Abelian gauge invariance of the WZ-type coupling in ABJM theory

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Abelian

gauge

invariance

of

the

WZ-type

coupling

in

ABJM

theory

Dongmin Jang

a

,

Yoonbai Kim

a

,

O-Kab Kwon

b

,

c

,

∗

,

D.D. Tolla

d

a_Department_of_Physics,_BK21_Physics_Research_Division,_Institute_of_Basic_Science,_Sungkyunkwan_University,_Suwon_440-746,_Republic_of_Korea b_Department_of_Physics,_Kyungpook_National_University,_Taegu_702-701,_Republic_of_Korea

c_Department_of_Physics,_Ewha_Womans_University,_Seoul_120-750,_Republic_of_Korea d_{International}_School_for_Advanced_Studies_(SISSA),_Via_Bonomea_265,₃₄₁₃₆_Trieste,_Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received9June2015

Receivedinrevisedform8July2015 Accepted13July2015

Availableonline17July2015 Editor:N.Lambert

WeconstructtheinteractiontermsbetweentheworldvolumeﬁeldsofmultipleM2-branesand3-form

gaugeﬁeldof11-dimensionalsupergravity,inthecontextofABJMtheory.Theobtained

Wess–Zumino-typecouplingissimultaneouslyinvariantundertheUL(N) ×UR(N)non-Abeliangaugetransformationof

theABJMtheoryandtheAbeliangaugetransformation ofthe3-formﬁeldin11-dimensionalsupergravity.

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

In type IIA and IIB string theories, the RR form fields in 10-dimensionalsupergravitiesarecoupledtotheD-branesthrough Wess–Zumino(WZ)-typeaction[1–3].Inthe effectivefield theory ofmultipleDp-branes, theWZ-typeactionincludesthecouplings to higher rank RR form fields, which are usually referred to as theMyers couplings [3]. Like theWZ-typecouplings ofD-branes instringtheory,WZ-typecouplingsofmultipleM2-branescanbe constructed [4–10] in the context of the effective field theories, forinstance,theBagger–Lambert–Gustavsson theory [11] andthe Aharony–Bergman–Jafferis–Maldacena (ABJM) theory [12]. These WZ-typecouplingsdescribethecouplingsbetweenM2-branesand 3- and6-formgaugefieldsin11-dimensionalsupergravity.

In [9]the invariance under the non-Abelian gauge symmetry, UL

(

N

)

×

UR

(

N

)

of the original ABJMtheory [12], was utilizedto determinetheWZ-typecouplings ontheM2-brane worldvolume. Theresultswereextendedtoincludenon-lineartermsoftheform ﬁelds[10]. Theproposed WZ-typeaction in[9]was putto some tests and proven to be consistent. First, in the particular case of N

=

1, it nicely reproduces the well-known coupling of the 3-formgaugeﬁeldtotheworldvolumeﬁeldsofasingleM2-brane

[13]. Second, under the circle compactification, the action gives the correct Myers coupling of the RR form fields to the world-volumefieldsofD2-branesin typeIIA stringtheory[3].Third, in the particularcase ofa 6-formgauge field withconstant 7-form

*

Correspondingauthor.

E-mailaddresses:[email protected](D. Jang),

[email protected]

(Y. Kim),

[email protected](O-K. Kwon),

[email protected]

(D.D. Tolla).

ﬁeld strength, the proposed WZ-type action in [7,9] reproduces the fullsupersymmetry-preserving quadraticmass-deformationof theABJMtheory[14,15].Lesssupersymmetriccasesof

N =

2 and

N =

4 inABJMtheory havealsobeeninvestigatedin[16,17].The aforementionedtestssupportthecorrectnessoftheproposed WZ-type couplingtoareasonable extent,however,thereremains one more important test to be passed, i.e. the invariance under the Abeliangaugetransformationoftheformﬁeldsin11-dimensional supergravity.Itisthemaingoalofthispapertoconductthistest.

The 11-dimensional supergravity action isinvariant underthe Abelian gauge transformation of theformﬁelds,

Cr

→

Cr

+

d

r−1

,

(1.1)

where

r

=

3

,

6.Therefore,theWZ-typecouplingsonthe worldvol-ume ofM2-branes should also satisfy the invariance under(1.1). For the Myers couplings of RR form fields, this issue was clari-fiedin[18–20].Inthispaper,weconsidertheWZ-typecouplings forthe3-formgauge fieldinthe viewpointofsuch

Abelian gauge

invariance. We show that the WZ-typecouplings in[9] is invari-ant under the Abelian gauge transformation (1.1) only when the ﬁeldstrengths,Fμν andFμν of

ˆ

thenon-Abeliangaugeﬁeldsofthe UL

(

N

)

×

UR

(

N

)

gaugesymmetry,arevanishing.Inthecaseof non-vanishing non-Abelianfield strengths,we show that thecoupling needs amodificationby apieceinvolvingthosefield strengths,in ordertobeinvariant undertheAbeliangaugetransformation.We findanexactformofthemodificationandproposeasimpleform ofthe3-formfieldcouplings,whichresemblethecaseofthe My-erscouplings[3]instringtheory.

This paper is organized as follows. In Section 2, we test the Abelian gauge invariance of the 3-form ﬁeld couplings inall or-ders oftheexpansionparameterwithvanishingnon-Abeliangauge

http://dx.doi.org/10.1016/j.physletb.2015.07.022

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fieldstrengths,proposedin[9].InSection 3,weproposeasimple formofthe3-formfieldcouplingswithnon-vanishingnon-Abelian gaugefield strengths andtest theproposalisinvariant underthe Abeliangauge transformation.In Section 4 we draw our conclu-sion.

