not choose ${𝒜}_{4}$algebra because its degrees of freedom are just four. We need an algebra with arbitrary dimensions N, which is taken to infinity to define M-theory. Here we choose the most simple indefinite metric Lie 3-algebra, so called the Lorentzian Lie 3-algebra associated with $u\left(N\right)$Lie algebra,

$\begin{array}{ccc}& & \left[{T}^{-1},{T}^{a},{T}^{b}\right]=0\hfill \\ & & \left[{T}^{0},{T}^{i},{T}^{j}\right]=\left[{T}^{i},{T}^{j}\right]={f}_{\phantom{\rule{1.em}{0ex}}k}^{ij}{T}^{k}\hfill \\ & & \left[{T}^{i},{T}^{j},{T}^{k}\right]={f}^{ijk}{T}^{-1}\hfill \end{array}$uid61

where $a=-1,0,i$($i=1,\cdots ,{N}^{2}$). ${T}^{i}$are generators of $u\left(N\right)$. A metric is defined by a symmetric bilinear form,

$\begin{array}{ccc}\hfill <{T}^{-1},{T}^{0}>& =& -1\hfill \\ \hfill <{T}^{i},{T}^{j}>& =& {h}^{ij}\hfill \end{array}$uid62

and the other components are 0. The action is decomposed as

$\begin{array}{c}\hfill S=\text{Tr}\left(-\frac{1}{4}{\left({x}_{0}^{K}\right)}^{2}{\left[{x}^{I},{x}^{J}\right]}^{2}+\frac{1}{2}{\left({x}_{0}^{I}\left[{x}_{I},{x}^{J}\right]\right)}^{2}-\frac{1}{2}{\left({x}_{0}^{I}{b}_{\mu }+\left[{a}_{\mu },{x}^{I}\right]\right)}^{2}-\frac{1}{2}{E}^{\mu \nu \lambda }{b}_{\mu }\left[{a}_{\nu },{a}_{\lambda }\right]\\ \hfill +i{\overline{\psi }}_{0}{\Gamma }^{\mu }{b}_{\mu }\psi -\frac{i}{2}\overline{\psi }{\Gamma }^{\mu }\left[{a}_{\mu },\psi \right]+\frac{i}{2}{x}_{0}^{I}\overline{\psi }{\Gamma }_{IJ}\left[{x}^{J},\psi \right]-\frac{i}{2}{\overline{\psi }}_{0}{\Gamma }_{IJ}\left[{x}^{I},{x}^{J}\right]\psi \right)\end{array}$uid63

where we have renamed ${X}_{0}^{I}\to {x}_{0}^{I}$, ${X}_{i}^{I}{T}^{i}\to {x}^{I}$, ${\Psi }_{0}\to {\psi }_{0}$, ${\Psi }_{i}{T}^{i}\to \psi$, $2{A}_{\mu 0i}{T}^{i}\to {a}_{\mu }$, and ${A}_{\mu ij}\left[{T}^{i},{T}^{j}\right]\to {b}_{\mu }$. ${a}_{\mu }$correspond to the target coordinate matrices ${X}^{\mu }$, whereas ${b}_{\mu }$are auxiliary fields.

In this action, ${T}^{-1}$mode; ${X}_{-1}^{I}$, ${\Psi }_{-1}$or ${A}_{-1a}^{\mu }$does not appear, that is they are unphysical modes. Therefore, the indefinite part of the metric () does not exist in the action and the Lie 3-algebra model of M-theory is ghost-free like a model in . This action can be obtained by a dimensional reduction of the three-dimensional $𝒩=8$BLG model , ,  with the same 3-algebra. The BLG model possesses a ghost mode because of its kinetic terms with indefinite signature. On the other hand, the Lie 3-algebra model of M-theory does not possess a kinetic term because it is defined as a zero-dimensional field theory like the IIB matrix model .

This action is invariant under the translation

$\delta {x}^{I}={\eta }^{I},\phantom{\rule{2.em}{0ex}}\delta {a}^{\mu }={\eta }^{\mu }$uid64

where ${\eta }^{I}$and ${\eta }^{\mu }$belong to $u\left(1\right)$. This implies that eigen values of ${x}^{I}$and ${a}^{\mu }$represent an eleven-dimensional space-time.

The action is also invariant under 16 kinematical supersymmetry transformations

$\stackrel{˜}{\delta }\psi =\stackrel{˜}{ϵ}$uid65

and the other fields are not transformed. $\stackrel{˜}{ϵ}$belong to $u\left(1\right)$and satisfy ${\Gamma }_{012}\stackrel{˜}{ϵ}=\stackrel{˜}{ϵ}$. $\stackrel{˜}{ϵ}$and $ϵ$should come from sixteen components of thirty-two $𝒩=1$supersymmetry parameters in eleven dimensions, corresponding to eigen values $±$1 of ${\Gamma }_{012}$, respectively, as in the previous subsection.

