Toward M5-branes from ABJM action

Based on going project with Seiji Terashima (YITP, Kyoto U. ）

Futoshi Yagi (YITP, Kyoto U.)

Type IIASuperstring theory

Fundamental string

D2 brane

D4 brane

NS5 brane

compactifyon S1

Wrap on S1

Unwrap on S 1

M Theory

M2 brane

M5 braneWrap on S1

Unwrap on S 1

From M theory to type IIA superstring theory

§1 　 Introduction1/18

N D2-branesWorld Volume Theory

of N D2 branes| |

3 dimensionalU(N) Supersymmetric

Gauge theory

ABJM model !!(Hopeful candidate)

N M2 branes

By Aharony, Bergman, Jaffris, Maldacena

ArXiv: 0806.1218 [hep-th]

World Volume Theoryof N M2 branes

↓

・・・

・・・

S1 compactification

IIA

M

2/18

Theme

Find multiple M5-branes action

3/18

“Toward” M5-branes from ABJM action

N M2-branes (N →∞)

ABJM modelN D2-branes (N →∞)

3 dim SYM

M5-brane(with non-zero flux)

S1 compactification

S1 compactification

Approach to M5-brane

NNijji iXX 1],[We found

a classical solution!!

4/18

D4-brane (with non-zero flux 1/Θ)∝

M2-branes

M5-branes

M5-branes

M2-branes

0 1 2 3 4 5 6 7 8 9 10M2 ○ ○ ○ M5 ○ ○ ○ ○ ○ ○

0 1 2 3 4 5 6 7 8 9 10M2 ○ ○ ○ 　　M5 ○ ○ 　 ○ ○ ○ ○

Our classical solution

A classical solution already studied

Terashima,Gomis, Rodriguez-Gomez, Van Raamsdonk, VerlindeHanaki, Lin Nastase, Papageorgakisb, Ramgoolamc

5/18

*Non-BPS solution

* M5 brane looks like D4 brane.

Plan of this talk

§1 Introduction

§2 Brief review of ABJM model

§3 Classical solution of the ABJM model

§4 Evidence for the claim

§5 Conclusion and discussion

6/18

)(1

1)(

)4,3,2,1(

)4,3,2,1(

)()(

)2(

)1(

AdjA

AdjA

NNA

NNAY

NUNU

A

A

potkinCS SSSS

By Aharony, Bergman, Jaffris, Maldacena

ArXiv: 0806.1218 [hep-th]