2. Abeliangaugeinvariance:Fμν

= ˆ

Fμν

=

0 case

In the ABJM theory of multiple M2-branes, the bosonic sec-toroftheM2-braneworldvolumeﬁeldscontainstwonon-Abelian gauge ﬁelds, Aμ and Aμ,

ˆ

and four complex scalar ﬁelds, YA

( A

=

1

,

2

,

3

,

4). TheWZ-typecouplingswere constructedbyusing four covariant building blocks and their complex conjugates [9]. Thesebuilding blocks are the 3-form gauge field C3, the 6-form gauge field C6, both of which are functionals of the complex scalarfields,thecovariantderivativesofthecomplexscalarfields

DμYA

= ∂

μ YA

+

i AμYA

−

iYAAμ,

ˆ

andtheanti-symmetrizedcubic

productofthecomplexscalarﬁelds,

β

_CA B

≡

1₂

(

YA_Y†

CYB

−

YBY † CYA

)

. Themanifestlycovariantobjects,butmissingfromthislist,arethe non-Abeliangauge ﬁeld strengths, Fμν and Fμν .

ˆ

In [9],the WZ-type couplings are constructed under the assumption that these gauge ﬁeld strengths are vanishing, which means that the cor-responding non-Abelian gauge ﬁelds were in pure gauge. In this section,wereconsiderthe3-formWZ-typecouplingsproposedin

[9]andshowthat those are invariantin all ordersofthe expan-sionparameterundertheAbeliangaugetransformation(1.1),when thenon-Abelian gauge ﬁelds, Aμ and Aμ,

ˆ

are in pure gauge, i.e.

Fμν

= ˆ

Fμν

=

0.

2.1. Deﬁnitions

Inorder toshow the Abeliangauge invariance forthe 3-form WZ-type couplings, let us consider such type of coupling for a generic p-form gauge ﬁeld, which does naturally couple to a

(

p

−

1

)

-brane. Eventually, we specializethe results to the p

=

3 case.ThespeciﬁcformofWZ-typecouplingsisgivenby

˜

Sp

=

μ

p−1

p

{

TrS

}

P

[

C(p)

]

=

μ

p−1 2

p dpx

{

TrS

}

1 p

!

μ1···μp

_P

[

_C (p)

]

[μ1···μp]

+ (

c.c.

)

,

(2.2) where

μ

p−1 represents the tension of

(

p

−

1

)

-brane, P

[· · ·]

is a non-Abelian pullback (see below or [9]),

{

TrS

}

denotes the sum overall possiblewaysthatthegaugeindicescanbecontractedto formasingletraceproductdividedbythenumberofindependent termsat a givenorder in theexpansion parameter

λ

.More pre-cisely,

{

TrS

}

= {

Tr

}/

nterms,where

{

Tr

}

isdeﬁnedin[9]and

n

termsis thenumberofindependenttermsatagivenorderin

λ

. General-izingthedeﬁnitionsgivenin[9],thenon-Abelianpullbackofthe

p-form gaugeﬁeldisgivenby P

[

C(p)

]μ

1···μp

=

C_A 1···AmB¯1··· ¯Bn

δ

A1 μ1IN

+ λ

Dμ1Y A1

· · ·

_δ

Am μmIN

+ λ

DμmY Am

×

δ

B¯1 μm+1IN

+ λ

Dμm+1Y † B1

· · ·

δ

B¯n μm+nIN

+ λ

Dμm+nY † Bn

=

p l

p

−

l k

C_A 1···AlB¯1··· ¯Bk[μl+k+1···μp

×

Dμ1Y A1

· · ·

_D μlY Al_D μl+1Y † B1

· · ·

Dμl+k]Y † Bk

,

(2.3) where

m n

=

m! (m−n)!n!,

λ

=

2

π

l 3/2

P (lP isthe Planck length), and

IN isthe

N

×

N unit matrix.Usingsuchdeﬁnitionofthepullback forthe p

=

3 case,theWZ-typecouplingin(2.2)gives

˜

S3

=

μ

2

d3x

μνρ 3

!

{

TrS

}

1 2Cμνρ

+

3

λ

CμνADρY A

+

3

λ

2

CμA BDνYADρYB

+

CμAB¯DνYADρY † B

+ λ

3

_C A BCDμYADνYBDρYC

+

3CA BC¯DμYADνYBDρY † C

+ (

c.c.

)

,

(2.4)

where

μ

2 isthe tensionof M2-brane. Thisform ofthe WZ-type couplingswasproposedin[9].

Notethatthebackgroundformfieldsarefunctionsofthe trans-verse coordinates in general, so they become functionals of the transversescalar fields,Y and Y†.Thedependence ofthe3-form gaugefield onthecomplexscalarfieldsisexpressedby meansof ageneralizedTaylorexpansion,

C

(

Y

,

Y†

)

=

r,s

λ

r+s r

!

s

!

Y A1

· · ·

_YAr_Y† B1

· · ·

Y † Bs

∂

A1

· · · ∂

Ar

∂

B¯1

· · · ∂

B¯sC 0

_,

(2.5) wherethe superscript‘0’ meansthat the corresponding field has nodependenceon thecomplexscalar fieldsandweomit the in-dices on 3-form gauge field, and

∂

A

≡

_∂(λ∂_YA₎,

∂

B¯

= ¯∂

B

≡

∂

∂(λY†_B),

(∂ ¯∂

· · ·)

C0

_{≡ (∂ ¯∂ · · ·)}

_C

₍

_Y

_,

_Y†

₎

_|

Y=Y†₌₀.Keeping(2.5)inmind,forthe

UL

(

N

)

×

UR

(

N

)

gauge invariance,each term ofthe WZ-type cou-plings in (2.4) should contain equal numbers of bifundamental ﬁelds (Y , DY ) and anti-bifundamental ﬁelds (Y†_, _DY†_). _In addi-tion,thegaugeindicesmustbecontractedappropriatelytoforma singletracecoupling.

Using(2.3)andinsertingtheexpandedp-form gaugeﬁeld(2.5)

into the action (2.2), we obtain the WZ-type couplings for the

p-form gaugeﬁeldintermsoftheexpansionparameter

λ

,

˜

Sp

=

μ

p−1 2

p dpx1 p

!