A commutation relation between the kinematical supersymmetry transformations is given by

${\stackrel{˜}{\delta }}_{2}{\stackrel{˜}{\delta }}_{1}-{\stackrel{˜}{\delta }}_{1}{\stackrel{˜}{\delta }}_{2}=0$uid66

The action is invariant under 16 dynamical supersymmetry transformations,

$\begin{array}{ccc}& & \delta {X}^{I}=i\overline{ϵ}{\Gamma }^{I}\Psi \hfill \\ & & \delta {A}_{\mu ab}\left[{T}^{a},{T}^{b},\phantom{\rule{1.em}{0ex}}\right]=i\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\left[{X}^{I},\Psi ,\phantom{\rule{1.em}{0ex}}\right]\hfill \\ & & \delta \Psi =-{A}_{\mu ab}\left[{T}^{a},{T}^{b},{X}^{I}\right]{\Gamma }^{\mu }{\Gamma }_{I}ϵ-\frac{1}{6}\left[{X}^{I},{X}^{J},{X}^{K}\right]{\Gamma }_{IJK}ϵ\hfill \end{array}$uid67

where ${\Gamma }_{012}ϵ=-ϵ$. These supersymmetries close into gauge transformations on-shell,

$\begin{array}{ccc}& & \left[{\delta }_{1},{\delta }_{2}\right]{X}^{I}={\Lambda }_{cd}\left[{T}^{c},{T}^{d},{X}^{I}\right]\hfill \\ & & \left[{\delta }_{1},{\delta }_{2}\right]{A}_{\mu ab}\left[{T}^{a},{T}^{b},\phantom{\rule{1.em}{0ex}}\right]={\Lambda }_{ab}\left[{T}^{a},{T}^{b},{A}_{\mu cd}\left[{T}^{c},{T}^{d},\phantom{\rule{1.em}{0ex}}\right]\right]\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}-{A}_{\mu ab}\left[{T}^{a},{T}^{b},{\Lambda }_{cd}\left[{T}^{c},{T}^{d},\phantom{\rule{1.em}{0ex}}\right]\right]+2i{\overline{ϵ}}_{2}{\Gamma }^{\nu }{ϵ}_{1}{O}_{\mu \nu }^{A}\hfill \\ & & \left[{\delta }_{1},{\delta }_{2}\right]\Psi ={\Lambda }_{cd}\left[{T}^{c},{T}^{d},\Psi \right]+\left(i{\overline{ϵ}}_{2}{\Gamma }^{\mu }{ϵ}_{1}{\Gamma }_{\mu }-\frac{i}{4}{\overline{ϵ}}_{2}{\Gamma }^{KL}{ϵ}_{1}{\Gamma }_{KL}\right){O}^{\Psi }\hfill \end{array}$uid68

where gauge parameters are given by ${\Lambda }_{ab}=2i{\overline{ϵ}}_{2}{\Gamma }^{\mu }{ϵ}_{1}{A}_{\mu ab}-i{\overline{ϵ}}_{2}{\Gamma }_{JK}{ϵ}_{1}{X}_{a}^{J}{X}_{b}^{K}$. ${O}_{\mu \nu }^{A}=0$and ${O}^{\Psi }=0$are equations of motions of ${A}_{\mu \nu }$and $\Psi$, respectively, where

$\begin{array}{ccc}\hfill {O}_{\mu \nu }^{A}& =& {A}_{\mu ab}\left[{T}^{a},{T}^{b},{A}_{\nu cd}\left[{T}^{c},{T}^{d},\phantom{\rule{1.em}{0ex}}\right]\right]-{A}_{\nu ab}\left[{T}^{a},{T}^{b},{A}_{\mu cd}\left[{T}^{c},{T}^{d},\phantom{\rule{1.em}{0ex}}\right]\right]\hfill \\ & & +{E}_{\mu \nu \lambda }\left(-\left[{X}^{I},{A}_{ab}^{\lambda }\left[{T}^{a},{T}^{b},{X}_{I}\right],\phantom{\rule{1.em}{0ex}}\right]+\frac{i}{2}\left[\overline{\Psi },{\Gamma }^{\lambda }\Psi ,\phantom{\rule{1.em}{0ex}}\right]\right)\hfill \\ \hfill {O}^{\Psi }& =& -{\Gamma }^{\mu }{A}_{\mu ab}\left[{T}^{a},{T}^{b},\Psi \right]+\frac{1}{2}{\Gamma }_{IJ}\left[{X}^{I},{X}^{J},\Psi \right]\hfill \end{array}$uid69

() implies that a commutation relation between the dynamical supersymmetry transformations is

${\delta }_{2}{\delta }_{1}-{\delta }_{1}{\delta }_{2}=0$uid70

up to the equations of motions and the gauge transformations.

