Noncommutative Nonlinear Supersymmetry

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hep-th/0212329CSULB–PA–02–6(Revised Version)Noncommutative Nonlinear Supersymmetry Hitoshi N ISHINO 1and Subhash R AJPOOT 2Department of Physics &Astronomy California State University 1250Bell?ower Boulevard Long Beach,CA 90840Abstract We present noncommutative nonlinear supersymmetric theories.The ?rst exam-

ple is a non-polynomial Akulov-Volkov-type lagrangian with noncommutative nonlin-ear global supersymmetry in arbitrary space-time dimensions.The second example is the generalization of this lagrangian to Dirac-Born-Infeld lagrangian with nonlinear supersymmetry realized in dimensions D =2,3,4,6and 10.

PACS:12.60.Jv,11.10.Lm,04.20.Gz

Key Words:Noncommutative Geometry,Nonlinear Supersymmetry,Akulov-Volkov,

Dirac-Born-Infeld Action

1.Introduction

The importance of noncommutative geometry has been widely recognized,motivated by the recent developments of M-theory,[1]open superstrings[2],or D-branes[3][4][5]leading to noncommutative space-time coordinates[6][7].In fact,the low energy e?ective theory of open strings attached to noncommutative branes becomes a noncommutative gauge theory [8].Another example is a recent study[9]showing the equivalence between Dirac-Born-Infeld (DBI)theory with noncommutative gauge?eld strength and the ordinary DBI theory under so-called Seiberg-Witten map[7].It has been also pointed out that type IIB matrix model with D-brane backgrounds can be interpreted as noncommutative Yang-Mills theory[10].

In noncommutative geometry,all the products in the theory are replaced by so-called?product involving the constant tensorθμν[11].The next natural step to be considered is to make such algebra consistent with general covariance,which could possibly lead to the consistent formulation of noncommutative supergravity.However,there seems to be some fundamental problem with such trials,due to the di?culty of choosing the right measure, and/or dealing with complex metric with the right degrees of freedom[12][13].There seems to be a persistent problem for unifying noncommutativity with the concept of metrics in gravity[12][13],not to mention supergravity with local supersymmetry.

As far as global supersymmetry is concerned,there has been considerable progress in noncommutative theories,e.g.,at quantum level[14],or in superspace with Moyal-Weyl deformations[11]for supersymmetric DBI theory[15].Also ten-dimensional(10D)super-symmetric Yang-Mills has been generalized to noncommutative case,including the F4-order corrections as the?rst non-trivial terms for supersymmetric DBI lagrangian[16]which can serve as an underlying theory of all noncommutative supersymmetric Yang-Mills theory in D≤9.

There has been,however,a di?erent realization of global supersymmetry,called nonlinear realization.About three decades have past,since Volkov and Akulov(VA)gave a lagrangian for nonlinear supersymmetry in4D in terms of Nambu-Goldstone fermion in1972[17]. Interestingly,it has been also known that nonlinear supersymmetries are not peculiar to 4D,but such formulations are universal in arbitrary space-time dimensions for both simple and extended supersymmetries[18].In such a universal formulation,the lagrangian is given in terms of‘vielbein’as a generalization of the vierbein in the original VA lagrangian in 4D[17].The inclusion of non-Abelian?eld strength as in DBI action is also shown to be straightforward[18].

Considering these long and recent developments as well,we realize the importance of combining the two concepts,i.e.,noncommutative algebra[6]and nonlinear supersymme-try[17][18].Such a trial is also strongly motivated by D-brane physics[3][4]related to superstrings[2]and M-theory[1].In fact,a typical example is the pioneering work on su-

2

persymmetric DBI action in 10D by Aganagic,Popescu and Schwarz [5],and it is a natural next question whether such a lagrangian can be compatible with noncommutativity.In our present paper,we establish explicit lagrangians which are noncommutative generalization of DBI action with nonlinear supersymmetries.

