Open access peer-reviewed chapter

3-Algebras in String Theory

By Matsuo Sato

Submitted: November 13th 2011Reviewed: April 19th 2012Published: July 11th 2012

DOI: 10.5772/46480

Downloaded: 2144

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 [1], [2], [3]. 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(N)u(N)Hermitian 3-algebra, we can describe a low energy effective action of N coincident supermembranes [4], [5], [6], [7], [8], which are fundamental objects in M-theory.

With this as motivation, 3-algebras with invariant metrics were classified [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. 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 𝒜4algebra, 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(N)u(M)and sp(2N)u(1)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 [23], [24], 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 [25]. 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 [26]. 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 [27] 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 [26], [28], [29], 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 [26], [28], as they should [30], [31], [32], [33], [34], [35].

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×VVdefined by (x,y,z)[x,y;z], where the 3-bracket is complex linear in the first two entries, whereas complex anti-linear in the last entry, equipped with a metric <x,y>, satisfying the following properties:

the fundamental identity

[[x,y;z],v;w]=[[x,v;w],y;z]+[x,[y,v;w];z]-[x,y;[z,w;v]]uid2

the metric invariance

<[x,v;w],y>-<x,[y,w;v]>=0uid3

and the anti-symmetry

[x,y;z]=-[y,x;z]uid4

for

x,y,z,v,wVuid5

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

D(x,y)z:=[z,x;y]uid6

From (), one can show that D(x,y)form a Lie algebra,

[D(x,y),D(v,w)]=D(D(x,y)v,w)-D(v,D(y,x)w)uid7

There is an one-to-one correspondence between the metric Hermitian 3-algebra and a class of metric complex super Lie algebras [19]. Such a class satisfies the following conditions among complex super Lie algebras S=S0S1, where S0and S1are even and odd parts, respectively. S1is decomposed as S1=VV¯, where Vis an unitary representation of S0: for aS0, u,vV,

[a,u]Vuid8

and

<[a,u],v>+<u,[a*,v]>=0uid9

v¯V¯is defined by

v¯=<,v>uid10

The super Lie bracket satisfies

[V,V]=0,[V¯,V¯]=0uid11

From the metric Hermitian 3-algebra, we obtain the class of the metric complex super Lie algebra in the following way. The elements in S0, V, and V¯are defined by (), (), and (), respectively. The algebra is defined by () and

[D(x,y),z]:=D(x,y)z=[z,x;y][D(x,y),z¯]:=-D(y,x)z¯=-[z,y;x]¯[x,y¯]:=D(x,y)[x,y]:=0[x¯,y¯]:=0uid12

One can show that this algebra satisfies the super Jacobi identity and ()-() as in [19].

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

[x,y;z]:=α[[y,z¯],x]uid13

where αis an arbitrary constant. One can also show that this algebra satisfies ()-() for () as in [19].

2.2. Classification of metric Hermitian 3-algebra

The classical Lie super algebras satisfying ()-() are A(m-1,n-1)and C(n+1). The even parts of A(m-1,n-1)and C(n+1)are u(m)u(n)and sp(2n)u(1), 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(m)u(n)and sp(2n)u(1)Hermitian 3-algebras.

First, we will construct the u(m)u(n)Hermitian 3-algebra from A(m-1,n-1), according to the relation in the previous subsection. A(m-1,n-1)is simple and is obtained by dividing sl(m,n)by its ideal. That is, A(m-1,n-1)=sl(m,n)when mnand A(n-1,n-1)=sl(n,n)/λ12n.