⎛

⎝

μ(p)

r,s p

l+k=0

λ

2qbl_k,_,r_s

+ (

c.c.

)

⎞

⎠ ,

(2.6) where

l, k are

thenumbersof DY , DY† _from_the_pullback_in_(2.3) and

r,

s are the numbers of Y , Y† from theTaylor expansion in

(2.5),and b_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

(

DμYA

)

(l)

(

DνY†B

)

( k)_YC(r)_Y†(s) D

(2.7) with Ul_k,_,r_s

=

1 r

!

s

!

p l

p

−

l k

∂

_C(r)

∂

_D¯(s)C0 A(l)_B¯(k)_μ(p−l−k)

.

(2.8) In order to avoida cluttering ofour expressions, we have intro-ducedthefollowingcompactnotationforourindexing

μ

(p)

_≡

_μ

1

· · ·

μ

p

,

∂

_A(r)

≡ ∂

A1

· · · ∂

Ar

,

(

DμYA

)

(l)

≡

Dμ1Y A1

· · ·

_D μlY Al

_,

YC(r)

≡

YC1

· · ·

_YCr

_,

_etc

_.

_(2.9) We alsouseindices(

μ

, A), (

ν

, B) onlywith DY , DY†,while the indices(C , D) areusedonlywith(Y ,

Y

†_).

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TheWZ-typecouplingsin(2.6)areoriginatedfromthe p-from

gaugeﬁeld

C

p.Therefore,thenumberofcovariant derivatives in-volvedmustbelessthanorequalto p, whichmeans

0

≤

l

+

k

≤

p

.

(2.10)

Thecouplingshouldalsobeinvariantunderthenon-Abeliangauge symmetry, UL

(

N

)

×

UR

(

N

)

, ofwhich realizationrequires that the number ofinvolved bifundamental and anti-bifundamental ﬁelds mustbethesame,i.e.

l

+

r

=

k

+

s

=

q

,

(2.11)

where

q is

thetotal numberof Y and DY (or Y† _and _DY†₎ _in_a giventermoftheWZ-typecoupling.Usingtheconstraintsin(2.10)

and(2.11),werewritetheWZ-typecoupling(2.6)as

˜

Sp

=

μ

p−1 2

×

p dpx1 p

!

⎛

⎝

μ(p)

∞ q=0 p

m=0 m

k=0

λ

2qbm_k_,−_q₋k,_kq−m+k

+ (

c.c.

)

⎞

⎠ .

(2.12)

2.2. Abelian gauge invariance

To prove the Abelian gauge invariance (1.1) for the WZ-type couplings in (2.6), we repeatedly integrate by parts the quantity

bl_k,_,r_s.Thentheexpression(2.7)canbewrittencompletelyinterms ofa

(

p

+

1

)

-formﬁeldstrengths,

F_μν0 (i)_A(j)_B¯(k)

= (

p

+

1

)∂

[μC_ν0(i)_A(j)_B¯(k)_]

,

i

+

j

+

k

=

p

,

(2.13) where we used the compact indexing notation deﬁned in (2.9). Oncethisprocedureisachieved, theresultingexpression is man-ifestlygauge invariant becauseoftheAbeliangaugeinvariance of the

(

p

+

1

)

-formﬁeldstrengths.

First,letusconsiderthecase

l

=

0.Integratingbyparts,

b

l_k,_,r_scan bewrittenas b_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

(

DμYA

)

(l−1)DμYA

(

DνY†B

)

( k)_YC(r)_Y†(s) D

=

Gl_k,_,r_s

−

El_k,_,r_s

−

r Al_k,_,r_s

−

sBl_k,_,r_s

,

(2.14) whereweomitthetotalderivativetermand

Gl_k,_,r_s

= −(

l

−

1

)

U_kl,_,r_s

{

TrS

}

×

(

DμYA

)

(l−2)DμDμYA YA

(

DνY†B

)

( k)_YC(r)_Y†(s) D

−

kUl_k,_,r_s

{

TrS

}

×

(

DμYA

)

(l−1)YA

(

DνY†B

)

( k−1)_D μDνY†BY C(r)_Y†(s) D

,

El_k,_,r_s

=

∂

_μUl_k,_,r_s

{

TrS

}

(

DμYA

)

(l−1)YA

(

DνY†B

)

(k)YC(r)Y †(s) D

,

A_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

(

DμYA

)

(l−1)YA

(

DνY†B

)

(k)YC(r−1)DμYC Y†_D(s)

,

B_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

(

DμYA

)

(l−1)YA

(

DνY†B

)

( k)_YC(r)_Y†(s−1) D DμY † D

.

(2.15) Hereworldvolume indicesare anti-symmetrizedbutarekept im-plicit.Therefore,thepresenceofthetwocovariantderivatives act-ing on a single object implies that such terms contain the non-Abeliangaugeﬁeldstrengthsduetotherelation,

Dμ

,

Dν

YA

=

i FμνYA

−

iYAF

ˆ

μν

.

(2.16)

Forthisreason, we seethat Gl_k,_,r_s termsin(2.14) are vanishingin the case of Fμν

= ˆ

Fμν

=

0.As a result, if the WZ-typecoupling in(2.6) canberewrittenintermsofF₍0_p₊₁₎ and

G

l,r

k,s,then thatis enough to prove the invarianceof (2.6) underthe Abelian gauge transformationwhenFμν

= ˆ

Fμν

=

0.

Theexpressionof

U

l,r

k,sin(2.8)contains

∂

C

0_._In_order_to_convert such terms to a

(

p

+

1

)

-form ﬁeld strength, we need to totally anti-symmetrizetheindiceson

∂

C0asfollows

∂

αCβ01···βp

= (

p

+

1

)∂

[αC 0

β1···βp]

+ ∂

β1C

0

αβ2···βp

+ · · · .

