From Basu-Harvey to Nahm equation and M2 to D2 via …physics.ipm.ac.ir/conferences/stringtheory/note/M.Aali.pdfFrom Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebra M

From Basu-Harvey to Nahm equation and M2 to D2via 3-Lie bialgebra

M. AaliSupervised by: A. Rezaei-Aghdam

Azarbaijan Shahid Madani University,Tabriz-Iran

May 26, 2016

M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 1 / 23

Outline

1 BLG model

2 3-Lie bialgebra

3 BLG model on 3-Lie bialgebraFrom Basu-Harvey to Nahm equation via 3-Lie bialgebraFrom M2 to D2


BLG model

Bagger-Lambert-Gustavsson model

Basu-Harvey equation

dX i

dx2 =12εijkl [X

j ,X k ,X l ], (1)

Nahm-equation

∂σX i =i2εijk [X j ,X k ]. (2)

A. Basu, J. A. Harvey, Nucl. Phys. B713 (2005) 136- W. Nahm, Phys. Lett. B90 (1980)413.M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 3 / 23

BLG model

Bagger-Lambert-Gustavsson model

Bagger- Lambert- Gostavsson developed (BLG model)1 a SCFT involving Chern-simon theory coupled to Matter2 world volume theory for 2 coincident M2-brane with 16

supersymmetricsupersymmetric transformations

δX Ia = i εΓIΨa,

δΨa = DµX IaΓµΓIε−

12

X IbX J

c X Kd f bcd

aΓIJK ε,

δ(Aµ)ab = i εΓµΓIX I

cΨd f cdab, (3)

J. Bagger, N. Lambert, Phys. Rev. D75 (2007) 045020.J. Bagger and N. Lambert, Phys. Rev. D77 (2008) 065008.J. Bagger, N. Lambert, JHEP 02 (2008) 105.A. Gustavsson, Nucl. Phys. B811 (2009) 66.


BLG model

BLG Model

equations of motion

ΓµDµΨa +12

ΓIJX IcX J

d Ψbf cdba = 0, (4)

D2X Ia −

i2

ΨcΓIJX J

d Ψbf cdba +

12

f bcdaf efg

dX Jb X K

c X IeX J

F X Kg = 0, (5)

(Fµν)ba + εµνλ(X J

c DλX Jd +

i2

ΨcΓλΨd )f cdba = 0, (6)

BLG Lagrangian

L = − 12

DµX A(I)DµX (I)A +

i2ψAΓµDµψA +

i4

fABCDψBΓIJX C(I)X D(J)ψA

− 112

fABCDfEFGDX A(I)X B(J)X C(K )X E(I)X F (J)X G(K )

+12εµνλ[fABCDAµAB∂νAλCD+

23

fAEFGfBCDGAµABAνCDAλEF ] (7)

I = 1, ...,8, µ, ν = 0,1,2.M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 5 / 23

3-Lie bialgebra

Lie bialgebras1 are algebraic structure of N = (2,2) and N = (4,4)may be N = (8,8) supersymmtric WZW models2.

Definitiona of 3-Lie bialgebraaarXiv:1604.04475

3-Lie algebra A with the co commutator map δ : A → A⊗A⊗A1 δ is a 1-cocycle of A with value in ⊗3A, i.e:

δ([Ta,Tb,Tc])=ad (3)Tb⊗Tcδ(Ta)−ad (3)

Ta⊗Tcδ(Tb)+ad (3)Ta⊗Tbδ(Tc), (8)

ad (3)Tb⊗Tc =adTb⊗Tc ⊗ 1⊗1+1⊗ adTb⊗Tc ⊗ 1+1⊗1⊗ adTb⊗Tc , (9)

2 the dual map tδ : ⊗3A∗ → A∗ is a 3-Lie bracket on A∗ with

(T a ⊗ T b ⊗ T c , δ(Td ))=(tδ(T a ⊗ T b ⊗ T c),Td )=([T a, T b, T c],Td ), (10)

1 Y. K. Schwarzbach, Lecture notes in physics 038, Springer-Verlag (2004)107.2 M.Aali-Javanangrouh, A. Rezaei-Aghdam, Arxiv:1402.5600v1].