Real sl(m,n)is defined by

h1cich2uid15

where h1and h2are m×mand n×nanti-Hermite matrices and cis an n×marbitrary complex matrix. Complex sl(m,n)is a complexification of real sl(m,n), given by

αβγδuid16

where α, β, γ, and δare m×m, n×m, m×n, and n×ncomplex matrices that satisfy

trα=trδuid17

Complex A(m-1,n-1)is decomposed as A(m-1,n-1)=S0VV¯, where

α00δS00β00V00γ0V¯uid18

() is rewritten as VV¯defined by

B=0β00B=00β0uid19

where BVand BV¯. () is rewritten as

[X,Y;Z]=α[[Y,Z],X]=α0yzx-xzy00uid20

for

X=0x00VY=0y00VZ=0z00Vuid21

As a result, we obtain the u(m)u(n)Hermitian 3-algebra,

[x,y;z]=α(yzx-xzy)uid22

where x, y, and zare arbitrary n×mcomplex matrices. This algebra was originally constructed in [8].

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

D(x,y)=(xy,yx)S0uid23

() and () are rewritten as

[(xy,yx),(x'y',y'x')]=([xy,x'y'],[yx,y'x'])[(xy,yx),z]=xyz-zyx[(xy,yx),w]=yxw-wxy[x,y]=(xy,yx)[x,y]=0[x,y]=0uid24

This algebra is summarized as

xyzwyx,x'y'z'w'y'x'uid25

which forms complex A(m-1,n-1).

Next, we will construct the sp(2n)u(1)Hermitian 3-algebra from C(n+1). Complex C(n+1)is decomposed as C(n+1)=S0VV¯. The elements are given by

α0000-α0000ab00c-aTS000x1x200000x2T000-x1T00V000000y1y2y2T000-y1T000V¯uid26

where αis a complex number, ais an arbitrary n×ncomplex matrix, band care n×ncomplex symmetric matrices, and x1, x2, y1and y2are n×1complex matrices. () is rewritten as VV¯defined by BB¯=UB*U-1, where BV, B¯V¯and

U=01001000000100-10uid27

Explicitly,

B=00x1x200000x2T000-x1T00B¯=000000x2*-x1*-x1000-x2000uid28

() is rewritten as

[X,Y;Z]:=α[[Y,Z¯],X]=α00y1y200000y2T000-y1T00,000000z2*-z1*-z1000-z2000,00x1x200000x2T000-x1T00=α00w1w200000w2T000-w1T00uid29

for

X=00x1x200000x2T000-x1T00VY=00y1y200000y2T000-y1T00VZ=00z1z200000z2T000-z1T00Vuid30

where w1and w2are given by

(w1,w2)=-(y1z1+y2z2)(x1,x2)+(x1z1+x2z2)(y1,y2)+(x2y1T-x1y2T)(z2*,-z1*)uid31

As a result, we obtain the sp(2n)u(1)Hermitian 3-algebra,

[x,y;z]=α((yz˜)x+(z˜x)y-(xy)z˜)uid32

for x=(x1,x2), y=(y1,y2), z=(z1,z2), where x1, x2, y1, y2, z1, and z2are n-vectors and

z˜=(z2*,-z1*)ab=a1·b2-a2·b1uid33

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 [36] is given by

SM2=d3σ(-G+i4ϵαβγΨ¯ΓMNαΨ(ΠβMΠγN+i2ΠβMΨ¯ΓNγΨ-112Ψ¯ΓMβΨΨ¯ΓNγΨ))uid35

where M,N=0,,10, α,β,γ=0,1,2, Gαβ=ΠαMΠβMand ΠαM=αXM-i2Ψ¯ΓMαΨ. Ψis a SO(1,10)Majorana fermion.

This action is invariant under dynamical supertransformations,

δΨ=ϵδXM=-iΨ¯ΓMϵuid36

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

[δ1,δ2]XM=-2iϵ1ΓMϵ2uid37
[δ1,δ2]Ψ=0uid38

The action is also invariant under the κ-symmetry transformations,

δΨ=(1+Γ)κ(σ)δXM=iΨ¯ΓM(1+Γ)κ(σ)uid39

where

Γ=13!-GϵαβγΠαLΠβMΠγNΓLMNuid40

If we fix the κ-symmetry () of the action by taking a semi-light-cone gauge [26]Advantages of a semi-light-cone gauges against a light-cone gauge are shown in [37], [38], [39]

Γ012Ψ=-Ψuid42

we obtain a semi-light-cone supermembrane action,

SM2=d3σ(-G+i4ϵαβγ(Ψ¯ΓμναΨ(ΠβμΠγν+i2ΠβμΨ¯ΓνγΨ-112Ψ¯ΓμβΨΨ¯ΓνγΨ)+Ψ¯ΓIJαΨβXIγXJ))uid43

where Gαβ=hαβ+ΠαμΠβμ, Παμ=αXμ-i2Ψ¯ΓμαΨ, and hαβ=αXIβXI.