(2.17) Here the ﬁrstterm of(2.17) is acomponent ofthe

(

p

+

1

)

-form ﬁeld strength andso it isinvariant underthe gauge transforma-tion (1.1). The reaming terms are not gauge invariant, therefore, thereshouldbeacompletecancellation ofsuch termsinorderto guaranteethegaugeinvariance.Thisiswhatwearegoingtoshow next.

Using the anti-symmetrization in(2.17), one can rewrite Al_k,_,r_s

and

B

l_k,_,r_s for

l

=

0 as Al_k,_,r_s

=

1 lb l,r k,s

+

p

+

1 l FA l,r k,s

+

(

l

+

1

)(

s

+

1

)

lr B l+1,r−1 k−1,s+1

−

l

+

1 lr E l+1,r−1 k,s

,

Bl_k,_,r_s

=

r

+

1 ls b l−1,r+1 k+1,s−1

+

p

+

1 k

+

1FB l,r k,s

−

(

l

−

1

)(

r

+

1

)

ls A l−1,r+1 k+1,s−1

−

1 sE l,r k+1,s−1

,

(2.18) where FAl_k,_,r_s

=

1 r

!

s

!

p l

p

−

l k

∂

C(r−1)

∂

_D¯(s)

∂

[CC0_···]

× {

TrS

}

(

DμYA

)

(l−1)YA

(

DνY†B

)

( k)_YC(r−1)_D μYC Y†_D(s)

,

FBl_k,_,r_s

=

1 r

!

s

!

p l

p

−

l k

∂

C(r)

∂

_D¯(s−1)

∂

_{[ ¯}_DC_···]0

× {

TrS

}

(

DμYA

)

(l−1)YA

(

DνY†B

)

( k)_YC(r)_Y†(s−1) D DμY † D

.

(2.19) Since the FA- and FB-terms dependon dC0 butnot on C0, they areinvariantunderthegaugetransformation(1.1).Herewenotice that theexpressions Al_k,_,r_s and Bl_k,_,r_sin(2.18) areobtained fromthe integrationbypartsusingthederivationoperatorin

DμY

A_, there-fore,suchintegrationbypartsdoesnotreducethenumberof

DY

†_. However, Al_k,_,r_s in (2.18)contains theexpression B_kl+₋1₁,_,r_s−₊1₁ with re-ducednumberof DY†,hence,oneshould be carefulinusingthe expression Al_k,_,r_s in(2.18).

In analyzing bl_k,_,r_s withl

=

0 in (2.14), we treat the two cases

k

=

0 and k

=

0, separately. For the case of k

=

0, we use the expression Al_k,_,r_s in(2.18) without Bl_k+₋1₁,_,r_s−₊1₁.Thenthefollowing re-cursionrelationisobtained,

bl₀,q_,_q−l

=

l qG l,q−l 0,q

−

(

q

−

l

)(

p

+

1

)

q FA l,q−l 0,q

−

l

(

p

+

1

)

FB l,q−l 0,q

+

l

+

1 q E l+1,q−l−1 0,q

+

l qE l,q−l 1,q−1

−

l qE l,q−l 0,q

+

(

l

−

1

)(

q

−

l

+

1

)

q A l−1,q−l+1 1,q−1

−

q

−

l

+

1 q b l−1,q−l+1 1,q−1

,

(2.20)

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wherewehaveset

r

=

q

−

l, s

=

q. Forthecase

k

=

0,we cannot usethe expression Al_k,_,r_s in (2.18) due to the term Bl_k+₋1₁,_,r_s−₊1₁ with reduced number of DY†. Instead, plugging B_kl,_,r_s from (2.18) into

(2.14),weobtaintheotherrecursionrelation, bl_k,_,q_q−₋l_k

=

Gl_k,_,q_q−₋l_k

−

(

p

+

1

)(

q

−

k

)

k

+

1 FB l,q−l k,q−k

−

El_k,_,q_q−₋l_k

+

E_kl,₊q−₁l_,_q₋_k₋₁

− (

q

−

l

)

Al_k,_,q_q−₋l_k

+

(

l

−

1

)(

q

−

l

+

1

)

l A l−1,q−l+1 k+1,q−k−1

−

q

−

l

+

1 l b l−1,q−l+1 k+1,q−k−1

,

(2.21) wherewe haveset r

=

q

−

l, s

=

q

−

k. Since the expressions in

(2.20) and(2.21) cannot cover thecase of

l

=

k

=

0,we haveto considerthiscaseseparately.Inthiscasethereappearonly

Y and

Y†originatedfromtheTaylorexpansionof

C

(

Y

,

Y

†

₎

_._When_we_set

r

=

s

=

q, weobtain b0₀,_,q_q

=

1

(

q

!)

2

p 0

∂

_C(q)

∂

_D¯(q)C_ρ0(p)

{

TrS

}

YC(q)Y†_D(q)

=

1

(

q

!)

2

p 0

∂

_C(q−1)

∂

_D_¯(q)

∂

CC_ρ0(p)

× {

TrS

}

YC(q−1)YCY†_D(q)

= (

p

+

1

)

F0₀_,,_qq

+

1 qE 1,q−1 0,q

.

(2.22)

Herewehavedeﬁnedagaugeinvariantquantity, F₀0_,,_qq

=

1

(

q

!)

2

p 0

∂

_C(q−1)

∂

_D¯(q)

∂

_[CC0_ρ(p)_]

× {

TrS

}

YC(q−1)YCY†_D(q)

(2.23) throughtheanti-symmetrizationofthe p-form gaugeﬁeld,

∂

CC0_ρ(p)

= (

p

+

1

)∂

[CC 0

ρ(p)_]

+ ∂ρ

1C

0

Cρ2···

+ · · · .

(2.24)

Now,foraﬁxedvalueof

q and m in

(2.12),therecursion rela-tions(2.20)and(2.21)leadto

m

k=0 bm_k_,−_q₋k,_kq−m+k

=

m

k=0 m

−

k q G m−k,q−m+k k,q−k

+

m

k=0 m

−

k

+

1 q E m−k+1,q−m+k−1 k,q−k

−

m

k=0 m

−

k q E m−k,q−m+k k,q−k

.