The 16 dynamical supersymmetry transformations () are decomposed as

$\begin{array}{ccc}& & \delta {x}^{I}=i\overline{ϵ}{\Gamma }^{I}\psi \hfill \\ & & \delta {x}_{0}^{I}=i\overline{ϵ}{\Gamma }^{I}{\psi }_{0}\hfill \\ & & \delta {x}_{-1}^{I}=i\overline{ϵ}{\Gamma }^{I}{\psi }_{-1}\hfill \\ \hfill \\ & & \delta \psi =-\left({b}_{\mu }{x}_{0}^{I}+\left[{a}_{\mu },{x}^{I}\right]\right){\Gamma }^{\mu }{\Gamma }_{I}ϵ-\frac{1}{2}{x}_{0}^{I}\left[{x}^{J},{x}^{K}\right]{\Gamma }_{IJK}ϵ\hfill \\ & & \delta {\psi }_{0}=0\hfill \\ & & \delta {\psi }_{-1}=-\text{Tr}\left({b}_{\mu }{x}^{I}\right){\Gamma }^{\mu }{\Gamma }_{I}ϵ-\frac{1}{6}\text{Tr}\left(\left[{x}^{I},{x}^{J}\right]{x}^{K}\right){\Gamma }_{IJK}ϵ\hfill \\ \hfill \\ & & \delta {a}_{\mu }=i\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\left({x}_{0}^{I}\psi -{\psi }_{0}{x}^{I}\right)\hfill \\ & & \delta {b}_{\mu }=i\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\left[{x}^{I},\psi \right]\hfill \\ & & \delta {A}_{\mu -1i}=i\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\frac{1}{2}\left({x}_{-1}^{I}{\psi }_{i}-{\psi }_{-1}{x}_{i}^{I}\right)\hfill \\ & & \delta {A}_{\mu -10}=i\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\frac{1}{2}\left({x}_{-1}^{I}{\psi }_{0}-{\psi }_{-1}{x}_{0}^{I}\right)\hfill \end{array}$uid71

and thus a commutator of dynamical supersymmetry transformations and kinematical ones acts as

$\begin{array}{ccc}& & \left({\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}\right){x}^{I}=i{\overline{ϵ}}_{1}{\Gamma }^{I}{\stackrel{˜}{ϵ}}_{2}\equiv {\eta }^{I}\hfill \\ & & \left({\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}\right){a}^{\mu }=i{\overline{ϵ}}_{1}{\Gamma }^{\mu }{\Gamma }_{I}{x}_{0}^{I}{\stackrel{˜}{ϵ}}_{2}\equiv {\eta }^{\mu }\hfill \\ & & \left({\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}\right){A}_{-1i}^{\mu }{T}^{i}=\frac{1}{2}i{\overline{ϵ}}_{1}{\Gamma }^{\mu }{\Gamma }_{I}{x}_{-1}^{I}{\stackrel{˜}{ϵ}}_{2}\hfill \end{array}$uid72

where the commutator that acts on the other fields vanishes. Thus, the commutation relation for physical modes is given by

${\stackrel{˜}{\delta }}_{2}{\delta }_{1}-{\delta }_{1}{\stackrel{˜}{\delta }}_{2}={\delta }_{\eta }$uid73

where ${\delta }_{\eta }$is a translation.

(), (), and () imply the $𝒩=1$supersymmetry algebra in eleven dimensions as in the previous subsection.

### 3.3. Hermitian 3-algebra model of M-theory

In this subsection, we study the Hermitian 3-algebra models of M-theory . Especially, we study mostly the model with the $u\left(N\right)\oplus u\left(N\right)$Hermitian 3-algebra ().

The continuum action () can be rewritten by using the triality of $SO\left(8\right)$and the $SU\left(4\right)×U\left(1\right)$decomposition , ,  as

$\begin{array}{ccc}\hfill {S}_{cl}& =& \int {d}^{3}\sigma \sqrt{-g}\left(-V-{A}_{\mu ba}\left\{{Z}^{A},{T}^{a},{T}^{b}\right\}{A}_{dc}^{\mu }\left\{{Z}_{A},{T}^{c},{T}^{d}\right\}\hfill \\ & & \phantom{\rule{2.em}{0ex}}+\frac{1}{3}{E}^{\mu \nu \lambda }{A}_{\mu ba}{A}_{\nu dc}{A}_{\lambda fe}\left\{{T}^{a},{T}^{c},{T}^{d}\right\}\left\{{T}^{b},{T}^{f},{T}^{e}\right\}\hfill \\ & & \phantom{\rule{2.em}{0ex}}+i{\overline{\psi }}^{A}{\Gamma }^{\mu }{A}_{\mu ba}\left\{{\psi }_{A},{T}^{a},{T}^{b}\right\}+\frac{i}{2}{E}_{ABCD}{\overline{\psi }}^{A}\left\{{Z}^{C},{Z}^{D},{\psi }^{B}\right\}-\frac{i}{2}{E}^{ABCD}{Z}_{D}\left\{{\overline{\psi }}_{A},{\psi }_{B},{Z}_{C}\right\}\hfill \\ & & \phantom{\rule{2.em}{0ex}}-i{\overline{\psi }}^{A}\left\{{\psi }_{A},{Z}^{B},{Z}_{B}\right\}+2i{\overline{\psi }}^{A}\left\{{\psi }_{B},{Z}^{B},{Z}_{A}\right\}\right)\hfill \end{array}$uid75

where fields with a raised $A$index transform in the 4 of SU(4), whereas those with lowered one transform in the $\overline{4}$. ${A}_{\mu ba}$($\mu =0,1,2$) is an anti-Hermitian gauge field, ${Z}^{A}$and ${Z}_{A}$are a complex scalar field and its complex conjugate, respectively. ${\psi }_{A}$is a fermion field that satisfies

${\Gamma }^{012}{\psi }_{A}=-{\psi }_{A}$uid76

and ${\psi }^{A}$is its complex conjugate. ${E}^{\mu \nu \lambda }$and ${E}^{ABCD}$are Levi-Civita symbols in three dimensions and four dimensions, respectively. The potential terms are given by