3-Lie bialgebra

Definition Manin triple

Triple of 3-Lie algebras (D,A,A∗)1 A and A∗ are 3-Lie subalgebras of D2 D = A⊕A∗ as a vector space3 A and A∗ are isotropic i.e. (T a, Tb) = δa

b , (Ta,T b) = (Ta, Tb) = 0.

By using

δ(T a) = fbcdaT b ⊗ T c ⊗ T d (11)

faefg fbcdg − fbef

g facdg + fcefg fabdg − fdef

g fabcg = 0, (12)

f aefg f bcdg − f bef

g f acdg + f cefg f abdg − f def

g f abcg = 0, (13)

fabcg f def

g = fgbcf f deg

a + fgbce f dfg

a − fgbcd f efg

a − fgacf f deg

b

+ fgace f dfg

b − fgacd f efg

b + fgabf f deg

c − fgabe f dfg

c

+ fgabd f efg

c . (14)


3-Lie bialgebra

Proposition

(AG ,A∗G∗) is a 3-Lie bialgebra and the structure constants f abcd and

fabcd satisfy mix fundamental identity if and only if (G,G∗) is Lie

bialgebra.

[TA,TB,TD] = F DABCTD (15)

An Example 3:

[T−,Ta,Tb] = 0,[T+,Ti ,Tj ] = fij kTk ,

[Ti ,Tj ,Tk ] = fijkT−, (16)

[T−, T a, T b] = 0,[T+, T i , T j ] = f ij

k T k ,

[T i , T j , T k ] = f ijkT−, (17)

f+ijk = f ij

k , f ijk− = f ijk , f−ab

c = 0, f abc+ = 0,

f+ijk = fij k , fijk− = fijk , f−ab

c = 0, fabc+ = 0, (18)

−f ijk flmk + f ik

l fkmj − f jk

m flk i − f jkl fkm

i + f ikm flk j = 0. (19)

3P-M Ho, Y. Imamura, Y. Matsuo, JHEP 07 (2008) 003, arXiv:0805.1202M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 8 / 23

BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra

Supersymmetric boundary conditions

Euler-Lagrange equations of a Lagrangian field theory∫M

dmx ∂µ

(δLδ∂µΦ

δΦ

), (20)

If this term goes to zero the action will be invariant. Different properties:

1 non-compact2 With baundary condition.

Boundary break translation symmetry and so the number ofsupersymmetry will be change. Global supersymmetry variation of theaction

δsusyS =

∫dmx ∂µKµ , (21)

A action with maximal supersymmetry at the boundary

Kn = 0. (22)



Supersymmetric boundary conditions

component of the supercurrent normal to the boundary

Jn|δM =δLδ∂nΦ

δΦ

∣∣∣∣∂M− Kn|∂M . (23)

Therefore boundary conditions for which:

Jn|∂M = 0 (24)

D. Gaiotto, E. Witten,J. Statist. Phys. 135(2009) 789, arXiv:0804.2902.D. S. Berman, M. J.Perry, E. Sezgin, D. C. Thompson,JHEP 1004:025,2010.



From Basu-Harvey to Nahm equation

supercurrent for BLG model on Manin triple

Jµ = −εDνX IAΓνΓIΓµΨA − 1

6εX I

AX JBX K

C F ABCDΓIJK ΓµΨD . (25)

0 =

(−εDνX I

AΓνΓIΓ2ΨA − 16εX I

AX JBX K

C F ABCDΓIJK Γ2Ψd

)|∂M . (26)

SO(1,10)→ SO(1,2)× SO(8) (27)SO(1,10)→ SO(1,1)× SO(4)× SO(4) (28)

X V = {X 3,X 4,X 5,X 6} (29)Y P = {X 7,X 8,X 9,X 10} (30)



0 = −εDνX VA ΓνΓV ΨA

−εDνY PA ΓνΓPΨA

−ε(

D2Y PA Γ2ΓPδDA +

16

Y PA Y Q

B Y RC F ABCDΓPQR

)ΨD

−ε(

D2X VA Γ2ΓV δDA +

16

X VA X U

B X WC F ABCDΓVUW

)ΨD

−ε(

12

X VA X U

B Y PC F ABCDΓVUP

)ΨD

−ε(

12

X VA Y P

B Y QC F ABCDΓAPQ

)ΨD (31)