In [26], it is shown under an approximation up to the quadratic order in αXμand αΨbut exactly in XI, that this action is equivalent to the continuum action of the 3-algebra model of M-theory,

Scl=d3σ-g(-112{XI,XJ,XK}2-12(Aμab{ϕa,ϕb,XI})2-13EμνλAμabAνcdAλef{ϕa,ϕc,ϕd}{ϕb,ϕe,ϕf}+12Λ-i2Ψ¯ΓμAμab{ϕa,ϕb,Ψ}+i4Ψ¯ΓIJ{XI,XJ,Ψ})uid44

where I,J,K=3,,10and {ϕa,ϕb,ϕc}=ϵαβγαϕaβϕbγϕcis the Nambu-Poisson bracket. An invariant symmetric bilinear form is defined by d3σ-gϕaϕbfor complete basis ϕain three dimensions. Thus, this action is manifestly VPD covariant even when the world-volume metric is flat. XIis a scalar and Ψis a SO(1,2)×SO(8)Majorana-Weyl fermion satisfying (). Eμνλis a Levi-Civita symbol in three dimensions and Λis a cosmological constant.

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

δXI=iϵ¯ΓIΨδAμ(σ,σ')=i2ϵ¯ΓμΓI(XI(σ)Ψ(σ')-XI(σ')Ψ(σ)),δΨ=-Aμab{ϕa,ϕb,XI}ΓμΓIϵ-16{XI,XJ,XK}ΓIJKϵuid45

where Γ012ϵ=-ϵ. These supersymmetries close into gauge transformations on-shell,

[δ1,δ2]XI=Λcd{ϕc,ϕd,XI}[δ1,δ2]Aμab{ϕa,ϕb,}=Λab{ϕa,ϕb,Aμcd{ϕc,ϕd,}}-Aμab{ϕa,ϕb,Λcd{ϕc,ϕd,}}+2iϵ¯2Γνϵ1OμνA[δ1,δ2]Ψ=Λcd{ϕc,ϕd,Ψ}+(iϵ¯2Γμϵ1Γμ-i4ϵ¯2ΓKLϵ1ΓKL)OΨuid46

where gauge parameters are given by Λab=2iϵ¯2Γμϵ1Aμab-iϵ¯2ΓJKϵ1XaJXbK. OμνA=0and OΨ=0are equations of motions of Aμνand Ψ, respectively, where

OμνA=Aμab{ϕa,ϕb,Aνcd{ϕc,ϕd,}}-Aνab{ϕa,ϕb,Aμcd{ϕc,ϕd,}}+Eμνλ(-{XI,Aabλ{ϕa,ϕb,XI},}+i2{Ψ¯,ΓλΨ,})OΨ=-ΓμAμab{ϕa,ϕb,Ψ}+12ΓIJ{XI,XJ,Ψ}uid47

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

δ2δ1-δ1δ2=0uid48

up to the equations of motions and the gauge transformations.

This action is invariant under a translation,

δXI(σ)=ηI,δAμ(σ,σ')=ημ(σ)-ημ(σ')uid49

where ηIare constants.

The action is also invariant under 16 kinematical supersymmetry transformations

δ˜Ψ=ϵ˜uid50

and the other fields are not transformed. ϵ˜is a constant and satisfy Γ012ϵ˜=ϵ˜. ϵ˜and ϵshould come from sixteen components of thirty-two 𝒩=1supersymmetry parameters in eleven dimensions, corresponding to eigen values ±1 of Γ012, respectively. This 𝒩=1supersymmetry consists of remaining 16 target-space supersymmetries and transmuted 16 κ-symmetries in the semi-light-cone gauge [26], [25], [40].