As a preliminary step in the next section,we ?rst present the noncommutative gener-alization of VA actions in arbitrary space-time dimensions.Based on this,we study non-commutative generalization of supersymmetric DBI action with nonlinear supersymmetry in dimensions D =2,3,4,6and 10in section 3.Section 4is for concluding remarks.Appendix A is devoted to the detailed explanations for ?ipping/hermiticity properties of fermions in general space-time dimensions ?D .Appendix B is for a lemma for general variations of non-commutative functionals.Appendix C is for a lemma related to the hermitian conjugation in (2.13).

2.Noncommutative VA Lagrangian in ?D

We ?rst present our result,and subsequently we explain its notational or technical details.Our total action is valid in ?D space-time dimensions with the usual signature (ηmn )=diag .(+,D ?1 ?,?,···,?):

I VA ≡

d D x sdet ?(E μm )≡ d D x E (2.1a)= d D x [(?1)D ?1sdet ?(g μν)]1

/2?

≡ d D x g 1/2?,(2.1b)where E μm is our vielbein and g μνis our ‘metric’:

E μm ≡δμm +S (i

λγ(μ??ν)λ)?(λ?γm ?νλ)]=g νμ.(2.2b)

The λis a (symplectic)(pseudo)Majorana spinor which is possible in any space-time dimension [19].3As usual,the symbol ?refers to a noncommutative product de?ned typically in terms of two arbitrary ?elds f (x )and g (x )by [11]

f ?

g ≡f exp (i ←?μθμν→?ν)g ≡

∞ n =1(+i )n 3The fermion λmay carry implicit Sp (1)indices,if it is symplectic (pseudo)Majorana spinor [19].See Appendix A for more details.

3

Any subscript symbol ?therefore refers to expressions containing such ?products,such as the determinants which are symmetrized by the symmetrization operator S :

E ≡sdet ?(E μm )≡S 1

D !

?μ1···μD ?ν1···νD G μ1ν1?···?G μD νD ,(2.4b) g ≡(?1)D ?1g .(2.4c)

The factor (?1)D ?1is need in (2.1b),due to det (ηmn )=(?1)D ?1.The S -operation is the total symmetrization of any ?product:

S (A 1?···?A n )≡1n !n f ?···?f (p ∈I R).(2.6)

Here p can be any real number,not necessarily 1/2or ?1.The split 1+f seems to be always needed,in such a way that this in?nite series makes sense.

As explained in [18]for commutative case,the distinction between the two groups of indices μ,ν,···and m,n,···is ‘formal’,in order to use the analogy with general coordinate transformations.The meaning of this becomes clearer,when we proceed.

The inverse vielbein is de?ned again as an in?nite series

E m μ≡[(I +Λ)?

1?]m μ≡(I ?Λ+Λ?Λ?Λ?Λ?Λ+···)m

μ≡(I ?Λ+Λ2??Λ3?+···)m μ(2.7)

De?ned in this fashion,E m μis unique,satisfying the ortho-normality conditions

S (E μm ?E m ν)=E μm ?E m ν=δμν,

S (E m μ?E μn )=E m μ?E μn =δm n .(2.8)The inverse metric G μνis de?ned by

g μν=S (E m μ?ηmn ?E n ν),(2.9)

satisfying the conditions

S(gμν?gνρ)=δμρ,S(gμν?gνρ)=δρμ.(2.10) It is sometimes important to set up the complex-conjugate acting on the?products,as

(A?B)?≡(B?)?(A?).(2.11)

This rule is valid,even though there is a complex exponent withθμνimplicitly in the ?product,due toθμν=?θνμ.Accordingly,our‘vielbein’de?ned by(2.2a)is real.To show this,we use the general hermiticity feature of(pseudo)Majorana spinors in D-dimensional space-time in the commutative case that

[i(λγμ?νχ),(2.12)

for the inner product of two(symplectic)(pseudo)Majorana spinorsχandλ.5Now in the noncommutative case,we can con?rm the generalization of this with the inclusion of the S-operator,as

{S[i(λ?γμ?νχ)].(2.13)

Here the details of the con?rmation is given in Appendix C.By supplying the Sp(1)-indices A,B,···=1,2,the proof goes in a parallel way for the symplectic(pseudo)Majorana case,too. Now the hermiticity of Eμm is transparent,because(Λμm)?=Λμm.