§2 　 Brief review of ABJM model7/18

Complex scalars

Dirac Spinors

Gauge fields

)(

)TrTr(

32

1

32

1Tr

2

fermbos3

pot

3kin

)2()2()2()2()2()1()1()1()1()1(3CS

VVxdS

DiYDYDxdS

AAAi

AAAAAi

AAxdk

S

AAA

A

††

††††

††††††††

DCBAABCDDCBA

ABCD

ABAB

BAB

ABAAB

BBA

A

ABCCBAABBBBA

YYYY

YYYYYYYYk

iV

YYYYYYYYYYYYk

V

22(Tr2

~~~~~~2

~)

~~()

~~(

~Tr

3

4

ferm

2

2

2

bos

0

0~†A

A

AY

YY

ABJM theory is proposed to be the world volume theory of N M2-branes probing C4/Zk

k: 　 Chern-Simons level

4

23

22

21

24321 ,,,),,,(: yeyeyeyeyyyyZ k

i

k

i

k

i

k

i

k

8/18

ABJM theory satisfies various property which are expected to the M2-branes probing C4/Zk

v : the distance between the M2 and singularity(v.e.v of a scalar field)

Scaling limitv → ∞, k → ∞, v / k : fixed

ABJM modelU(N) ×U(N)

3dim SYM theoryU(N)

Mukhi et.al. , ABJM, Homma-Iso-Sumitomo-Zhang

v

9/18

k

2

k

v

C4/ZkR7×S1

World volume theory of D2 branes (3dim SYM) is obtained from ABJM model by S1 compactification.

N M2-branes (N →∞)

ABJM modelN D2-branes (N →∞)

3 dim SYM

M5-brane(with non-zero flux)

D4-brane (with non-zero flux 1/Θ)∝

S1 compactificationv → ∞, k → ∞,

v / k : fixed

S1 compactification

We found a classical solution!! v → ∞

§3 　 Classical solution of ABJM model10/18

NNijji iXX 1],[

Bosonic potential of the ABJM action

0

0~~~~~~~2

~)

~~()

~~(

~Tr

3

4 2

2

2

bos †A

A

AABCCBAABBBBAY

YYYYYYYYYYYYYY

kV

Ansatz (the solution becomes D2-D4 in the limit v → ∞)

,0

,0

,ˆ

,0,),/1(ˆ)ˆ,ˆ(

)2()1(

43

2

111

1

AA

YY

yY

YYYvOxvyxY † iyx ˆ,ˆ

)0ˆˆ

ˆˆ,ˆˆ(

yx

yyxx

††

11/18

0)()(

0)(

1

2

12221

121221

21

211

2

12221

y

Y

x

YyY

yx

Y

x

YyY

x

Yy

x

YY

x

YyY

221

1

)( yY

v

x

Y

],)[(,,])(,[,)(0

],)[(,,])(,[,)(0

12212212212

bos

22121221221

bos

YYYYYYYY

V

YYYYYYYY

V

～

～

0

x

f

y

g

y

g

x

fyxgyxf )ˆ,ˆ(),ˆ,ˆ(

e.o.m.

,,, 22

11

YY

VY

Y

VDue to the special relation a perturbative solution exists.

Solution in the limit Θ→0

12/18

We found a “M5-brane solution”, whose configuraiton is

Claim

SubtletyWe cannot see the S1 direction manifestly.

(Similar situation for M2 M5 bound state)⊥

0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10M2 ○ ○ ○ M5 ○ ○ ○ ○ 　○ 　　○

Compactified S1 direction

0,', 4321 YYerYreY ii

13/18

・ Corresponding configuration of M5-brane with constant flux is a solution of the e.o.m from the single M5-brane action.

・ We find the agreement between the tension of the M5-brane solution in the ABJM action and the one computed from single M5-brane action.

§4 　 Evidence for the claim14/18

0,', 4321 YYerYreY ii

Configuration of M5-brane

with constant flux

)0,1(),1,0(),(0~

,0~

2012 jiFconstF ij

satisfies the equations of motion from the M5-brane action!!

1st Evidence: Satisfying the e.o.m

34)(

34

34)(

345

11

2~

IIA

IIA

FB

FF

～～

～

By dimensional reduction of the M5-brane world volume theory

(4+1)dim Non-commutative SYM (Seiberg - Witten)

15/18

012

012345

~1

~~

F

FF

～ (Non-linear self-duality condition)

0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10M2 ○ ○ ○ M5 ○ ○ ○ ○ 　○ 　　○

22/2

0

2

2

2

2

)0(

~~~~~~2

~)

~~()

~~(

~Tr

1

rrrddrdk

v

dxdyk

v

YYYYYYYYYYYYk

V

k

ABCCBAABBBBA

～

～

～

2nd Evidence: Matching of TensionTension of the M5-brane from ABJM action

0

0~†A

A

AY

YY

yY

YY

v

x

Y

2

22

21

1

)()(

16/18

Volume factorTension

1~

,0~

1 01266

0125

Fxdk

vxdFSM ～～

Tension of the M5-brane world volume action

kvFBFF

F

FF

IIA

IIA /

1,

11,2

~,~

1

~~

34)(

34

34)(

345

012

012345 ～～～～～

Tension

・ We gave a classical solution of the ABJM model, which reduce to D4-brane solution [X1,X2] = iΘ in the scaling limit.

We interpret this solution as a “M5-brane solution” from ABJM model.

・ We gave a several consistency checks that it indeed represents M5-brane.

・ Corresponding configuration with constant magnetic flux is a solution of the e.o.m of M5-brane world volume action.

・ We find the agreement between the tension of the M5-brane solution in the ABJM action and the one computed from M5-brane world volume action.

§5 Conclusion and discussion

Conclusion

17/18

Discussion

・ Multiple M5 branes

・ Fluctuation from the classical solution 　→ World volume theory of M5-branes

・ S1 direction which M5-brane is wrapping

・ Contribution of monopole operators

・ Relation to three algebra

18/18

Thus, there exist perturbative solutions for these equations.

Perturbative solution is

We interpret that this solution represents M5-brane

52242246

44235

3224

223

21

1ˆ

4

3ˆˆ

2

3ˆˆ

3

11ˆ

16

211ˆˆ

8

3ˆˆ

12

23ˆ

8

7

1ˆˆˆ

8

51ˆˆ

2

1ˆ

2

1

2

ˆˆ

vxcyxyxx

vyxyxx

vyxx

vyxx

v

xxvY

* The product in this equation is Weyl ordered product.