A commutation relation between the kinematical supersymmetry transformations is given by

δ˜2δ˜1-δ˜1δ˜2=0uid51

A commutator of dynamical supersymmetry transformations and kinematical ones acts as

(δ˜2δ1-δ1δ˜2)XI(σ)=iϵ¯1ΓIϵ˜2η0I(δ˜2δ1-δ1δ˜2)Aμ(σ,σ')=i2ϵ¯1ΓμΓI(XI(σ)-XI(σ'))ϵ˜2η0μ(σ)-η0μ(σ')uid52

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

δ˜2δ1-δ1δ˜2=δηuid53

where δηis a translation.

If we change a basis of the supersymmetry transformations as

δ'=δ+δ˜δ˜'=i(δ-δ˜)uid54

we obtain

δ2'δ1'-δ1'δ2'=δηδ˜2'δ˜1'-δ˜1'δ˜2'=δηδ˜2'δ1'-δ1'δ˜2'=0uid55

These thirty-two supersymmetry transformations are summarised as Δ=(δ',δ˜')and () implies the 𝒩=1supersymmetry algebra in eleven dimensions,

Δ2Δ1-Δ1Δ2=δηuid56

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 [26], [28], 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 [21], [22],

d3σ-g{ϕa,ϕb,ϕc}[Ta,Tb,Tc]uid58

we obtain the Lie 3-algebra model of M-theory [26], [28],

S0=<-112[XI,XJ,XK]2-12(Aμab[Ta,Tb,XI])2-13EμνλAμabAνcdAλef[Ta,Tc,Td][Tb,Te,Tf]-i2Ψ¯ΓμAμab[Ta,Tb,Ψ]+i4Ψ¯ΓIJ[XI,XJ,Ψ]>uid59

We have deleted the cosmological constant Λ, which corresponds to an operator ordering ambiguity, as usual as in the case of other matrix models [27], [41].

This model can be obtained formally by a dimensional reduction of the 𝒩=8BLG model [4], [5], [6],

S𝒩=8BLG=d3x<-112[XI,XJ,XK]2-12(DμXI)2-Eμνλ(12AμabνAλcdTa[Tb,Tc,Td]+13AμabAνcdAλef[Ta,Tc,Td][Tb,Te,Tf])+i2Ψ¯ΓμDμΨ+i4Ψ¯ΓIJ[XI,XJ,Ψ]>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=4super Yang-Mills in four dimensions, the BFSS matrix theory [27], and the IIB matrix model [41]. 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 Taas XI=XaITa, Ψ=ΨaTaand Aμ=AabμTaTb, where I=3,,10and μ=0,1,2. <>represents a metric for the 3-algebra. Ψis a Majorana spinor of SO(1,10) that satisfies Γ012Ψ=Ψ. Eμνλis a Levi-Civita symbol in three-dimensions.

Finite dimensional Lie 3-algebras with an invariant metric is classified into four-dimensional Euclidean 𝒜4algebra and the Lie 3-algebras with indefinite metrics in [9], [10], [11], [21], [22]. We do not choose 𝒜4algebra 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(N)Lie algebra,

[T-1,Ta,Tb]=0[T0,Ti,Tj]=[Ti,Tj]=fkijTk[Ti,Tj,Tk]=fijkT-1uid61

where a=-1,0,i(i=1,,N2). Tiare generators of u(N). A metric is defined by a symmetric bilinear form,

<T-1,T0>=-1<Ti,Tj>=hijuid62

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

S=Tr(-14(x0K)2[xI,xJ]2+12(x0I[xI,xJ])2-12(x0Ibμ+[aμ,xI])2-12Eμνλbμ[aν,aλ]+iψ¯0Γμbμψ-i2ψ¯Γμ[aμ,ψ]+i2x0Iψ¯ΓIJ[xJ,ψ]-i2ψ¯0ΓIJ[xI,xJ]ψ)uid63

where we have renamed X0Ix0I, XiITixI, Ψ0ψ0, ΨiTiψ, 2Aμ0iTiaμ, and Aμij[Ti,Tj]bμ. aμcorrespond to the target coordinate matrices Xμ, whereas bμare auxiliary fields.