(2.25) Weomittedthe dependenceof FA- and FB-termsin(2.25) since theyaregenericallygaugeinvariantundertheAbeliangauge trans-formation. We notice that (2.25) does not involve the A-terms

becausethose terms are nicely canceled out between(2.20) and

(2.21).Therelation(2.25)isstillvalidforthe

m

=

0 casesinceone canexactlyreproducetherelation(2.22)bysetting

m

=

0 in(2.25). Theexpression(2.25)isnotgaugeinvariantduetothepresenceof

E-terms. However, summingover allpossible

m we

ﬁndthat the dependenceof

E-terms

does cancelout.Eventually,weobtainthe followinggaugeinvariantrelationforaﬁxed

q,

p

m=0 m

k=0 bm_k_,−_q₋k,_kq−m+k

=

p

m=0 m

k=0 m

−

k q G m−k,q−m+k k,q−k

,

(2.26)

by omitting FA- and FB-terms. Since the G-terms vanish in the case of Fμν

= ˆ

Fμν

=

0, inserting (2.26) into (2.12), proves the AbeliangaugeinvarianceofourWZ-typecoupling.

We considered the pullback of Cp to the worldvolume of the ABJM theory forour calculational convenience. However, we haven’t consider the interior product of the p-form gauge field with the complex scalar fields Y and Y†_, _which _are _needed _to couple gauge fields with rank higher than p

+

1 to multiple

p-dimensional-branes. InM-theory,weneedsuchinteriorproduct to write the WZ-typecoupling of6-form gauge ﬁeld to multiple M2-branes. The absence of such interior products in our analy-sis in thissection implies that our resultsare applicable onlyto the 3-form gauge ﬁeld WZ-type couplingin (2.4). Obviously, the Abelian gauge invariance of (2.4) follows from (2.26) by setting

p

=

3.

3. Abeliangaugeinvariance:Fμν

=

0 &Fμν

ˆ

=

0 case

Intheprevioussection,we showedthattheWZ-typecoupling

(2.6) withvanishing gauge ﬁeld strengths is invariant underthe Abelian gauge transformation (1.1). Once the non-Abelian gauge ﬁeldstrengthsareturnedon,i.e.

Fμν

=

0 & Fμν

ˆ

=

0,the

G-terms

in (2.26), which are apparently not invariant under the Abelian gauge transformation, are non-vanishing. Therefore, for the con-struction of gauge invariant WZ-type coupling, one has to de-formtheWZ-typecouplingin (2.6)to cancelout thegauge non-invariantpiece,speciﬁcallythe

G-terms

in(2.26).Tothatend,we startbyrewritingthe

G-term

as

Gl_k,_,r_s

= −

i 2H l,r k,s

+

i 2H

ˆ

l,r k,s (3.27) with Hl_k,_,r_s

= (

l

−

1

)

Ul_k,_,r_s

{

TrS

}

×

(

DμYA

)

(l−2)

FμμYA

YA

(

DνY†B

)

(k)YC(r)Y †(s) D

−

kUl_k,_,r_s

{

TrS

}

×

(

DμYA

)

(l−1)YA

(

DνY†B

)

(k−1)

Y†_BFμν

YC(r)Y_D†(s)

,

ˆ

Hl_k,_,r_s

= (

l

−

1

)

Ul_k,_,r_s

{

TrS

}

×

(

DμYA

)

(l−2)

YAF

ˆ

_μ_μ

YA

(

DνY†B

)

(k)YC(r)Y †(s) D

−

kUl_k,_,r_s

{

TrS

}

×

(

DμYA

)

(l−1)YA

(

DνY†B

)

(k−1)

ˆ

F_μ_νY†_B

YC(r)Y_D†(s)

.

(3.28) Inthis section,we alsofollow thenotation forcompactindexing explained in the previous section. Since the ﬁeld strengths, Fμν

and Fμν ,

ˆ

appearinasymmetricway,we onlydealwith Hl_k,_,r_s,for simplicity.Werewrite Hl_k,_,r_s byusingthepropertyof

{

TrS

}

as

Hl_k,_,r_s

=

1 q

k

(

l

−

1

)

J_kl,_,r_s

−

kKl_k,_,r_s

+

s

(

l

−

1

)

M_kl,_,r_s

−

krN_kl,_,r_s

,

(3.29) where J_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

D_μYA

(l−2)YA

D_νY†_B

(k−1)YC(r)Y†_D(s)D_μ

×

Y_B†FμνYA

,

K_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

DμYA

(l−1)

DνY†B

(k−1) YC(r)Y_D†(s)

×

Y_B†FμνYA

,

(5)

M_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

DμYA

(l−2) YA

DνYB†

(k) YC(r)Y†_D(s−1)

×

Y†_DFμμYA

,

N_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

DμYA

(l−1)YA

DνYB†

(k−1) YC(r−1)Y†_D(s)

×

Y†_BFμνYC

.

(3.30)

Wewouldliketonotethat Jl_k,_,r_sand

K

_kl,_,r_scontain

Y

†_{F Y -terms}_with indices( A, B) whichweretheindicesof(DY ,DY†)before inte-grationbyparts.Forthisreason,theindices( A,

B

)arecontracted withtheindicesoftheformﬁelds

C

0_in_the_{representation}_of

_U

l,r

k,s deﬁnedin(2.8).Ontheotherhand,theindices(C, D)inthe ex-pressionof Nl_k,_,r_sand

M

_kl,_,r_sin(3.30)aretheindicesof(Y ,

Y

†₎_in_the Taylorexpansion(2.5).Therefore,thoseindicesarecontractedwith theindicesofpartialderivatives

∂

and

¯∂

in therepresentationof

U_kl,_,r_s.Subsequently werewrite M_kl,_,r_s andNl_k,_,r_s intermsof

Y

†_BF YA

throughanti-symmetrization,

∂

D¯C0A(l−2)_AAB¯(k)_ρ(p−l−k)