Our action I VA(2.1)is invariant under global nonlinear supersymmetry

δQλ=?+S[i(?γμλ).(2.14) Relevantly,Eμm and gμνtransform as

δQ Eμm=S[ξν??νEμm+(?μξν)?Eνm],(2.15a)

=S[ξρ??ρgμν+(?μξρ)?gρν+(?νξρ)?gμρ].(2.15b)δQ g

μν

In other words,these?elds are transforming formally the same as‘general coordinate trans-formations’.

Since our Eμm and gμνare transforming,as if they were under general coordinate transformations,the invariance con?rmation of our action I AV can be con?rmed as follows: First,consider the variation

δE≡δ[sdet?(Eμm)]=S[E?E mμ?(δEμm)],(2.16)

for an arbitrary variation δE μm ,con?rmed as

(LHS)=δE =δ

1

(D ?1)!?μ1···μD ?m 1···m D S [(δE μ1m 1)?E μ2m 2?···?E μD m D ]

=

1

(D ?1)!

?m 1···m D ?n 1m 2···m D S [E ?E n 1μ1?(δE μ1m 1)]

=S [E ?E m μ?(δE μm )]=(RHS).

(2.17)

Here use is made of the relationships (2.8),?mr 1···r D ?1?n r 1···r D ?1=(?1)D ?1(D ?1)!,and

?μ1···μD =S (?m 1···m D E ?E m 1μ1?···?E m D μD ),

(2.18)

which in turn is con?rmed as

(RHS)=S [?m 1···m D E ?E m 1μ1?···?E m D μD ]

=S [1

(D !)2

?i 1···i D ?j 1···j D ?k 1···k D ?l 1···l D (A i 1j 1?···?A i D j D )?(A k 1l 1?···?A k D l D )]

=S [

1D !

?i 1···i D ?j 1···j D (A ?B )i 1j 1?···?(A ?B )i D j D ]

=sdet ?(A ?B )=(RHS),

(2.22)

6

where we used the indices i,j,···,because it is common to the indices

μ,ν,···

and m,n,···.

Using (2.21),we can prove the relationship

g ≡(?1)D ?1g ≡(?1)D ?1sdet ?(g μν)=E 2

?,

(2.23)

as

(LHS)=(?1)D ?1sdet ?(g μν)=(?1)D ?1sdet ?[S (E μm ?ηmn ?E νn )]

=(?1)D ?1sdet ?(E μm ?ηmn ?E νn )

=(?1)D ?1[sdet ?(E μm )]?[sdet ?(ηmn )]?[sdet ?(E νn )T ]=+[sdet ?(E μm )]?[sdet ?(E νn )]

=E ?E =E 2?=(RHS).

(2.24)

Even though (2.23)does not necessarily imply the equality g

1/2

?=E ,it can be con?rmed

by help of the following lemma:

(A p ?)?(A q ?)=A p +q

?

(p,q ∈I R),(2.25)

where an arbitrary real scalar A itself can contain some ?products in it.The lemma

(2.25)can be con?rmed by splitting A ≡1+a ,and

(LHS of (2.25))=A p ??A q ?=(1+a )p ??(1+a )q

?

= ∞ n =0

p (p ?1)(p ?2)···(p ?n +1) n !

m

a ?···?a

=∞ m =0∞ n =0

p (p ?1)···(p ?n +1)

m !

m +n

a ?···?a

=

∞ N =0N n =0

p (p ?1)···(p ?n +1)

(N ?n )!