In this action, T-1mode; X-1I, Ψ-1or A-1aμdoes 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 [42]. This action can be obtained by a dimensional reduction of the three-dimensional 𝒩=8BLG model [4], [5], [6] 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 [41].

This action is invariant under the translation

δxI=ηI,δaμ=ημuid64

where ηIand ημbelong to u(1). This implies that eigen values of xIand aμrepresent an eleven-dimensional space-time.

The action is also invariant under 16 kinematical supersymmetry transformations

δ˜ψ=ϵ˜uid65

and the other fields are not transformed. ϵ˜belong to u(1)and satisfy Γ012ϵ˜=ϵ˜. ϵ˜and ϵshould come from sixteen components of thirty-two 𝒩=1supersymmetry parameters in eleven dimensions, corresponding to eigen values ±1 of Γ012, respectively, as in the previous subsection.

A commutation relation between the kinematical supersymmetry transformations is given by

δ˜2δ˜1-δ˜1δ˜2=0uid66

The action is invariant under 16 dynamical supersymmetry transformations,

δXI=iϵ¯ΓIΨδAμab[Ta,Tb,]=iϵ¯ΓμΓI[XI,Ψ,]δΨ=-Aμab[Ta,Tb,XI]ΓμΓIϵ-16[XI,XJ,XK]ΓIJKϵuid67

where Γ012ϵ=-ϵ. These supersymmetries close into gauge transformations on-shell,

[δ1,δ2]XI=Λcd[Tc,Td,XI][δ1,δ2]Aμab[Ta,Tb,]=Λab[Ta,Tb,Aμcd[Tc,Td,]]-Aμab[Ta,Tb,Λcd[Tc,Td,]]+2iϵ¯2Γνϵ1OμνA[δ1,δ2]Ψ=Λcd[Tc,Td,Ψ]+(iϵ¯2Γμϵ1Γμ-i4ϵ¯2ΓKLϵ1ΓKL)OΨuid68

where gauge parameters are given by Λab=2iϵ¯2Γμϵ1Aμab-iϵ¯2ΓJKϵ1XaJXbK. OμνA=0and OΨ=0are equations of motions of Aμνand Ψ, respectively, where

OμνA=Aμab[Ta,Tb,Aνcd[Tc,Td,]]-Aνab[Ta,Tb,Aμcd[Tc,Td,]]+Eμνλ(-[XI,Aabλ[Ta,Tb,XI],]+i2[Ψ¯,ΓλΨ,])OΨ=-ΓμAμab[Ta,Tb,Ψ]+12ΓIJ[XI,XJ,Ψ]uid69

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

δ2δ1-δ1δ2=0uid70

up to the equations of motions and the gauge transformations.

The 16 dynamical supersymmetry transformations () are decomposed as

δxI=iϵ¯ΓIψδx0I=iϵ¯ΓIψ0δx-1I=iϵ¯ΓIψ-1δψ=-(bμx0I+[aμ,xI])ΓμΓIϵ-12x0I[xJ,xK]ΓIJKϵδψ0=0δψ-1=-Tr(bμxI)ΓμΓIϵ-16Tr([xI,xJ]xK)ΓIJKϵδaμ=iϵ¯ΓμΓI(x0Iψ-ψ0xI)δbμ=iϵ¯ΓμΓI[xI,ψ]δAμ-1i=iϵ¯ΓμΓI12(x-1Iψi-ψ-1xiI)δAμ-10=iϵ¯ΓμΓI12(x-1Iψ0-ψ-1x0I)uid71

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

(δ˜2δ1-δ1δ˜2)xI=iϵ¯1ΓIϵ˜2ηI(δ˜2δ1-δ1δ˜2)aμ=iϵ¯1ΓμΓIx0Iϵ˜2ημ(δ˜2δ1-δ1δ˜2)A-1iμTi=12iϵ¯1ΓμΓIx-1Iϵ˜2uid72

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

δ˜2δ1-δ1δ˜2=δηuid73

where δηis a translation.