mnmn

pnlp

lmnm FigxaFFxa

agxdS

~det)()(

)(4

1 *2

6

M5-brane action

)()(

1~,

6

1,

det,

*

2

* xaFa

FFg

FAAAF

ggXXGg

lmnlmnpqr

lmnpqrlmnijkkijjkiijk

mnnmmn

auxiliary field:)(xa

Gauge symmetry

mn

pqpq

mnmnp

mnpmn

nmmn

nmmn

F

Fig

g

aVVaF

a

xAxxa

xxaAxa

xAxa

~)

~det()(

2,)(2

)(),()(.3

)()(2

1,0)(.2

)(,0)(.1

2

2

][

][

0)3( C for simplicity

Pasti, Sorokin, Tonin (‘97)

Non-linear self duality condition = Equations of motion for a(x)

0)( 2

qrs

qrsp

nlmnpqr VaF

a

a

Crucial problem of our work

We cannot see the extension to the compactified S1 direction!!

0,', 4321 YYerYreY ii

Compactified S1 direction

0,', 4321 YYrYrY

The M5-brane should extend not only to the direction of r and r’ but also to S1 direction

Similar problem inTerashima (‘08)Nastase, Papageorgakis, Ramgoolam (‘09)(another M2-M5 bound state)

Why we cannot see the extension of the M5-brane to the compactified S11 direction?

Explanation 0Because three algebra structure is not manifest in theABJM model .

Structure like

cyx ],ˆ,ˆ[ is needed ?

Explanation 1Because we calculate the solution perturbatively from the D4-brane solution.

Wrapping on S1 is the non-perturbative effect

Ｃｆ Nambu-Poisson bracket (Ho,Matsuo ‘08)

3,2,1,,

)(),(),(

kjix

h

x

g

x

fxhxgxf

kjiijkiii ～

Explanation 2Because we take large N limit.

N →∞ with λ=N/k: fixed k → ∞ Type IIA limit(S1 cannot be seen)

Although we cannot see the S1 direction,we still claim that this classical solution correspond to M5-braneand that some aspects of M5 brane can be seen from this solution.

(Nastase et.al ‘09)

N →∞ with k: fixed λ → ∞ Strong coupling limit(Classical solutionis no more reliable)

Or

e.o.m. (infinite dimensional nonlinear PDE)

Comment on BLG model Bagger, Lambert (‘07)Gustavson (‘07)

・ Candidate of multiple M2-branes proposed before the ABJM model

・ Gauge symmetry is based on three algebra

・ Only one three algebra! 　→ Two M2 branes case Thus, we need ABJM model (N M2 branes are describable )!!(Or we should release the constraint to three algebra)

Not positive definite → Reduce to D2 brane after removing the ghost

Not totally antisymmetric → Turn out to include the ABJM model (Bagger, Lambert)

Infinite dimensional (Nambu-Poisson bracket) → Correspond to M5-brane !! (Ho, Matsuo)

dd

abccba TfTTT ],,[

Three Algebra

dd

abccba TfTTT ,,

ede

abcabcdbaab hffTTTrh ),,(

dabc

dabc ff ][

Metric is positive definite

Anti-symmetry

Fundamental identity

M Theory

Type AⅡSuperstirng

TypeⅠSuperstirng

(S1 compactified)

Compactifyon S1 × S1/Z2

(cylinder)

SO(32)Heterotic

Superstirng(S1 compactified)

Type BⅡSuperstring

(S1 compactified)

E8×E8

HeteroticSuperstirng

Compactifyon S1

Compactifyon S1/Z2

Compactifyon S1×S1 （ T2 ）

Low energy limit(no compactification)

11 dimensionalSupergravity

Compactifyon S1/Z2 × S1

(cylinder)

N Dp branes in 10 dimensional Minkowski Space

・・・

N Dp-branes(p+1 dim. object)

Quantize theoscillation modeof the stringand pick upmassless mode

World Volume Theoryof N Dp branes

||p+1 dimensional

U(N) SupersymmetricGauge theory

Open string ：End points are on the Dp branes

0sl

Non-perturbative aspects of superstring theorycan be captured by studying this SYM theory!!

ABJM model !!(Hopeful candidate)

N M2 branes

What is the low energy effective theory on multiple M2-branes?

・・・

Quantizationof open membrane???

??

??

CommentIf you believe the strongest version ofAdS/CFT correspondence,world volume theory gives M theoryin AdS4×S7 background.

By Aharony, Bergman, Jaffris, Maldacena

ArXiv: 0806.1218 [hep-th]

How about multiple M5-branes ?

But ！！Formulation of M theory has not been

established yet.Quantization of membranes are difficult.

(partially because there are no free parameters.)

It is expected that 5 types of superstring theories can be understood in a unified manner through ``M theory’’

To study “quantum M-theory”is important and challenging problem!!