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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
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 2 / 23
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.
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 4 / 23
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].
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 6 / 23
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)
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 7 / 23
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)
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 9 / 23
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra
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.
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 10 / 23
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra
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)
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 11 / 23
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra
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)
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 12 / 23
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra
−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)
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 13 / 23
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra
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
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 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 (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
BLG model on 3-Lie bialgebra From Basu-Harvey to Nahm equation via 3-Lie bialgebra
∂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)
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 16 / 23
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)
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 17 / 23
BLG model on 3-Lie bialgebra From M2 to D2
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)
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 18 / 23
BLG model on 3-Lie bialgebra From M2 to D2
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
BLG model on 3-Lie bialgebra From M2 to D2
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.
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 20 / 23
BLG model on 3-Lie bialgebra From M2 to D2
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
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 21 / 23
BLG model on 3-Lie bialgebra From M2 to D2
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
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 22 / 23
BLG model on 3-Lie bialgebra From M2 to D2
Thank You
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 23 / 23
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