(), (), and () imply the 𝒩=1supersymmetry 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 [26]. Especially, we study mostly the model with the u(N)u(N)Hermitian 3-algebra ().

The continuum action () can be rewritten by using the triality of SO(8)and the SU(4)×U(1)decomposition [8], [43], [44] as

Scl=d3σ-g(-V-Aμba{ZA,Ta,Tb}Adcμ{ZA,Tc,Td}+13EμνλAμbaAνdcAλfe{Ta,Tc,Td}{Tb,Tf,Te}+iψ¯AΓμAμba{ψA,Ta,Tb}+i2EABCDψ¯A{ZC,ZD,ψB}-i2EABCDZD{ψ¯A,ψB,ZC}-iψ¯A{ψA,ZB,ZB}+2iψ¯A{ψB,ZB,ZA})uid75

where fields with a raised Aindex transform in the 4 of SU(4), whereas those with lowered one transform in the 4¯. Aμba(μ=0,1,2) is an anti-Hermitian gauge field, ZAand ZAare a complex scalar field and its complex conjugate, respectively. ψAis a fermion field that satisfies

Γ012ψA=-ψAuid76

and ψAis its complex conjugate. Eμνλand EABCDare Levi-Civita symbols in three dimensions and four dimensions, respectively. The potential terms are given by

V=23ΥBCDΥCDBΥBCD={ZC,ZD,ZB}-12δBC{ZE,ZD,ZE}+12δBD{ZE,ZC,ZE}uid77

If we replace the Nambu-Poisson bracket with a Hermitian 3-algebra's bracket [19], [20],

d3σ-g{ϕa,ϕb,ϕc}[Ta,Tb;T¯c¯]uid78

we obtain the Hermitian 3-algebra model of M-theory [26],

S=<-V-Aμb¯a[ZA,Ta;T¯b¯]Ad¯cμ[ZA,Tc;T¯d¯]¯+13EμνλAμb¯aAνd¯cAλf¯e[Ta,Tc;T¯d¯][Tb,Tf;T¯e¯]¯+iψ¯AΓμAμb¯a[ψA,Ta;T¯b¯]+i2EABCDψ¯A[ZC,ZD;ψ¯B]-i2EABCDZ¯D[ψ¯A,ψB;Z¯C]-iψ¯A[ψA,ZB;Z¯B]+2iψ¯A[ψB,ZB;Z¯A]>uid79

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

V=23ΥBCDΥ¯CDBΥBCD=[ZC,ZD;Z¯B]-12δBC[ZE,ZD;Z¯E]+12δBD[ZE,ZC;Z¯E]uid80

This matrix model can be obtained formally by a dimensional reduction of the 𝒩=6BLG action [8], which is equivalent to ABJ(M) action [7], [45]The authors of [46], [47], [48], [49] studied matrix models that can be obtained by a dimensional reduction of the ABJM and ABJ gauge theories on S3. They showed that the models reproduce the original gauge theories on S3in planar limits.,

S𝒩=6BLG=d3x<-V-DμZADμZA¯+Eμνλ(12Aμc¯bνAλd¯aT¯d¯[Ta,Tb;T¯c¯]+13Aμb¯aAνd¯cAλf¯e[Ta,Tc;T¯d¯][Tb,Tf;T¯e¯]¯)-iψ¯AΓμDμψA+i2EABCDψ¯A[ZC,ZD;ψB]-i2EABCDZ¯D[ψ¯A,ψB;Z¯C]-iψ¯A[ψA,ZB;Z¯B]+2iψ¯A[ψB,ZB;Z¯A]>uid82