N

a ?···?a (N ≡m +n ).(2.26)

We now use the identity for the usual commutative product (1+a )p (1+a )q =(1+a )p +q :

N n =0

p (p ?1)···(p ?n +1)

(N ?n )!

≡

(p +q )(p +q ?1)···(p +q ?N +1)

N !

N

a ?···?a

=(1+a )p +q ?=A p +q

?

=(RHS of (2.25)).(2.28)

7

Once (2.25)is established,it is clear that

g 1/2?

=E ,(2.29)

because it satis?es

g 1/2?? g 1/2

?= g 1?= g

=E ?E ,(2.30)

by (2.25),as desired.As a corollary,the reality of the integrand g

1/2

?in (2.1b)is easily

seen.

The relationship (2.29)also provides an alternative con?rmation of the invariance δQ I VA ,via δQ G μνinstead of δQ E μm .First,note the lemma for an arbitrary variation δ:

δ[(F [?])p ?]=p S [(δF [?])?(F [?])p ?1

?

](p ∈I R),(2.31)

con?rmed by the lemma (B.3)in Appendix B for a general variation of a noncommutative

functional of ?.Second,we use the relationship

δ g

=S [ g ?(δg μν)?g μν],(2.32)

con?rmed as

(LHS)=δ g

=δ (?1)D ?1

(D ?1)!

?μ1···μD ?ν1···νD S [(δg μ1ν1)?g μ2ν2?···?g μD νD ]

=

(?1)D ?1

(D ?1)!?m 1···m D ?n 1···n D S [E ?E ?(δg μ1ν1)?E m 1μ1?E n 1ν1

?(E m 2μ2?g μ2ν2?E n 2ν2)?···?(E m D μD ?g μD νD ?E n D νD )]

=

(?1)D ?1

yielding(2.15b).The invariance of our action I VA can be now throughδQ gμνinstead of δQ Eμm,as

δQ g1/2?=+12S[ g?1/2?? g?(δQ gμν)?gμν]

=+1

S[ g1/2??ξρ?(?ρgμν)?gμν+2(?μξρ)?gρν?gμν]

2

=S[1

?2γμ?1)for translation arising out of the commutator of two supersymmetries.In the commutative case[17][18],we know that this‘constant’vector poses no problem for the same reason given above.As such, all the e?ect of constantθμνdoes not upset the basic structure of transformation(2.15) mimicking a‘general coordinate transformation’.Once this point is understood,we have no worry about the compatibility between the constantθμνand general covariance,because the latter is just a‘fake’symmetry of the system.

9

3.Noncommutative Supersymmetric DBI Lagrangian

Once we have understood a noncommutative generalization of VA lagrangian in?D di-mensions,it is relatively easy to generalize it to a DBI lagrangian[5]with nonlinear super-symmetry.The only caveat is that due to the Fierz arrangement involved for quartic fermion terms,the space-time dimensions will be restricted to be D=2,3,4,6and10,as we will see shortly.

The generalization from the VA case occurs in the de?nition of the metric.Our action is now in terms of a new metric Gμν:

I DBI≡ d D x[(?1)D?1sdet?(Gμν)]1/2?≡ d D x G1/2?,(3.1a)

G≡(?1)D?1G≡(?1)D?1sdet?(Gμν),(3.1b) where the previous metric(2.2)is now generalized to the new metric

Gμν≡S[ημν+2i(λ?γm?μλ)(

λ?γ??μ?ν??λ)+Fμν].(3.2b)

Compared with g

μν,the di?erence is in the last two terms in(3.2b).A special case of this

lagrangian in10D corresponds to the lagrangian in[5].The new?eld Aμundergoes the supersymmetry transformation rule:

δQ Aμ=S[ξν??νAμ+(?μξν)?Aν+ξμ+iλ?γρ?μλ)],

δQλ=?+S[i(

?γμλ)is the same as the last section.This is a noncommutative and multi-dimensional generalization of the commutative case in10D[5].