= (

p

+

1

)∂

_{[ ¯}_DC0 A(l−2)_AAB¯(k)_ρ(p−l−k)_]

+ (

l

−

2

) ∂

_D_¯C0 A(l−2)_A_AB¯(k)_ρ(p−l−k)

_¯ D↔A

+ · · · ,

(3.31) where

∂

α C...β...0

α↔β

≡ ∂

βC...0α....Tobespeciﬁcthe

anti-symmetriza-tion(3.31)leadsto Ml_k,_,r_s

= (

p

+

1

)

FMl_k,_,r_s

−

(

k

+

1

)(

r

+

1

)

ls K l−1,r+1 k+1,s−1

+

(

k

+

1

)(

r

+

1

)

ls N l−1,r+1 k+1,s−1

+

k Q_kl,_,r_s

+

(

l

−

2

)(

k

+

1

)(

r

+

1

)

ls R l−1,r+1 k+1,s−1

+

k

+

1 s S l,r k+1,s−1

,

(3.32) where FMl_k,r_,_s

=

1 r

!

s

!

p l

p

−

l k

∂

_C(r)

∂

_D¯(s−1)

∂

_{[ ¯}_DC_A0(l−2)_AAB¯(k)_ρ(p−l−k)_]

× {

TrS

}

DμYA

(l−2)YA

×

DνY†B

(k) YC(r)Y_D†(s−1)

Y_D†FμμYA

,

Q_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

D_μYA

(l−2)YA

×

DνY†B

(k−1) YC(r)Y_D†(s−1)DμYD†

Y†_BFμνYA

,

R_kl−₊1₁,_,r_s+₋1₁

=

U_kl−₊1₁,_,r_s+₋1₁

{

TrS

}

DμYA

(l−3)YA

×

DνYB†

(k) YC(r)D_μYCY_D†(s−1)

Y†_BFμνYA

,

S_kl,₊r₁_,_s₋₁

= ∂μ

U_kl,₊r₁_,_s₋₁

{

TrS

}

DμYA

(l−2) YA

×

DνYB†

(k) YC(r)Y†_D(s−1)

Y†_BFμνYA

.

(3.33) Manifestly,theﬁrsttermintheright-handsideof(3.32) is invari-antunderthegaugetransformation(1.1).Nowinserting(3.32)into

(3.29) and integratingthe K -term by parts with the help ofthe derivationoperatorin

DY ,

werewrite Hl_k,_,r_sas

H_kl,_,r_s

=

1 q

s

(

l

−

1

)(

p

+

1

)

FMl_k,_,r_s

−

k

(

l

−

1

)(

l

−

2

)

T_kl,_,r_s

−

k

(

l

−

1

)(

k

−

1

)

V_kl,_,r_s

−

klK_kl,_,r_s

−

(

l

−

1

)(

k

+

1

)(

r

+

1

)

l K l−1,r+1 k+1,s−1

−

krNl_k,_,r_s

+

(

l

−

1

)(

k

+

1

)(

r

+

1

)

l N l−1,r+1 k+1,s−1

−

rk

(

l

−

1

)

Rl_k,_,r_s

+

(

l

−

1

)(

l

−

2

)(

k

+

1

)(

r

+

1

)

l R l−1,r+1 k+1,s−1

−

k

(

l

−

1

)

S_kl,_,r_s

+ (

l

−

1

)(

k

+

1

)

Sl_k,₊r₁_,_s₋₁

,

(3.34) where T_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

×

DμYA

(l−3)DμDμYA YA

DνYB†

(k−1) YC(r)Y†_D(s)

×

Y†_BFμνYA

,

V_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

×

D_μYA

(l−2)YA

D_νY†_B

(k−2)D_μD_νY_B†YC(r)YD†(s)

×

Y†_BFμνYA

.

(3.35)

Using the relation (3.34) and following the procedure to the resultin(2.25),weobtain p

m=0 m

k=0 m

−

k q H m−k,q−m+k k,q−k

= (

p

+

1

)

p

m=2 m

−2 k=0

(

q

−

k

)(

m

−

k

)(

m

−

k

−

1

)

q2 FM m−k,q−m+k k,q−k

−

p

m=4 m

−3 k=1 k

(

m

−

k

)(

m

−

k

−

1

)(

m

−

k

−

2

)

q2 T m−k,q−m+k k,q−k

−

p

m=4 m

−2 k=2 k

(

k

−

1

)(

m

−

k

)(

m

−

k

−

1

)

q2 V m−k,q−m+k k,q−k

−

p

m=2 m

−1 k=1 k

(

m

−

k

)

q K m−k,q−m+k k,q−k

.

(3.36) Itturnsoutthatthe

N-, R-,

and

S-terms

in(3.34)disappearwhen the summation is taken over all possible k and m in (3.36). For the p

=

3 case of our consideration in this paper, the T - and V -terms in(3.36)donotappearsincethenumberofworldvolume indices cannot exceedthree.Taking intoaccount thisobservation andplugging(3.27)and(3.36)into(2.26),weobtain

3

m=0 m

k=0 bm_k_,−_q₋k,_kq−m+k

=

i 2 3

m=2 m

−1 k=1 k

(

m

−

k

)

q

K_km_,−_q₋k_k,q−m+k

− ˆ

Km_k_,_q−₋k,_kq−m+k

+ (

gauge invariant terms

),

(3.37) where K

ˆ

l_k,_,r_sisdeﬁnedas

ˆ

K_kl,_,r_s

=

U_kl,_,r_s

{

TrS

}

×

(

DμYA

)

(l−1)

(

DνYB†

)

( k−1)_YC(r)_Y†(s) D

YAF

ˆ

μνY†B

.

(3.38) Since the K -terms in (3.37) are not invariant under the Abelian gaugetransformation,theWZ-typecoupling(2.2)isalsonotgauge invariant.Therefore,inordertomaketheWZ-typecouplinggauge

(6)

invariant,wehaveto subtractthe K -terms in(3.37)fromthe ac-tionin(2.2).

