Multiple M5-branes and ABJM Theory

Multiple M5-branes and ABJM Theory

Seiji Terashima (YITP, Kyoto)

based on the works (JHEP 0912 (2009) 059 and arXiv:1012:3961, to appear in JHEP)

with Futoshi Yagi (IHES)

2011 February 18 at NTU

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1. Introduction

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Recent exciting progress in string theory:

Low energy actions of multiple Membranes in M-theory

was found !

Why this is so exciting?

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For string theory, perturbation theory is well understood and

we can compute, for example, scattering amplitudes of gravitons

But, for M-theory, we do NOT have well defined perturbative description.

(because quantization of membrane have serious problems, for example,

no coupling constant and presence of continuous spectrum.)

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For non-perturbative aspects of string theory, D-branes have been very important objects to understand

Because D-brane is described by perturbative open strings

although they are non-perturbative objects

→ Yang-Mills action as multiple D-brane action!AdS/CFT, Matrix Model, MQCD, etc

Why D-branes are so useful?

On the other hand, until very recently, multiple M2-brane action had not been obtained.

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We will understand many aspects of M-theory (and string theory) !

Recently, Bagger and Lambert (BLG) proposed

multiple membrane actions,then

Aharony, Bergman, Jafferis and Maldacena (ABJM)found different multiple membrane actions.

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Many possible applications,

ex. AdS4/CFT3 (3+1)d gravity theory ↔ (2+1)d field theory

relevant to condensed matter physics,because the membrane action are Chern-Simons theories

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M5-branes are more mysterious and interesting

For example,

•On M5-branes, there is self dual 3-form field strength.

•M5-branes on torus give N=4 SYM and S-duality should be manifest.

•Seiberg-Witten curve is obtained for M5 on curveThus, it is very interesting to find

low energy action of multiple M5-branes

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Single M5 → effective action is known (ex. Pasti-Sorokin-Tonin)

From BLG action, single M5-action was obtained by Ho-Matsuo-Imamura-Shiba

N M5-branes → effective action is NOT known. N³ degree of freedom

We will consider effective action for multiple M5-branes

via ABJM action

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Bound states of M2-branes and M5-branes should be constructed in the M2-brane actions.

(M-theory lift of D2-D4 bound state in IIA)

We will have M5-brane action by consideringfluctuations around the backgroundrepresenting M2-M5 bound states!

Is ABJM action useful to understand M5-branes?

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Indeed, we found solutions of the BPS equations of ABJM which describe the M5-branes

M2-branes

M5-branes

Fuzzy 3-sphere appears

ST, GRRV

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This is an M-theory lift of D2-D4described by t’Hooft-Polyakov Monopole or Nahm equation

D2-branes

D4-branes


M2-branes

M5-branes


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How about the M-theory lift of usual D2-D4 bound state?

This bound state is described by D4-brane with magnetic flux or noncommutative R²

which would be easier to be analyzed.

D4-branes with nonzero magnetic field F

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D4-branes with nonzero magnetic field F

We construct such M2-M5 bound state in ABJM action!

The bound state is M5-branes with nonzero 3-form flux

Yagi-ST

D4-brane with magnetic flux or noncommutative R² = D2-D4 bound state

M-theory lift of this?

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N M2-branes (N →∞)ABJM model

N D2-branes (N →∞)3 dim SYM

M5-brane(with non-zero 3-form flux)

S1 compactification

S1 compactification

NNijji iXX 1],[ ?

D4-brane (with non-zero flux 1/Θ)∝

Strategy to construct it

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M5-brane(with non-zero 3-form flux)

S1 compactification

S1 compactification

NNijji iXX 1],[


Strategy to construct it

We found a classical solution!

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Lie 3-bracket = self-dual 3-form flux and Nambu bracket is hidden.

→ 3-algebra may describe multiple M5-brane action.

We also calculate fluctuations from M5-brane solution.

D4-brane-like action but the gauge coupling constant depends on the spacetime coordinate obtained.