$\begin{array}{ccc}\hfill V& =& \frac{2}{3}{Υ}_{B}^{CD}{Υ}_{CD}^{B}\hfill \\ \hfill {Υ}_{B}^{CD}& =& \left\{{Z}^{C},{Z}^{D},{Z}_{B}\right\}-\frac{1}{2}{\delta }_{B}^{C}\left\{{Z}^{E},{Z}^{D},{Z}_{E}\right\}+\frac{1}{2}{\delta }_{B}^{D}\left\{{Z}^{E},{Z}^{C},{Z}_{E}\right\}\hfill \end{array}$uid77

If we replace the Nambu-Poisson bracket with a Hermitian 3-algebra's bracket , ,

$\begin{array}{ccc}& & \int {d}^{3}\sigma \sqrt{-g}\to ⟨\phantom{\rule{1.em}{0ex}}⟩\hfill \\ & & \left\{{\varphi }^{a},{\varphi }^{b},{\varphi }^{c}\right\}\to \left[{T}^{a},{T}^{b};{\overline{T}}^{\overline{c}}\right]\hfill \end{array}$uid78

we obtain the Hermitian 3-algebra model of M-theory ,

$\begin{array}{ccc}\hfill S& =& <-V-{A}_{\mu \overline{b}a}\left[{Z}^{A},{T}^{a};{\overline{T}}^{\overline{b}}\right]\overline{{A}_{\overline{d}c}^{\mu }\left[{Z}_{A},{T}^{c};{\overline{T}}^{\overline{d}}\right]}+\frac{1}{3}{E}^{\mu \nu \lambda }{A}_{\mu \overline{b}a}{A}_{\nu \overline{d}c}{A}_{\lambda \overline{f}e}\left[{T}^{a},{T}^{c};{\overline{T}}^{\overline{d}}\right]\overline{\left[{T}^{b},{T}^{f};{\overline{T}}^{\overline{e}}\right]}\hfill \\ & & +i{\overline{\psi }}^{A}{\Gamma }^{\mu }{A}_{\mu \overline{b}a}\left[{\psi }_{A},{T}^{a};{\overline{T}}^{\overline{b}}\right]+\frac{i}{2}{E}_{ABCD}{\overline{\psi }}^{A}\left[{Z}^{C},{Z}^{D};{\overline{\psi }}^{B}\right]-\frac{i}{2}{E}^{ABCD}{\overline{Z}}_{D}\left[{\overline{\psi }}_{A},{\psi }_{B};{\overline{Z}}_{C}\right]\hfill \\ & & -i{\overline{\psi }}^{A}\left[{\psi }_{A},{Z}^{B};{\overline{Z}}_{B}\right]+2i{\overline{\psi }}^{A}\left[{\psi }_{B},{Z}^{B};{\overline{Z}}_{A}\right]>\hfill \end{array}$uid79

where the cosmological constant has been deleted for the same reason as before. The potential terms are given by

$\begin{array}{ccc}\hfill V& =& \frac{2}{3}{Υ}_{B}^{CD}{\overline{Υ}}_{CD}^{B}\hfill \\ \hfill {Υ}_{B}^{CD}& =& \left[{Z}^{C},{Z}^{D};{\overline{Z}}_{B}\right]-\frac{1}{2}{\delta }_{B}^{C}\left[{Z}^{E},{Z}^{D};{\overline{Z}}_{E}\right]+\frac{1}{2}{\delta }_{B}^{D}\left[{Z}^{E},{Z}^{C};{\overline{Z}}_{E}\right]\hfill \end{array}$uid80

This matrix model can be obtained formally by a dimensional reduction of the $𝒩=6$BLG action , which is equivalent to ABJ(M) action , The authors of , , ,  studied matrix models that can be obtained by a dimensional reduction of the ABJM and ABJ gauge theories on ${S}^{3}$. They showed that the models reproduce the original gauge theories on ${S}^{3}$in planar limits.,

$\begin{array}{ccc}\hfill {S}_{𝒩=6BLG}& =& \int {d}^{3}x<-V-{D}_{\mu }{Z}^{A}\overline{{D}^{\mu }{Z}_{A}}+{E}^{\mu \nu \lambda }\left(\frac{1}{2}{A}_{\mu \overline{c}b}{\partial }_{\nu }{A}_{\lambda \overline{d}a}{\overline{T}}^{\overline{d}}\left[{T}^{a},{T}^{b};{\overline{T}}^{\overline{c}}\right]\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}+\frac{1}{3}{A}_{\mu \overline{b}a}{A}_{\nu \overline{d}c}{A}_{\lambda \overline{f}e}\left[{T}^{a},{T}^{c};{\overline{T}}^{\overline{d}}\right]\overline{\left[{T}^{b},{T}^{f};{\overline{T}}^{\overline{e}}\right]}\right)\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}-i{\overline{\psi }}^{A}{\Gamma }^{\mu }{D}_{\mu }{\psi }_{A}+\frac{i}{2}{E}_{ABCD}{\overline{\psi }}^{A}\left[{Z}^{C},{Z}^{D};{\psi }^{B}\right]-\frac{i}{2}{E}^{ABCD}{\overline{Z}}_{D}\left[{\overline{\psi }}_{A},{\psi }_{B};{\overline{Z}}_{C}\right]\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}-i{\overline{\psi }}^{A}\left[{\psi }_{A},{Z}^{B};{\overline{Z}}_{B}\right]+2i{\overline{\psi }}^{A}\left[{\psi }_{B},{Z}^{B};{\overline{Z}}_{A}\right]>\hfill \end{array}$uid82