Theformsof K_kl,_,r_s and K

ˆ

_kl,_,r_s are obtainedaftercarryingoutthe pullback(2.3) andthe Taylorexpansion (2.5).We want toﬁnd a compact expression of these terms before the pullback and the Taylorexpansion.Todothat,werewritethe

K

_kl,_,r_sas

K_kl,_,r_s

=

1 r

!

s

!

p l

p

−

l k

∂

C(r)

∂

_D¯(s)C0_A(l−1)_B¯(k−1)₍_AB¯)ρ(p−l−k)

× {

TrS

}

(

DμYA

)

(l−1)

(

DνY†B

)

(k−1)YC(r)Y †(s) D

×

Y†_BFμνYA

,

(3.39)

wherewehavereplaced

m

−

k by l in (3.37).Thenusingthe rela-tion kl

p l

p

−

l k

=

2

p 2

p

−

2 l

−

1

(

p

−

2

)

− (

l

−

1

)

k

−

1

,

(3.40) werewritethegaugenon-invariantquantityin(3.37)as

∞

q=1 3

m=2 m

−1 k=1 k

(

m

−

k

)λ

2(q+1) q K m−k,q−m+k k,q−k

=

2 ∞

q=0 1

m=0 m

k=0

λ

2(q+1) q

+

1 1

(

q

−

m

+

k

)

!(

q

−

k

)

!

×

p 2

p

−

2 m

−

k

(

p

−

2

)

−

m

+

k k

× ∂

C(q−m+k)

∂

_D¯(q−k)C0 A(m−k)_B¯(k)₍_A_B¯₎_ρ(p−2−m)

× {

TrS

}

(

DμYA

)

(m−k)

(

DνYB†

)

( k)_YC(q−m+k)_Y†(q−k) D

×

Y_B†FμνYA

.

(3.41)

Theexpression(3.41)involvesthescalarﬁelds

Y and Y

† _originated fromtheTaylorexpansion(YC,

Y

†_D)andthepullback(YA,

Y

†_B)of

formﬁelds.Since theworldvolume ﬁeld strengths, Fμν and Fμν ,

ˆ

appearvia the integrationby parts ofcovariant derivatives, they canonly couplewith thescalar ﬁeldsfrom thepullback ofform ﬁelds.Keepinginmindthisobservation,wehavetherelation,

1 q

+

1

{

TrS

}

DμYA

(m−k)

DνY†B

(k) YC(q−m+k)Y†_D(q−k)

×

Y_B†FμνYA

= {

TrS

}

D_μYA

(m−k)

D_νY†_B

(k)YC(q−m+k)Y†_D(q−k)

×

Y_B†YA

Fμν

.

(3.42)

Substitutionof(3.42)into(3.41)gives

∞

q=1 3

m=2 m

−1 k=1 k

(

m

−

k

)λ

2(q+1) q K m−k,q−m+k k,q−k

=

2 1

m=0 m

k=0

λ

l+k+2

p 2

p

−

2 m

−

k

p

−

2

−

m

+

k k

× {

TrS

}

iYiY†C_A(m−k)_B¯(k)_ρ(p−m)

(

DμYA

)

(m−k)

(

DνY†B

)

( k)_F μν

=

2

λ

2

p 2

{

TrS

}

P

[

iYiY†C_ρ(p)

]

F_μ_ν

=

2

{

TrS

}

P

[λ

2iYiY†C(p)

]

(p−2)

∧

F

,

(3.43)

whereweintroduceaninteriorproductfora

p-form

ﬁeld

(p),

iYiY†

(p)

=

iY

(_{··· ¯}p_B)Y † B

=

(p) ···AB¯Y A_Y† B

= −

iY†iY

(p)

.

(3.44) From(2.12),(3.37),and(3.43),wereadthecountertermto can-celoutthegaugedependentpieceinacompactformwith

p

=

3,

Sc.t.

= −μ

2

3

{

TrS

}[

P

[

i

λ

2

(

iYiY†

)

C(3)

] ∧ (

F

− ˆ

F

)

].

(3.45)

Hereexplicitexpressionsincluding

{

TrS

}

in(3.45)aregivenby

{

TrS

}[

C_μAB¯YAY † BFνρ] =CμAB¯_aabdˆ_bc_ˆ YA aaˆY†B ˆ b bFμνcd

,

{

TrS

}[

CμAB¯YAY † BF

ˆ

νρ] =CμAB¯abˆ ˆ d abˆcˆY A a ˆ aY † B ˆ b bF

ˆ

μνˆcdˆ

.

(3.46)

Formoredetailsof _Cμ_A_B_¯abdˆ

abcˆ andCμAB¯

ˆ

abdˆ

abˆcˆ,see(2.7) in[9].Similar formofcountertermwith(3.45)wasalsoobtainedin[7],inwhich theformfieldsarenotfunctionals ofscalarfields.Additionofthe counter term (3.45) to the action (2.2) finally defines the gauge invariantWZ-typecouplingforthe3-formgaugefield,

S3

=

μ

2 2

3

{

TrS

}

P

[

C(3)

] + (

c.c.

)

−

μ

2 2

3

{

TrS

}

P

[

i

λ

2

(

iYiY†

)

C(3)

] ∧ (

F

− ˆ

F

)

+ (

c.c.

)

.

(3.47) 4. Conclusion

Thispaperisacomplementoftheprogramstartedin[9].The objectiveoftheprogramistoconstructtheWZ-typecouplings de-scribingthedynamicsofmultipleM2-branesinnon-trivial3- and 6-formﬁeldsin11-dimensionalsupergravity.