→ consistent with the properties of M5-brane action.

Interestingly, Our solution is closely related to the Lie 3-algebra,

although this is in ABJM, not BLG.

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2. M2-branes and ABJM action

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M2 wrapping S¹ = fund. string in IIA M2 at a point in S¹ = D2 in IIA M5 wrapping S¹ = D4 in IIA M5 at a point in S¹ = NS5 in IIA

M2-M5 ← D2-D4

Consider M2-branes in M-theory compactified on S¹

M-theory on S¹ = IIA string in 10d (Radius of S¹ ~ string coupling)

Thus, M-theory is the strong coupling limit of IIA string, and

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D2-brane effective action is (2+1)d N=8 Yang-Mills theory

which has

7 scalars = location of D2-brane

16 SUSY and SO(7) global symmetry

Not Conformal (Yang-Mills coupling is not dimensionless)

low energy limit = l_s → 0 with Yang-Mills coupling fixed

(cut-off: 1/l_s , g_YM^2: g_s/l_s )

21We want to find a conformal action for M2-brane

effective action of M2-brane on flat space should have

8 scalars = location of M2-brane

16 SUSY and SO(8) global symmetry

Conformal symmetry (=not Yang-Mills theory)

For (2+1)d Yang-Mills theory,Strong coupling limit = low energy limit

M2-brane action = low energy limit of D2-brane action. Thus, we should solve the strong coupling dynamics. → very difficult.

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Fields in ABJM action:

4 complex scalars (A=1,2,3,4)bi-fundamental rep. of U(N) x U(N)

4 (2+1)d Dirac spinors bi-fundamental rep. of U(N) x U(N)

(2+1)d U(N) x U(N) gauge fields

,

,

,

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ABJM action:

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( (2+1)d N=6 ) SUSY transformation:

Gaiotto-Giobi-Yin, Hosomichi et.el, Bagger-Lambert, ST, Bandres-Lipstein-Schwarz

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ABJM action has

12 SUSY and SU(4)xU(1) global symmetryand

Conformal symmetry

(1)This action with U(N)xU(N) gauge group describes N M2-branes on

(2) ABJM derived this action as a limit of a D-brane configuration

c.f. BLG is SU(2)xSU(2)

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(3) Bagger and Lambert showed that ABJM action also has Lie 3-algebra structure defined by

Structure constant: which satisfy (i) and (ii)

(i) fundamental identities

(ii) NOT total anti-symmetric

However, meaning or importance of the 3-algebrahad been unclear for ABJM action.

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3. ABJM to 3d YM and M2-M5 bound state

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Orbifold to R^7 x S¹

Scaling limit v → ∞, k → ∞, v / k : fixedwhere v is the distance between the M2 and singularity

M2-branes probing M2-branes probing R^7 x S¹ = D2-branes probing R^7

(2+1)d ABJM theory (Chern-Simon)

(2+1)d SuperYM theory

θ= 2 π / k2 π v / k

Mukhi et.al.ABJM

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Bosonic part of ABJM

Consider and take a linear combination

where and

is the 3-bracket

then,

This v.e.v gives mass to gauge field

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is massive and can be integrated out. Then we have

3D YM from CS theory through Higgsing!

M2 → D2 in the limit

From the known D4-D2 bound state solution,we want to find a M-theory lift of this solution

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Potential of the ABJM action

Ansatz

(i.e. forget gauge fields and only consider Hermite and constant part of Y¹ and Y²)

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e.o.m.

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

where f → 0 for v →∞

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M5-brane(with non-zero flux)

S1 compactification

S1 compactification

NNijji iXX 1],[


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

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Two equations for one function f(x,y).Are these really consistent?

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Two equations for one function f(x,y).Are these really consistent?

We can show a following identity,which guarantees the existence of the solutions !

Thus, there exist perturbative solutions for these equations.

Anologue for the D2-brane is

This is followed from Jacobi identity.

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This identity is shown fromthe fundamental identity of Lie 3-algebra

and following identities including both 2-bracket and 3-bracket:

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a perturbative solution is

We can show that the solutions have only one parameter,although there seem two parameters.

