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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. 1. Introduction. Recent exciting progress in string theory:. Low energy actions of

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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



Recent exciting progress in string theory
Recent exciting progress in string theory:

Low energy actions of

multiple Membranes in M-theory

was found !

Why this is so exciting?


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.)


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

Why D-branes are so useful?

Because

D-brane is described by perturbativeopen strings

although they are non-perturbative objects

→ Yang-Mills action as multiple D-brane action!

AdS/CFT, Matrix Model, MQCD, etc

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


Recently,

Bagger and Lambert (BLG) proposed

multiple membrane actions,

then

Aharony, Bergman, Jafferis and Maldacena (ABJM)

found different multiple membrane actions.

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


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


M5 branes are more mysterious and interesting
M5-branesare 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 curve

Thus, it is very interesting to find

low energy action of multiple M5-branes


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


Is ABJM action useful to understand M5-branes?

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 considering

fluctuations around the background

representing M2-M5 bound states!


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

ST, GRRV

M2-branes

M5-branes

Fuzzy 3-sphere appears


M2-branes which describe the M5-branes

M5-branes

Fuzzy 3-sphere appears

This is an M-theory lift of D2-D4

described by t’Hooft-Polyakov Monopole or Nahm equation

D2-branes

D4-branes

Fuzzy 2-sphere appears


How about the which describe the M5-branesM-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


D4-brane with magnetic flux or noncommutative R² which describe the M5-branes

= D2-D4 bound state

D4-branes with nonzero magnetic field F

M-theory lift of this?

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

Yagi-ST

The bound state is

M5-branes with nonzero 3-form flux


Strategy to construct it which describe the M5-branes

N D2-branes (N →∞)

3 dim SYM

N M2-branes (N →∞)

ABJM model

S1 compactification

?

M5-brane

(with non-zero 3-form flux)

D4-brane

(with non-zero flux ∝ 1/Θ)

S1 compactification


Strategy to construct it which describe the M5-branes

N D2-branes (N →∞)

3 dim SYM

N M2-branes (N →∞)

ABJM model

S1 compactification

We found

a classical solution!

M5-brane

(with non-zero 3-form flux)

D4-brane

(with non-zero flux ∝ 1/Θ)

S1 compactification


Interestingly, which describe the M5-branes

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

although this is in ABJM, not BLG.

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.


2. M2-branes and ABJM action which describe the M5-branes


Consider M2-branes in M-theory compactified on S which describe the M5-branes¹

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

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


D2-brane which describe the M5-branes 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 )


effective action of M2-brane which describe the M5-branes 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.

We want to find a conformal action for M2-brane


Fields in abjm action
Fields in ABJM action: which describe the M5-branes

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

,

,

,


ABJM action: which describe the M5-branes


( (2+1)d N=6 ) SUSY transformation: which describe the M5-branes

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


ABJM action has which describe the M5-branes

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

and

Conformal symmetry

  • 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)


(3) Bagger and Lambert showed that which describe the M5-branes

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-algebra

had been unclear for ABJM action.


3. ABJM to 3d YM which describe the M5-branes

and M2-M5 bound state


Orbifold to R^7 x S which describe the M5-branes¹

M2-branes probing R^7 x S¹

= D2-branes probing R^7

M2-branes probing

(2+1)d ABJM theory

(Chern-Simon)

(2+1)d SuperYM theory

2 π v / k

θ= 2 π / k

Mukhi et.al.

ABJM

Scaling limit

v → ∞, k → ∞, v / k : fixed

where v is the distance between the M2 and singularity


Bosonic part of ABJM which describe the M5-branes

where and

is the 3-bracket

Consider and take a linear combination

then,

This v.e.v gives mass to gauge field


is massive and can be integrated out. which describe the M5-branes

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


Potential of the ABJM action which describe the M5-branes

Ansatz

(i.e. forget gauge fields and

only consider Hermite and constant part of Y¹ and Y²)


e.o.m. which describe the M5-branes

additional Ansatz

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

where f→ 0 for v →∞


N which describe the M5-branes M2-branes (N →∞)

ABJM model

N D2-branes (N →∞)

3 dim SYM

S1 compactification

M5-brane

(with non-zero flux)

D4-brane

(with non-zero flux ∝ 1/Θ)

S1 compactification


e.o.m. (infinite order nonlinear PDE) which describe the M5-branes


Two equations for one function which describe the M5-branesf(x,y).

Are these really consistent?


Two equations for one function which describe the M5-branesf(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.


This identity is shown from which describe the M5-branes

the fundamental identity of Lie 3-algebra

and following identities

including both 2-bracket and 3-bracket:


a perturbative solution is which describe the M5-branes

We can show that the solutions have only one parameter,

although there seem two parameters.

Another remark: solution is real


We claim that which describe the M5-branes

the solution represents

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

0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10

M2 ○ ○ ○

M5 ○ ○ ○ ○  ○   ○

Compactified S1 direction

although we can not see the S1 direction manifestly.

This will been seen by non-perturbative effects,

like monopole operator (vortex) in ABJM.

Instanton particle in D4 (?)


We can find which describe the M5-branes

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!:


general expression of the solutions with Poisson bracket which describe the M5-branes

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

take a following ansatz:

then the solution is


Relation to Nambu-Poisson bracket which describe the M5-branes

The M5-branes wrap the space with

Poisson bracket for the KK reduced space

is

This is not consistent with our solution

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

i.e. we can choose the normalization such that

means

Thus, we should define


The induced metric on the M5-brane is which describe the M5-branes

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.


The M5-brane will have which describe the M5-branes

a constant flux which implies

by the non-linear self duality.

This is expected because

non-commutative parameter of D4-brane is constant


Then, we can show that the metric which describe the M5-branes

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!


Lie 3-algebra and 3-form flux which describe the M5-branes

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

where


4. Multiple M5-brane action which describe the M5-branes

from ABJM


We will consider fluctuations around which describe the M5-branesΘ→ 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.


The result is which describe the M5-branes


Now, we assume order of the fluctuations as follows: which describe the M5-branes

This was chosen such that

all fluctuations are same order, thus remain in the Θ → 0.

Then, we find

leading order of the potential (assuming only p have v.e.v):


Parameterization of fluctuations which describe the M5-branes

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

the solution (from D2 point of view) is


the fluctuations are conveniently parameterized: which describe the M5-branes

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 +fluctuations)

(classical solution)


In Poisson bracket approximation (leading order in which describe the M5-branesΘ) ,

where we defined

We can also see that a combination of scalars

disappears in the action (Higgs mechanism).

Take unitary gauge.


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

where

and using “open string metric”

and coupling constant is

which is not constant


5. which describe the M5-branesConclusion


  • M2-M5 bound state in ABJM action is obtained. which describe the M5-branes

  • 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


Many important things are left! which describe the M5-branes

  • 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


Fin. which describe the M5-branes


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