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

S=Tr(-(2π)2k2V-(ZAAμR-AμLZA)(ZAARμ-ALμZA)-k2πi3Eμνλ(AμRAνRAλR-AμLAνLAλL)-ψ¯AΓμ(ψAAμR-AμLψA)+2πk(iEABCDψ¯AZCψBZD-iEABCDZDψ¯AZCψB-iψ¯AψAZBZB+iψ¯AZBZBψA+2iψ¯AψBZAZB-2iψ¯AZBZAψB))uid83

where AμR-k2πiAμb¯aTb¯Taand AμL-k2πiAμb¯aTaTb¯are N×NHermitian matrices. In the algebra, we have set α=2πk, where kis an integer representing the Chern-Simons level. We choose k=1in order to obtain 16 dynamical supersymmetries. Vis given by

V=+13ZAZAZBZBZCZC+13ZAZAZBZBZCZC+43ZAZBZCZAZBZC-ZAZAZBZCZCZB-ZAZAZBZCZCZBuid84

By redefining fields as

ZAk2π13ZAAμ2πk13AμψAk2π16ψAuid85

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

S=Tr(-V-(ZAAμR-AμLZA)(ZAARμ-ALμZA)-i3Eμνλ(AμRAνRAλR-AμLAνLAλL)-ψ¯AΓμ(ψAAμR-AμLψA)+iEABCDψ¯AZCψBZD-iEABCDZDψ¯AZCψB-iψ¯AψAZBZB+iψ¯AZBZBψA+2iψ¯AψBZAZB-2iψ¯AZBZAψB)uid86

as opposed to three-dimensional Chern-Simons actions.

If we rewrite the gauge fields in the action as AμL=Aμ+bμand AμR=Aμ-bμ, we obtain

S=Tr(-V+([Aμ,ZA]+{bμ,ZA})([Aμ,ZA]-{bμ,ZA})+iEμνλ(23bμbνbλ+2AμAνbλ)+ψ¯AΓμ([Aμ,ψA]+{bμ,ψA})+iEABCDψ¯AZCψBZD-iEABCDZDψ¯AZCψB-iψ¯AψAZBZB+iψ¯AZBZBψA+2iψ¯AψBZAZB-2iψ¯AZBZAψB)uid87

where [,]and {,}are the ordinary commutator and anticommutator, respectively. The u(1)parts of Aμdecouple because Aμappear only in commutators in the action. bμcan be regarded as auxiliary fields, and thus Aμcorrespond to matrices Xμthat represents three space-time coordinates in M-theory. Among N×Narbitrary complex matrices ZA, we need to identify matrices XI(I=3,10) representing the other space coordinates in M-theory, because the model possesses not SO(8)but SU(4)×U(1)symmetry. Our identification is

ZA=iXA+2-XA+6,XI=X^I-ixI1uid88

where X^Iand xIare su(N)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 [50], [7], [51]. We will see that this identification works also in our case. We should note that while the su(N)part is Hermitian, the u(1)part is anti-Hermitian. That is, an eigen-value distribution of Xμ, ZA, and not XIdetermine the spacetime in the Hermitian model. In order to define light-cone coordinates, we need to perform Wick rotation: a0-ia0. After the Wick rotation, we obtain

A0=A0^-ia01uid89

where A0^is a su(N)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 [30], [31], [32], [33], [34], [35]. This fact is a strong criterion for a model of M-theory. In [26], [28], 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-x-+2πR, where x±=12(x10±x0), and Lorentz boost in x10direction with an infinite momentum. After appropriate scalings of fields [26], [28], we define light-cone coordinate matrices as

X0=12(X+-X-)X10=12(X++X-)uid91

We integrate out bμby using their equations of motion.