Another remark: solution is real

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We claim thatthe solution represents

an M5-brane with 3-from flux wrapping following space

although we can not see the S1 direction manifestly.This will been seen by non-perturbative effects,

like monopole operator (vortex) in ABJM.

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

Compactified S1 direction

0,', 4321 YYerYreY ii

Instanton particle in D4 (?)

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Commutator and anti-commutator is simplified in this limit.

Then, the e.o.m. is reduced to 3rd order non-linear PDE

This is still difficult. Nevertheless, we found a solution!:

We can find

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general expression of the solutions with Poisson bracket

take a following ansatz:

The e.o.m. is approximated in the limit as

then the solution is

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Relation to Nambu-Poisson bracket

On the other hand, Nambu-Poisson bracket on the space is

The M5-branes wrap the space with Poisson bracket for the KK reduced space is

This is not consistent with our solution

Thus, we should define

i.e. we can choose the normalization such that

means

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The induced metric on the M5-brane is

The potential is evaluated as

In the star-product representation, Tr is given as then, we have

where we inserted

This indeed corresponds to the M5-brane volume factor,the cofficient is (a part of effective) tension of the M5-brane.

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The M5-brane will have

a constant flux which implies by the non-linear self duality.

This is expected because

non-commutative parameter of D4-brane is constant

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Then, we can show that the metric

with the constant flux is the solution of the single M5-brane action,

which is essentially Nambu-Goto action.

Furthermore, tension of the M5-brane computed from the M5-brane action

match with the one from the ABJM action!

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The potential can be written by the 3-bracket:

Now, substituting our solution we have

From the U(1) gauge transformation, we recover θdependence as

In the real coordinates, we have

This matches with the 3-form flux

Lie 3-algebra and 3-form flux

where

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4. Multiple M5-brane action from ABJM

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We will consider fluctuations around Θ→ 0 solution

First, decompose Y to Hermite and anti-Hermite parts

Since 3-bracket is a combination of commutator and anti-commutator:

Potential is also written by them.

We will expand the potential by the number of commutators.

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The result is

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Now, we assume order of the fluctuations as follows:

This was chosen such thatall fluctuations are same order, thus remain in the Θ → 0.

Then, we findleading order of the potential (assuming only p have v.e.v):

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Parameterization of fluctuations

For A =1,2, let us remember the D2-D4 case.

the solution (from D2 point of view) is

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the fluctuations are conveniently parameterized:

where

This is the covariant derivative operator, satisfies

For our M2-M5 case, the scalars are complex, thus it is natural to define

Then, the fluctuations are introduced by

(classical solution) (classical solution +fluctuations)

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In Poisson bracket approximation (leading order in Θ) ,

where we defined

We can also see that a combination of scalars

disappears in the action (Higgs mechanism). Take unitary gauge.

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Finally, we have action of multiple M5-branes (with flux)

and using “open string metric”

which is not constant

where

and coupling constant is

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5. Conclusion

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• M2-M5 bound state in ABJM action is obtained. • This solution reduces to D4-brane solution [X,X] = iΘ in the scaling

limit.– Corresponding configuration with magnetic flux is a solution of

the e.o.m of M5-brane world volume action.– the correct tension from ABJM action.

• Action of Multiple M5-branes, which are D4-brane action like, is obtained by considering the fluctuation..

• Lie 3-bracket evaluated for the solution becomes self-dual 3-form flux from M5-brane point of view.

• Nambu-Poisson bracket is hidden• The integrability of the e.o.m. with the ansatz is assured by some

non-trivial identities related to the 3-algebra

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• To see S¹ direction which M5-brane is wrapping: Contribution of monopole operators• Relation to 3-algebra, relation to M5 in BLG• Singularity at origin• stability (non-BPS)

• Our result support that recent argument that D4 action = M5 action

Many important things are left!

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Fin.

Documents

Multiple M5-branes and ABJM Theory