In [9] we constructed the WZ-type couplings preserving the UL

(

N

)

×

UR

(

N

)

non-Abelian gauge symmetry of the ABJM the-ory.This was achieved by appropriately choosing the scalarfield dependence of the form fields and selecting single traces from all possible contractions of non-Abelian gauge indices. After cir-clecompactification,theserestrictionssuccessfullyreproducedthe Myerscouplingswithsymmetrized-traceintypeIIAstringtheory. The WZ-type couplings should preserve not only the non-Abeliangaugesymmetriesoftheworldvolumetheorybutalsothe Abeliangaugesymmetriesofthebulk11-dimensionalsupergravity. Theactionshouldbeinvariant under

the Abelian gauge

transforma-tions(1.1)ofthe3- andthe6-formgaugeﬁelds.Theveriﬁcationof thisinvarianceiswhatwasmissingin[9].

Inthispaper,we concentrateontheWZ-typecouplingforthe 3-formgauge ﬁeld andshowedthat theWZ-typecoupling in[9]

is invariant under the Abelian gauge transformation only when the non-Abeliangauge fieldstrengths vanish.In thecaseof non-vanishingnon-Abelianfieldstrengths,weidentifiedamodification bythetermsinvolvingthosefieldstrengths,inordertomakethe WZ-typecoupling invariant under theAbelian gauge transforma-tion. We also found that the constructed gauge invariant 3-form couplingis expressedin acompact form(3.47). Extensionof our studyinthispapertothecasesofthe6-formgaugefieldandthe non-linearformfieldswouldbeinteresting.

(7)

Acknowledgements

This work was supported by the Korea Research Foundation Grant funded by the Korean Government with grant numbers NRF-2014R1A1A2057066 (Y.K.), NRF-2014R1A1A2059761 (O.K.) and the Mid-career Researcher Program through the NRF grant funded by the Korean Government (MEST) (No. 2014-051185) (O.K.),thegrantMIUR2010YJ2NYW_001(D.D.T).

References

[1]M.Li,BoundarystatesofD-branesandDy-strings,Nucl.Phys.B460(1996) 351,arXiv:hep-th/9510161.

[2]M.R.Douglas,Braneswithinbranes,arXiv:hep-th/9512077;

M.B.Green,J.A.Harvey,G.W.Moore,I-braneinﬂowandanomalouscouplings onD-branes,Class.QuantumGravity14(1997)47,arXiv:hep-th/9605033.

[3]R.C.Myers,Dielectric-branes,J.HighEnergyPhys.9912(1999)022, arXiv:hep-th/9910053.

[4]M. Li,T.Wang,M2-branescoupledto antisymmetricﬂuxes,J. HighEnergy Phys.0807(2008)093,arXiv:0805.3427[hep-th].

[5]M.A.Ganjali,Ondielectricmembranes,J.HighEnergyPhys.0905(2009)047, arXiv:0901.2642[hep-th].

[6]Y.Kim,O.K.Kwon,H.Nakajima,D.D.Tolla,CouplingbetweenM2-branesand form ﬁelds, J. High Energy Phys. 0910 (2009) 022, arXiv:0905.4840 [hep-th].

[7]N. Lambert,P.Richmond, M2-branesand backgroundﬁelds,J. HighEnergy Phys.0910(2009)084,arXiv:0908.2896[hep-th].

[8]S.Sasaki,Onnon-linearactionforgaugedM2-brane,J.HighEnergyPhys.1002 (2010)039,arXiv:0912.0903[hep-th].

[9]Y.Kim, O-K.Kwon,H. Nakajima,D.D. Tolla,Interactionbetween M2-branes andbulkformﬁelds,J.HighEnergyPhys.1011(2010)069,arXiv:1009.5209 [hep-th].

[10]J.P.Allen,D.J.Smith,CouplingM2-branestobackgroundﬁelds,J.HighEnergy Phys.1108(2011)078,arXiv:1104.5397[hep-th].

[11]J. Bagger, N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes,Phys.Rev.D77(2008)065008,arXiv:0711.0955[hep-th]; A.Gustavsson,AlgebraicstructuresonparallelM2-branes,Nucl.Phys.B811 (2009)66,arXiv:0709.1260[hep-th].

[12]O. Aharony,O. Bergman,D.L. Jafferis, J. Maldacena, N=6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals, J. High EnergyPhys.0810(2008)091,arXiv:0806.1218[hep-th].

[13]E. Bergshoeff, E. Sezgin, P.K. Townsend, Supermembranes and eleven-dimensionalsupergravity,Phys.Lett.B189(1987)75.

[14]K.Hosomichi,K.M.Lee,S.Lee,S.Lee,J.Park,N=5,6 superconformalChern– SimonstheoriesandM2-branesonorbifolds,J.HighEnergyPhys.0809(2008) 002,arXiv:0806.4977[hep-th].

[15]J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk, H. Verlinde, A mas-sive study of M2-brane proposals, J. High Energy Phys. 0809 (2008) 113, arXiv:0807.1074[hep-th].

[16]Y.Kim,O-K.Kwon,D.D.Tolla,Mass-deformedsuperYang–Millstheoriesfrom M2-braneswithﬂux,J.HighEnergyPhys.1109(2011)077,arXiv:1106.3866 [hep-th].

[17]Y.Kim,O-K.Kwon,D.D.Tolla,PartiallysupersymmetricABJMtheorywithﬂux, J.HighEnergyPhys.1211(2012)169,arXiv:1209.5817[hep-th].

[18]C. Ciocarlie, On the gauge invariance of the Chern–Simons action for N D-branes,J.HighEnergyPhys.0107(2001)028,arXiv:hep-th/0105253.

[19]J.Adam,J.Gheerardyn,B.Janssen,Y.Lozano,Thegaugeinvarianceofthe non-AbelianChern–SimonsactionforD-branesrevisited,Phys.Lett.B589(2004) 59,arXiv:hep-th/0312264.

[20]J.Adam,I.A.Illan,B.Janssen,Onthegaugeinvarianceandcoordinate transfor-mationsofnon-AbelianD-braneactions,J.HighEnergyPhys.0510(2005)022, arXiv:hep-th/0507198.