The Hermitian 3-algebra models of M-theory are classified into the models with $u\left(m\right)\oplus u\left(n\right)$Hermitian 3-algebra () and $sp\left(2n\right)\oplus u\left(1\right)$Hermitian 3-algebra (). In the following, we study the $u\left(N\right)\oplus u\left(N\right)$Hermitian 3-algebra model. By substituting the $u\left(N\right)\oplus u\left(N\right)$Hermitian 3-algebra () to the action (), we obtain

$\begin{array}{ccc}\hfill S& =& \text{Tr}\left(\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\frac{{\left(2\pi \right)}^{2}}{{k}^{2}}V\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\left({Z}^{A}{A}_{\mu }^{R}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{A}_{\mu }^{L}{Z}^{A}\right){\left({Z}^{A}{A}^{R\mu }\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{A}^{L\mu }{Z}^{A}\right)}^{†}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\frac{k}{2\pi }\frac{i}{3}{E}^{\mu \nu \lambda }\left({A}_{\mu }^{R}{A}_{\nu }^{R}{A}_{\lambda }^{R}-{A}_{\mu }^{L}{A}_{\nu }^{L}{A}_{\lambda }^{L}\right)\hfill \\ & & -{\overline{\psi }}^{A}{\Gamma }^{\mu }\left({\psi }_{A}{A}_{\mu }^{R}-{A}_{\mu }^{L}{\psi }_{A}\right)+\frac{2\pi }{k}\left(i{E}_{ABCD}{\overline{\psi }}^{A}{Z}^{C}{\psi }^{†B}{Z}^{D}-i{E}^{ABCD}{Z}_{D}^{†}{\overline{{\psi }^{†}}}_{A}{Z}_{C}^{†}{\psi }_{B}\hfill \\ & & -i{\overline{\psi }}^{A}{\psi }_{A}{Z}_{B}^{†}{Z}^{B}+i{\overline{\psi }}^{A}{Z}^{B}{Z}_{B}^{†}{\psi }_{A}+2i{\overline{\psi }}^{A}{\psi }_{B}{Z}_{A}^{†}{Z}^{B}-2i{\overline{\psi }}^{A}{Z}^{B}{Z}_{A}^{†}{\psi }_{B}\right)\right)\hfill \end{array}$uid83

where ${A}_{\mu }^{R}\equiv -\frac{k}{2\pi }i{A}_{\mu \overline{b}a}{T}^{†\overline{b}}{T}^{a}$and ${A}_{\mu }^{L}\equiv -\frac{k}{2\pi }i{A}_{\mu \overline{b}a}{T}^{a}{T}^{†\overline{b}}$are $N×N$Hermitian matrices. In the algebra, we have set $\alpha =\frac{2\pi }{k}$, where $k$is an integer representing the Chern-Simons level. We choose $k=1$in order to obtain 16 dynamical supersymmetries. $V$is given by

$\begin{array}{ccc}\hfill V& =& +\frac{1}{3}{Z}_{A}^{†}{Z}^{A}{Z}_{B}^{†}{Z}^{B}{Z}_{C}^{†}{Z}^{C}+\frac{1}{3}{Z}^{A}{Z}_{A}^{†}{Z}^{B}{Z}_{B}^{†}{Z}^{C}{Z}_{C}^{†}+\frac{4}{3}{Z}_{A}^{†}{Z}^{B}{Z}_{C}^{†}{Z}^{A}{Z}_{B}^{†}{Z}^{C}\hfill \\ & & -{Z}_{A}^{†}{Z}^{A}{Z}_{B}^{†}{Z}^{C}{Z}_{C}^{†}{Z}^{B}-{Z}^{A}{Z}_{A}^{†}{Z}^{B}{Z}_{C}^{†}{Z}^{C}{Z}_{B}^{†}\hfill \end{array}$uid84

By redefining fields as

$\begin{array}{ccc}\hfill {Z}^{A}& \to & {\left(\frac{k}{2\pi }\right)}^{\frac{1}{3}}{Z}^{A}\hfill \\ \hfill {A}^{\mu }& \to & {\left(\frac{2\pi }{k}\right)}^{\frac{1}{3}}{A}^{\mu }\hfill \\ \hfill {\psi }^{A}& \to & {\left(\frac{k}{2\pi }\right)}^{\frac{1}{6}}{\psi }^{A}\hfill \end{array}$uid85

we obtain an action that is independent of Chern-Simons level:

$\begin{array}{ccc}\hfill S& =& \text{Tr}\left(-V-\left({Z}^{A}{A}_{\mu }^{R}-{A}_{\mu }^{L}{Z}^{A}\right){\left({Z}^{A}{A}^{R\mu }-{A}^{L\mu }{Z}^{A}\right)}^{†}-\frac{i}{3}{E}^{\mu \nu \lambda }\left({A}_{\mu }^{R}{A}_{\nu }^{R}{A}_{\lambda }^{R}-{A}_{\mu }^{L}{A}_{\nu }^{L}{A}_{\lambda }^{L}\right)\hfill \\ & & -{\overline{\psi }}^{A}{\Gamma }^{\mu }\left({\psi }_{A}{A}_{\mu }^{R}-{A}_{\mu }^{L}{\psi }_{A}\right)+i{E}_{ABCD}{\overline{\psi }}^{A}{Z}^{C}{\psi }^{†B}{Z}^{D}-i{E}^{ABCD}{Z}_{D}^{†}{\overline{{\psi }^{†}}}_{A}{Z}_{C}^{†}{\psi }_{B}\hfill \\ & & -i{\overline{\psi }}^{A}{\psi }_{A}{Z}_{B}^{†}{Z}^{B}+i{\overline{\psi }}^{A}{Z}^{B}{Z}_{B}^{†}{\psi }_{A}+2i{\overline{\psi }}^{A}{\psi }_{B}{Z}_{A}^{†}{Z}^{B}-2i{\overline{\psi }}^{A}{Z}^{B}{Z}_{A}^{†}{\psi }_{B}\right)\hfill \end{array}$uid86

as opposed to three-dimensional Chern-Simons actions.

If we rewrite the gauge fields in the action as ${A}_{\mu }^{L}={A}_{\mu }+{b}_{\mu }$and ${A}_{\mu }^{R}={A}_{\mu }-{b}_{\mu }$, we obtain

$\begin{array}{ccc}\hfill S& =& \text{Tr}\left(-V+\left(\left[{A}_{\mu },{Z}^{A}\right]+\left\{{b}_{\mu },{Z}^{A}\right\}\right)\left(\left[{A}^{\mu },{Z}_{A}\right]-\left\{{b}^{\mu },{Z}_{A}\right\}\right)+i{E}^{\mu \nu \lambda }\left(\frac{2}{3}{b}_{\mu }{b}_{\nu }{b}_{\lambda }+2{A}_{\mu }{A}_{\nu }{b}_{\lambda }\right)\hfill \\ & & +{\overline{\psi }}^{A}{\Gamma }^{\mu }\left(\left[{A}_{\mu },{\psi }_{A}\right]+\left\{{b}_{\mu },{\psi }_{A}\right\}\right)+i{E}_{ABCD}{\overline{\psi }}^{A}{Z}^{C}{\psi }^{†B}{Z}^{D}-i{E}^{ABCD}{Z}_{D}^{†}{\overline{{\psi }^{†}}}_{A}{Z}_{C}^{†}{\psi }_{B}\hfill \\ & & -i{\overline{\psi }}^{A}{\psi }_{A}{Z}_{B}^{†}{Z}^{B}+i{\overline{\psi }}^{A}{Z}^{B}{Z}_{B}^{†}{\psi }_{A}+2i{\overline{\psi }}^{A}{\psi }_{B}{Z}_{A}^{†}{Z}^{B}-2i{\overline{\psi }}^{A}{Z}^{B}{Z}_{A}^{†}{\psi }_{B}\right)\hfill \end{array}$uid87

where $\left[\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right]$and $\left\{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right\}$are the ordinary commutator and anticommutator, respectively. The $u\left(1\right)$parts of ${A}^{\mu }$decouple because ${A}^{\mu }$appear only in commutators in the action. ${b}^{\mu }$can be regarded as auxiliary fields, and thus ${A}^{\mu }$correspond to matrices ${X}^{\mu }$that represents three space-time coordinates in M-theory. Among $N×N$arbitrary complex matrices ${Z}^{A}$, we need to identify matrices ${X}^{I}$($I=3,\cdots 10$) representing the other space coordinates in M-theory, because the model possesses not $SO\left(8\right)$but $SU\left(4\right)×U\left(1\right)$symmetry. Our identification is

$\begin{array}{ccc}\hfill {Z}^{A}& =& i{X}^{A+2}-{X}^{A+6},\hfill \\ \hfill {X}^{I}& =& {\stackrel{^}{X}}^{I}-i{x}^{I}1\hfill \end{array}$uid88

where ${\stackrel{^}{X}}^{I}$and ${x}^{I}$are $su\left(N\right)$Hermitian matrices and real scalars, respectively. This is analogous to the identification when we compactify ABJM action, which describes N M2 branes, and obtain the action of N D2 branes , , . We will see that this identification works also in our case. We should note that while the $su\left(N\right)$part is Hermitian, the $u\left(1\right)$part is anti-Hermitian. That is, an eigen-value distribution of ${X}^{\mu }$, ${Z}^{A}$, and not ${X}^{I}$determine the spacetime in the Hermitian model. In order to define light-cone coordinates, we need to perform Wick rotation: ${a}^{0}\to -i{a}^{0}$. After the Wick rotation, we obtain

${A}^{0}=\stackrel{^}{{A}^{0}}-i{a}^{0}1$uid89

where $\stackrel{^}{{A}^{0}}$is a $su\left(N\right)$Hermitian matrix.

### 3.4. DLCQ Limit of 3-algebra model of M-theory

It was shown that M-theory in a DLCQ limit reduces to the BFSS matrix theory with matrices of finite size , , , , , . This fact is a strong criterion for a model of M-theory. In , , it was shown that the Lie and Hermitian 3-algebra models of M-theory reduce to the BFSS matrix theory with matrices of finite size in the DLCQ limit. In this subsection, we show an outline of the mechanism.