The invariance of our action I DBI can be con?rmed in a way parallel to the previous case for I AV,with the aid of the lemma

δQ Fμν=S[ξρ??ρFμν+(?μ?ξρ)?Fρν+(?νξρ)?Fμρ?2i(

3

???μ|[(λγρ?|ν??λ)].(3.4)

The con?rmation of this lemma needs special care,associated with a Fierz rearrangement. This is because we need the equality

S[(λγρ?νλ)]?(μ?ν)=S[1?γρλ)?(

β)δ

6Here the indicesα,γmay contain also the Sp(1)indices for symplectic(pseudo)Majorana spinors.

10

This identity holds only in space-time dimensions D=2,3,4,6and10[20],so that the invariance of our action I DBI is valid only in these dimensions.The important ingredient here is the Fierz identity in its very universal form(3.6),which does not depend on the dimensionality of spinorial components.

The actual invariance con?rmation of I DBI under(3.3)is parallel to that for(2.1), because the metric Gμνtransforms under(3.2)exactly as(2.15b):

=S[ξρ??ρGμν+(?μξρ)?Gρν+(?νξρ)?Gμρ].(3.7)δQ G

μν

Since the rest of the proof is parallel to that for the action(2.1),we will skip it here.

As an independent consistency check,we study the commutator of two supersymmetries on Aμ:

??δ1,δ2??Aμ=ζν?νAμ+ζμ(ζμ≡2i(

ζν(

?2γνλ)?(

4

?2γμ?1)poses no problem in our formulation,neither does the constant tensorθμν.Needless to say,the result in[5]is a special case in10D.

4.Concluding Remarks

In this paper,we have established the noncommutative version of VA lagrangian,and that of DBI lagrangian with nonlinear supersymmetry in space-time dimensions2,3,4,6and 10.The invariance of our actions under nonlinear supersymmetry has been con?rmed by the use of various lemma,involving the symmetrized noncommutative determinants.The important new ingredient is that our noncommutative VA-type action is valid in?D,while our noncommutative DBI action is valid in D=2,3,4,6and10.

The di?culty of noncommutative generalization of supergravity has been well recognized for some time[12][21].This is caused by the compatibility question between the enlarged complexi?ed Lorentz symmetry such as U(1,3)and spinor structure of such space-time man-ifolds.Even though our lagrangians have‘formal’metrics or vielbeins,we do not encounter

11

such a problem,because gμν,Gμνor Eμm de?ned by(2.2)or(3.2)have the S-operator which makes these?elds real.Therefore no complexi?cation of Lorentz symmetries,such as U(1,3),is needed.

We have also seen that the existence of the‘constant’tensorθμνdoes not pose any problem with the‘general coordinate transformation’like(2.15)or(3.7).This is because general covariance is a‘fake’symmetry that does not actually exist in the system,but this is just for analogy that simpli?es the computation for invariance con?rmation.As a matter of fact,we have already encountered similar situations in the commutative case,such as the‘constant’Kronecker’s deltaδμm in Eμm,or the‘constant’vectorial parameter ζμ≡2i(

Appendix A:Flipping and Hermiticity Properties for Fermions

In this appendix,we analyze the ?ipping and hermiticity properties of fermionic bilinears,as promised in sections two and three.

Consider the general

D -dimensional space-time of dimension with the signature

(+,D ?1

?,?,···,?)with the Cli?ord algebra

{γm ,γn }=+2ηmn =+2diag.(+,D ?1 ?,?,···,?).(A .1)For general treatment of spinors,we follow [19],where the relevant equations are such as

(γ0)?=+γ0,(γi )?=?γi (i =1,2,···,D ?1),

(γm )?=Aγm A ?1,

A ≡γ0,A ?=A ,(γm )?=ηBγm

B ?1,B ≡(A T )?1

C ?1,A ≡(B T )?1C ,

(γm )T =+ηCγm C ?1,C ?C =+I ,C T =?ηC .