A matrix compactification [52] on a circle with a radius R imposes the following conditions on X-and the other matrices Y:

X--(2πR)1=UX-UY=UYUuid92

where Uis a unitary matrix. In order to obtain a solution to (), we need to take Nand consider matrices of infinite size [52]. A solution to () is given by X-=X¯-+X˜-, Y=Y˜and

U=0100101001n×nU(N)uid93

Backgrounds X¯-are

X¯-=-T3x¯0-T0-(2πR)diag(,s-1,s,s+1,)1n×nuid94

in the Lie 3-algebra case, whereas

X¯-=-i(T3x¯-)1-i(2πR)diag(,s-1,s,s+1,)1n×nuid95

in the Hermitian 3-algebra case. A fluctuation x˜that represents u(N)parts of X˜-and Y˜is

x˜(0)x˜(1)x˜(2)x˜(-1)x˜(0)x˜(1)x˜(2)x˜(-2)x˜(-1)x˜(0)x˜(1)x˜(2)x˜(-2)x˜(-1)x˜(0)x˜(1)x˜(2)x˜(-2)x˜(-1)x˜(0)x˜(1)x˜(-2)x˜(-1)x˜(0)uid96

Each x˜(s)is a n×nmatrix, where sis an integer. That is, the (s, t)-th block is given by x˜s,t=x˜(s-t).

We make a Fourier transformation,

x˜(s)=12πR˜02πR˜dτx(τ)eisτR˜uid97

where x(τ)is a n×nmatrix in one-dimension and RR˜=2π. From ()-(), the following identities hold:

tx˜s,tx'˜t,u=12πR˜02πR˜dτx(τ)x'(τ)ei(s-u)τR˜tr(s,tx˜s,tx'˜t,s)=V12πR˜02πR˜dτtr(x(τ)x'(τ))[x¯-,x˜]s,t=12πR˜02πR˜dττx(τ)ei(s-t)τR˜uid98

where tris a trace over n×nmatrices and V=s1.

Next, we boost the system in x10direction:

X˜'+=1TX˜+X˜'-=TX˜-uid99

The DLCQ limit is achieved when T, where the "novel Higgs mechanism" [50] is realized. In T, the actions of the 3-algebra models of M-theory reduce to that of the BFSS matrix theory [27] with matrices of finite size,

S=1g2-dτtr(12(D0xP)2-14[xP,xQ]2+12ψ¯Γ0D0ψ-i2ψ¯ΓP[xP,ψ])uid100

where P,Q=1,2,,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 [53] (see also [54], [55], [56]). If we add mass terms and a flux term,

Sm=-12μ2(XI)2-i2μΨ¯Γ3456Ψ+HIJKL[XI,XJ,XK]XLuid102

such that

HIJKL={-μ6ϵIJKL(I,J,K,L=3,4,5,6or7,8,9,10)0(otherwise)uid103

to the action (), the total action S0+Smis invariant under dynamical 16 supersymmetries,

δXI=iϵ¯ΓIΨδAμab[Ta,Tb,]=iϵ¯ΓμΓI[XI,Ψ,]δΨ=-16[XI,XJ,XK]ΓIJKϵ-Aμab[Ta,Tb,XI]ΓμΓIϵ+μΓ3456XIΓIϵuid104

From this action, we obtain various interesting solutions, including fuzzy sphere solutions [53].

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(m)u(n)and sp(2n)u(1)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=1supersymmetry in eleven dimensions. In the DLCQ limit, both the models reduce to the BFSS matrix theory with matrices of finite size as they should.

Acknowledgements

We would like to thank T. Asakawa, K. Hashimoto, N. Kamiya, H. Kunitomo, T. Matsuo, S. Moriyama, K. Murakami, J. Nishimura, S. Sasa, F. Sugino, T. Tada, S. Terashima, S. Watamura, K. Yoshida, and especially H. Kawai and A. Tsuchiya for valuable discussions.

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Matsuo Sato (July 11th 2012). 3-Algebras in String Theory, Linear Algebra - Theorems and Applications, Hassan Abid Yasser, IntechOpen, DOI: 10.5772/46480. Available from:

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