DLCQ limit of M-theory consists of a light-cone compactification, ${x}^{-}\approx {x}^{-}+2\pi R$, where ${x}^{±}=\frac{1}{\sqrt{2}}\left({x}^{10}±{x}^{0}\right)$, and Lorentz boost in ${x}^{10}$direction with an infinite momentum. After appropriate scalings of fields , , we define light-cone coordinate matrices as

$\begin{array}{ccc}& & {X}^{0}=\frac{1}{\sqrt{2}}\left({X}^{+}-{X}^{-}\right)\hfill \\ & & {X}^{10}=\frac{1}{\sqrt{2}}\left({X}^{+}+{X}^{-}\right)\hfill \end{array}$uid91

We integrate out ${b}^{\mu }$by using their equations of motion.

A matrix compactification  on a circle with a radius R imposes the following conditions on ${X}^{-}$and the other matrices $Y$:

$\begin{array}{ccc}& & {X}^{-}-\left(2\pi R\right)1={U}^{†}{X}^{-}U\hfill \\ & & Y={U}^{†}YU\hfill \end{array}$uid92

where $U$is a unitary matrix. In order to obtain a solution to (), we need to take $N\to \infty$and consider matrices of infinite size . A solution to () is given by ${X}^{-}={\overline{X}}^{-}+{\stackrel{˜}{X}}^{-}$, $Y=\stackrel{˜}{Y}$and

$U=\left(\begin{array}{ccccc}\ddots & \ddots & & & \\ & 0& 1& & 0& \\ & & 0& 1& & \\ & & & 0& 1& \\ & 0& & & 0& \ddots \\ & & & & & \ddots \end{array}\right)\otimes {1}_{n×n}\in U\left(N\right)$uid93

Backgrounds ${\overline{X}}^{-}$are

${\overline{X}}^{-}=-{T}^{3}{\overline{x}}_{0}^{-}{T}^{0}-\left(2\pi R\right)\text{diag}\left(\cdots ,s-1,s,s+1,\cdots \right)\otimes {1}_{n×n}$uid94

in the Lie 3-algebra case, whereas

${\overline{X}}^{-}=-i\left({T}^{3}{\overline{x}}^{-}\right)1-i\left(2\pi R\right)\text{diag}\left(\cdots ,s-1,s,s+1,\cdots \right)\otimes {1}_{n×n}$uid95

in the Hermitian 3-algebra case. A fluctuation $\stackrel{˜}{x}$that represents $u\left(N\right)$parts of ${\stackrel{˜}{X}}^{-}$and $\stackrel{˜}{Y}$is

$\left(\begin{array}{cccccccc}\ddots & \ddots & \ddots & & & & & \\ \ddots & \stackrel{˜}{x}\left(0\right)& \stackrel{˜}{x}\left(1\right)& \stackrel{˜}{x}\left(2\right)& & & \ddots & \\ \ddots & \stackrel{˜}{x}\left(-1\right)& \stackrel{˜}{x}\left(0\right)& \stackrel{˜}{x}\left(1\right)& \stackrel{˜}{x}\left(2\right)& & & \\ & \stackrel{˜}{x}\left(-2\right)& \stackrel{˜}{x}\left(-1\right)& \stackrel{˜}{x}\left(0\right)& \stackrel{˜}{x}\left(1\right)& \stackrel{˜}{x}\left(2\right)& & \\ & & \stackrel{˜}{x}\left(-2\right)& \stackrel{˜}{x}\left(-1\right)& \stackrel{˜}{x}\left(0\right)& \stackrel{˜}{x}\left(1\right)& \stackrel{˜}{x}\left(2\right)& \\ & & & \stackrel{˜}{x}\left(-2\right)& \stackrel{˜}{x}\left(-1\right)& \stackrel{˜}{x}\left(0\right)& \stackrel{˜}{x}\left(1\right)& \ddots \\ & \ddots & & & \stackrel{˜}{x}\left(-2\right)& \stackrel{˜}{x}\left(-1\right)& \stackrel{˜}{x}\left(0\right)& \ddots \\ & & & & & \ddots & \ddots & \ddots \end{array}\right)$uid96

Each $\stackrel{˜}{x}\left(s\right)$is a $n×n$matrix, where $s$is an integer. That is, the (s, t)-th block is given by ${\stackrel{˜}{x}}_{s,t}=\stackrel{˜}{x}\left(s-t\right)$.