(A .2)Here the matrix B is related to the complex conjugation of fermions,and C is for the usual charge conjugation,both in the same notation as in [19],while ?and ηare ±1,depending on the di?erence D ?2between the space-like and time-like coordinates.There are in total four cases:D ?2=1,2,8(mod 8),or 6,7,8(mod 8),or 4,5,6(mod 8),or 2,3,4(mod 8)[19],tabulated equivalently as

?

Fermions

2,3,4(mod 8)

?1+1

Pseudo-Majorana 6,7,8(mod 8)

?1?1Symplectic Pseudo-Majorana In the case of ‘symplectic (pseudo)Majorana’spinors,we have an additional Sp (1)indices A,B,···=1,2on these fermions.

We next study the ?ipping property

(χγm 1···m n ψ).(A .3)

This can be proven by taking the transposition of the l.h.s.,which is a scalar and intact under such an operation.As for symplectic (pseudo)-Majorana spinors,these includes also the Sp (1)indices,e.g.,the l.h.s.is (

1,2,3,4and8(mod8),we have the desirable antisymmetry(?1γm?2).In the case of symplectic(pseudo)Majorana spinors in D=4,5,6,7and8(mod8),we need to

multiply an extra Sp(1)metric(?

AB )≡ 0+1

?10 ,like(?A2γm?B1)?BA=

?(

ψ=ψ?A,

ψγm1···m nχ)?=??ηn+1(

ψA=ψ?A A,

ψAγm1···m nχ

B

)?=??ηn+1(

λγμ?νλ),i(

λAγμ?νλA),i(

λγm?μλ)for(pseudo)Majorana spinors,

i(

n!

?n(a n∈I R).(B.1)

14

Then a noncommutative generalization is

H?[?]≡

∞ n=0a n

n!δ(?n?)=

∞

n=0a n

m!

?m? ?δ? =S[H′?[?]?δ?]=(RHS).(B.4)

This lemma is general enough to cover the variations needed such as(2.31),when H[?]≡(F[?])p.

Appendix C:Con?rmation of(2.13)

Here we give the detailed con?rmation of(2.13).Note that our metric and vielbein are hermitian but not complex de?ned in a peculiar way with the S-operator.Since this aspect was not covered in references in the past[12],it is better to demonstrate the details of its con?rmation:

(LHS of(2.13))={S[i(λ?γμ?νχ)}?]

=+1λ?γμ?νχ)?i(?ν

2 +i∞ n=0(+i)nλ←?ρ1···←?ρnγμθρ1σ1···θρnσn?σ1···?σn?νχ ?

+1

n!

(?ν

2(?i)

∞

0(?i)nλ?

+1

n!(?σ

1

···?σnλ?)(γμ)?θρ1σ1···θρnσn?ρ

1

···?ρn?ν

2

∞

0(?i)nχA?1)(AγμA?1)θρ1σ1···θρnσn?ρ1···?ρn(Aλ)

15

+i

n!(?σ

1

···?σn

2

∞

0(?i)nχ)γμθρ1σ1···θρnσn?ρ1···?ρnλ

+i

n!(?σ

1

···?σn

2

∞

0(+i)nχ)γμθσ1ρ1···θσnρn?ρ1···?ρnλ

+i

n!(?σ

1

···?σn

2(?ν

2

(λ?γμ?νχ)]=(RHS of(2.13)).(C.1)

Here use is also made of(A.2),and(A.4)for(pseudo)Majorana spinorsλandχ.

In the case of symplectic(pseudo)Majorana spinors,we can con?rm

{S[i(λA?γμ?νχ

A

)],(C.2)

in a similar way.The special caseχ=λorχ

A =λA leads to our conclusion(Λμm)?=

Λμm as in section two.

This result is in a sense expected,because the S-operation is e?ectively equivalent to adding the hermitian conjugate of the original expression.However,we emphasize that each step in(C.1)is the result of subtle interplay between?ipping and hermiticity properties for (pseudo)Majorana spinors andγ-matrices.

16

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