−εDνX VA ΓνΓV ΨA = 0

−εDνY PA ΓνΓPΨA = 0

−ε(

D2Y PA Γ2ΓPδDA +

16

Y PA Y Q

B Y RC F ABCDΓPQR

)ΨD = 0

−ε(

D2X VA Γ2ΓV δDA +

16

X VA X U

B X WC F ABCDΓVUW

)ΨD = 0

−ε(

12

X VA X U

B Y PC F ABCDΓVUP

)ΨD = 0

−ε(

12

X VA Y P

B Y QC F ABCDΓAPQ

)ΨD = 0 (32)



From Basu-Harvey to Nahm equation

Dirichlet conditionsDµY P = 0 (33)

and the simplest solutionY P = 0 (34)

0 = εD2Y PΓ2ΓPΨ , (35)0 = εDνX V ΓνΓV Ψ , (36)

0 = ε

(D2X V

A δADΓ2ΓV +

16

F ABCDX VA X U

B X WC ΓVUW

)ΨD . (37)

ΓV =16εVUWZ ΓUWZ Γ3456 (38)

0 = D2X VA +

16εVUWZ X U

B X WC X Z

D F BCDA, (39)

Basu-Harvey type equations by considering scalar fields X V .M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 14 / 23


Proposition

(AG ,A∗G∗) is a 3-Lie bialgebra and the structure constants f abcd and

fabcd satisfy mix fundamental identity if and only if (G,G∗) is Lie

bialgebra.

[TA,TB,TD] = F DABCTD (40)

An Example 4:

[T−,Ta,Tb] = 0,[T+,Ti ,Tj ] = fij kTk ,

[Ti ,Tj ,Tk ] = fijkT−, (41)

[T−, T a, T b] = 0,[T+, T i , T j ] = f ij

k T k ,

[T i , T j , T k ] = f ijkT−, (42)

f+ijk = f ij

k , f ijk− = f ijk , f−ab

c = 0, f abc+ = 0,

f+ijk = fij k , fijk− = fijk , f−ab

c = 0, fabc+ = 0, (43)

−f ijk flmk + f ik

l fkmj − f jk

m flk i − f jkl fkm

i + f ikm flk j = 0. (44)

4P-M Ho, Y. Imamura, Y. Matsuo, JHEP 07 (2008) 003, arXiv:0805.1202M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 15 / 23


∂σX IA =

12εIJK X J

BX KC F BC

A

∂σX Ii =

12εIJK X J

j X Kk f jk

i +12εIJK X JjX K

k f ki j . (45)

BPS bound

E ≥ 16εJIKLTr(∂sX (I), [X (J),X (K ),X (L)])

≥ 16εJIKLTr(∂sX (I)

+ T+ + ∂sX (I)− T− + ∂sX (I)

i T i + ∂sX (I)+

T + + ∂sX (I)− T −

+∂sX (I)i

T i ,X (J)+ X (K )

i X (L)j [T+,T i ,T j ] + X (J)

+ X (K )i X (L)

j[T+,T i ,T j ]

+X (J)+

X (K )

iX (L)

j[T +,T i ,T j ] + X (J)

+X (K )

iX (L)

j [T +,T i ,T j ])

≥ 12g2

YM

∫dσεIJK∂sX I [X J ,X K ]. (46)


BLG model on 3-Lie bialgebra From M2 to D2

M2 to D2 and vice versaequations of motion of BLG mode

ΓµDµΨA +12

ΓIJX ICX J

DΨBF CDBA = 0,

D2X IA −

i2

ΨCΓIJX J

DΨBF CDBA +

12

F BCDAF EFG

DX JBX K

C X IEX J

F X KG = 0,

∂2X I+ = 0, (47)

Γµ∂µΨ+ = 0, (48)∂2X I

+ = 0, (49)Γµ∂µΨ+ = 0, (50)