We make a Fourier transformation,

$\stackrel{˜}{x}\left(s\right)=\frac{1}{2\pi \stackrel{˜}{R}}{\int }_{0}^{2\pi \stackrel{˜}{R}}d\tau x\left(\tau \right){e}^{is\frac{\tau }{\stackrel{˜}{R}}}$uid97

where $x\left(\tau \right)$is a $n×n$matrix in one-dimension and $R\stackrel{˜}{R}=2\pi$. From ()-(), the following identities hold:

$\begin{array}{ccc}& & \sum _{t}{\stackrel{˜}{x}}_{s,t}{\stackrel{˜}{{x}^{\text{'}}}}_{t,u}=\frac{1}{2\pi \stackrel{˜}{R}}{\int }_{0}^{2\pi \stackrel{˜}{R}}d\tau \phantom{\rule{0.166667em}{0ex}}x\left(\tau \right){x}^{\text{'}}\left(\tau \right){e}^{i\left(s-u\right)\frac{\tau }{\stackrel{˜}{R}}}\hfill \\ & & \text{tr}\left(\sum _{s,t}{\stackrel{˜}{x}}_{s,t}{\stackrel{˜}{{x}^{\text{'}}}}_{t,s}\right)=V\frac{1}{2\pi \stackrel{˜}{R}}{\int }_{0}^{2\pi \stackrel{˜}{R}}d\tau \phantom{\rule{0.166667em}{0ex}}\text{tr}\left(x\left(\tau \right){x}^{\text{'}}\left(\tau \right)\right)\hfill \\ & & {\left[{\overline{x}}^{-},\stackrel{˜}{x}\right]}_{s,t}=\frac{1}{2\pi \stackrel{˜}{R}}{\int }_{0}^{2\pi \stackrel{˜}{R}}d\tau \phantom{\rule{0.166667em}{0ex}}{\partial }_{\tau }x\left(\tau \right){e}^{i\left(s-t\right)\frac{\tau }{\stackrel{˜}{R}}}\hfill \end{array}$uid98

where $\text{tr}$is a trace over $n×n$matrices and $V={\sum }_{s}1$.

Next, we boost the system in ${x}^{10}$direction:

$\begin{array}{ccc}& & {\stackrel{˜}{X}}^{\text{'}+}=\frac{1}{T}{\stackrel{˜}{X}}^{+}\hfill \\ & & {\stackrel{˜}{X}}^{\text{'}-}=T{\stackrel{˜}{X}}^{-}\hfill \end{array}$uid99

The DLCQ limit is achieved when $T\to \infty$, where the "novel Higgs mechanism"  is realized. In $T\to \infty$, the actions of the 3-algebra models of M-theory reduce to that of the BFSS matrix theory  with matrices of finite size,

$S=\frac{1}{{g}^{2}}{\int }_{-\infty }^{\infty }d\tau \text{tr}\left(\frac{1}{2}{\left({D}_{0}{x}^{P}\right)}^{2}-\frac{1}{4}{\left[{x}^{P},{x}^{Q}\right]}^{2}+\frac{1}{2}\overline{\psi }{\Gamma }^{0}{D}_{0}\psi -\frac{i}{2}\overline{\psi }{\Gamma }^{P}\left[{x}_{P},\psi \right]\right)$uid100

where $P,Q=1,2,\cdots ,9$.

### 3.5. Supersymmetric deformation of Lie 3-algebra model of M-theory

A supersymmetric deformation of the Lie 3-algebra Model of M-theory was studied in  (see also , , ). If we add mass terms and a flux term,

${S}_{m}=⟨-\frac{1}{2}{\mu }^{2}{\left({X}^{I}\right)}^{2}-\frac{i}{2}\mu \overline{\Psi }{\Gamma }_{3456}\Psi +{H}_{IJKL}\left[{X}^{I},{X}^{J},{X}^{K}\right]{X}^{L}⟩$uid102

such that

${H}_{IJKL}=\left\{\begin{array}{cc}-\frac{\mu }{6}{ϵ}_{IJKL}& \left(I,J,K,L=3,4,5,6\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{or}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}7,8,9,10\right)\hfill \\ 0& \left(\text{otherwise}\right)\hfill \end{array}$uid103

to the action (), the total action ${S}_{0}+{S}_{m}$is invariant under dynamical 16 supersymmetries,

$\begin{array}{ccc}& & \delta {X}^{I}=i\overline{ϵ}{\Gamma }^{I}\Psi \hfill \\ & & \delta {A}_{\mu ab}\left[{T}^{a},{T}^{b},\phantom{\rule{1.em}{0ex}}\right]=i\overline{ϵ}{\Gamma }_{\mu }{\Gamma }_{I}\left[{X}^{I},\Psi ,\phantom{\rule{1.em}{0ex}}\right]\hfill \\ & & \delta \Psi =-\frac{1}{6}\left[{X}^{I},{X}^{J},{X}^{K}\right]{\Gamma }_{IJK}ϵ-{A}_{\mu ab}\left[{T}^{a},{T}^{b},{X}^{I}\right]{\Gamma }^{\mu }{\Gamma }_{I}ϵ+\mu {\Gamma }_{3456}{X}^{I}{\Gamma }_{I}ϵ\hfill \end{array}$uid104

From this action, we obtain various interesting solutions, including fuzzy sphere solutions .

## 4. Conclusion

The metric Hermitian 3-algebra corresponds to a class of the super Lie algebra. By using this relation, the metric Hermitian 3-algebras are classified into $u\left(m\right)\oplus u\left(n\right)$and $sp\left(2n\right)\oplus u\left(1\right)$Hermitian 3-algebras.

The Lie and Hermitian 3-algebra models of M-theory are obtained by second quantizations of the supermembrane action in a semi-light-cone gauge. The Lie 3-algebra model possesses manifest $𝒩=1$supersymmetry in eleven dimensions. In the DLCQ limit, both the models reduce to the BFSS matrix theory with matrices of finite size as they should.