L = − 12

DµX A(I)DµX (I)A +

i2ψAΓµDµψA +

i4

FABCDψBΓIJX C(I)X D(J)ψA

− 112

FABCDFEFGDX A(I)X B(J)X C(K )X E(I)X F (J)X G(K )

+12εµνλ[FABCDAµAB∂νAλCD+

23

FAEFGFBCDGAµABAνCDAλEF ] (51)



εµνλF ABCDAµAB∂νAλCD = εµνλ fij kAijµ(∂νA+

λk − ∂λA+νk ) + εµνλf ij

kAµij(∂νAk+λ

− ∂λAk+ν )− 2εµνλ fik jAi

µj(∂νAk+λ − ∂λAk+

ν )

− 2εµνλf jkiAiµj(∂νA+

λk − ∂λA+νk ), (52)

and

εµνλ F AEFG F BCDG AµABAνCDAλEF=ε

µνλ fij kAijλ[ A+

µl ,A+νm]

+εµνλ f ijkAλij [ A+l

µ ,A+mν ] + εµνλ fik jAi

λj [Al+µ ,A

m+ν ] + εµνλ f ik

jAjλi [A

+µl ,A

+νm]

(53)

A+µl = Aµl ,A+l

µ = Alµ, (54)

A+µl = A

′

µl ,A+lµ = A

′lµ, (55)

f ijkAµij ≡ Cµk , fij kAij

µ ≡ Ckµ (56)



General form of the BLG Lagrangian on the especial 3-Lie bialgebra

L=−2g2YMCa

µCµa−2gYMCµaDµX (8)a+2 fµνλ CµaF a

νλ + 2 fµνλ CaµBνλa + ....

(57)Integration of Cµk and Ck

µ :

L =12

FνλkF νλk +14

BνλkBνλk + ..., (58)

that

Bνλk = ∂νA′

λk − ∂λA′

νk − [A′

νk ,A′

λk ], (59)F kνλ = ∂νAk

λ − ∂λAkν − [Ak

ν ,Akλ]. (60)

is a result of expanding Dirac Born Infeld action5.

5D. Tong, February 2012.M. Aali Supervised by: A. Rezaei-Aghdam (Azarbaijan Shahid Madani University,Tabriz-Iran)From Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebraMay 26, 2016 19 / 23


Dirac-Born-Infeld action

S =

∫dp+1ζ

√−det(GµνhAB + 2πα′FµνAB + BµνAB), (61)

1 DBI action gives equation of motion proportional to beta-functiongained from sigma-model and conformal gauge action.

2 Supersymmetric WZW model is a result of defining sigma modelon Lie group and its algebraic structure is Lie bialgrbra.



Relation between D-brane and WZW model

SWZW =

∫d3xεαβγLµILνJLK

λ ∂αXµ∂βX ν∂γXλTr([TI ,TJ ],TK ) (62)

Xµ = XµATA (63)

SWZW−like=

∫d3xεαβγLµLLνMLN

λ ∂αX Iµ∂βX Jν∂γX KλTr([TITL,TJTM ],TKTN)

SWZW−like=

∫d2x {1

6εβγBνµQ∂βX Jν∂γX IµTr(TJTITQ)

16εαγBµλQ∂αX Iµ∂γX KλTr(TITKTQ)}+1

6εαγBνλQ∂αX Jν∂γX KλTr(TJTKTQ) + ...

(64)

BνµQ = LνLLλN fNLPxJ fPJ

Q (65)

, LLµX IµTITL|boundry = xLTL|boundry



Using the concept of 3-Lie bialgebra; we construct BLG model on theManin triple D of the especial 3-Lie bialgebra and shown that:Nahm equation can be obtained from Basu-Harvey equation and viceversa. arXiv:1604.05181One can construct M2-brane from a D2-brane and vice versa.arXiv:1604.05183Similar works have been done for BL model with N = 6 supersymmetryand multiple membrane which detail are arXiv:1604.05890



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From Basu-Harvey to Nahm equation and M2 to D2 via …physics.ipm.ac.ir/conferences/stringtheory/note/M.Aali.pdfFrom Basu-Harvey to Nahm equation and M2 to D2 via 3-Lie bialgebra M