MultipleParallelStrings and WrappedM2 Branes...

MultipleParallel Stringsfrom ABJM?

1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7

• Bulk vs.boundary

4. Effectiveaction

• (N + 1) →(N) + (1)

• Explicitcomputation

• Large N

5. Discussion

• Summary

Multiple Parallel Strings andWrapped M2 Branes from

the ABJM Model ?

Tamiaki Yoneya

University of Tokyo - Komaba

KEK 2009 Theory Workshop


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Contents

1. Introduction

2. Wrapped M2 branes from ABJM

3. Parallel strings

4. Effective action for parallel strings

5. Discussion


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

1. Introduction

Revisiting the M-theory conjecture

First recall the conjecture.

spacetimedimensions

11

9

Perturbative Theories

Type I HeteroSO(32)

Hetero E8 x E8

Type IIA Type IIB10

M Theory

Circle Circle/Z2

HeteroType ITypeIIAB

S duality

T duality

Cirlce Compactification

Unfortunately, no substantial progress, from the end of the previouscentury, on what the M theory really is.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

I Radius of the circle direction :

R11 = gs`s

M2 brane as gs → 0

I ”longitudinal”: wrapped along the 11-th circle direction

⇒ (fundamental) string

I ”transverse” : extended along directions orthogonal to the 11-thcircle

⇒ D2 brane

I Fundamental length scale of M theory = Planck scale

`P = g1/3s `s R11

and`P `s

as gs → 0.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

⇓

In the weak coupling (∼ 10 dimensional) limit, M2 branes shouldsmoothly reduce to perturbative strings of type IIA theory.

We would like to discuss this question in the context of the ABJMmodel, a candidate low-enegy theory for multiple M2 branes, in thesimplest possible setting.

• So far, almost all previous works have been focused on the”transverse” configurations of M2 branes:

weak coupling limit k ∼ ∞

m

S7, which is transverse to M2 branes, into CP3

S7/Zk → CP3, Zk → S1 ∼ M-theory circle

2π/k = Chern-Simons coupling constant


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• Remark: case of single M2 brane

The dynamics of a single M2 brane is already quite non-trivial, andhence the reduction to string(s) is not completely understood,quantum-mechanically.

Sekino-TY, hep-th/0108176 , Asano-Sekino-TY, hep-th/0308024

I wrapped M2 brane⇓ directly

matrix-string theory

[1 + 1D SYM with coupling 1/gs (N →∞)]

I large N limit with gYM = 1/√

gs →∞ can be studied by usingGKPW relation in the PP (BMN)-wave limit, under theassumption of gauge/gravity correspondence.

I The result of two-point correlators shows that the effectivescaling dimension of scalar fields is

∆eff =2

5− p

∣∣∣p=1

=1

2

This is consistent with the existence of 3D CFT description of M2branes.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Main features of the ABJM model

I Susy Chern-Simons U(N)×U(N) gauge theory in 3D withSO(6)(∼SU(4)) R-symmetry

I (super)Conformal invariant

I CS coupling = 2π/k with level number k⇔ C4/Zk = transverse space of M2 branes

I AdS/CFT correspondence at k = 1:AdS4 × S7 ⇔ effective CFT of N M2 branes in flat 11D

I But, only with N = 6 susy, manifestly.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• Notations(following Bandres-Lipstein-Schwarz, 0807.0880)

I bosonic fields: (XA, X A) (4, 4 of SU(4))

I fermionic fields: (ΨA, ΨA)(4, 4 of SU(4), 3D 2-component spinor)

I Chern-Simons U(N)×U(N) gauge fields: (Aµ, Aµ)

• Action

SABJM =k

2π

∫d3x Tr

[− DµX ADµXA + iΨAγ

µDµΨA]

+SCS +k

2π

∫d3x (L6 + L2,2)

L6 = potential term of O(X 6),

L2,2 = XΨcoupling terms of O(X 2Ψ2)

SCS =k

4π

∫d3x εµνλTr

[Aµ∂νAλ+

2i

3AµAνAλ−Aµ∂νAλ−

2i

3AµAνAλ)

]


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• Classical moduli space = (C4/Zk )N/SN

I residual gauge symmetry: (U(1)×U(1))N/SN

I X A → diagonal matrices with identification X A = e2πi/kX A

I At k = 1, R8/SN ⇔ N M2 branes in flat space

Would like to study, in the case k = 1, whether we can understandordinary strings by wrapping M2 branes along the M-circle.

But, that is in the strong-coupling regime!

Will however see that after the reduction due to wrapping,the effective coupling constant is

N

kr2

r = transverse distance scale among strings


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

2. Wrapped M2 brane from ABJM

Double dimensional reduction

The ABJM model implicitly assumes the static gauge for world-volumecoordinates: world 3-coordinates xµ = longitudinal 3 directions of 11D

⇓

Wrapping along the M-circle in 11-th direction can be performed bythe “double” dimensional reduction (gs 1)

I Recover the length dimension with respect to target space by

(X A,ΨA, xµ)→ `−1P (X A,ΨA, xµ), (Aµ, Aµ)→ `P (Aµ, Aµ)

I gauge fixing along the periodic direction

∂2A2 = 0 = ∂2A2, (A2, A2) ≡ R−111 (B, B)

I x2 = x2 + 2πR11, R11 = gs`s∫d3x → R11

∫d2x , ∂2 → 0 for all fields


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• Reduced 2D action (µ, ν, λ . . . ∈ (0, 1)) 5

SPS =k

2π`2s

∫d2x Tr

[−DµX ADµXA+

1

g2s `

2s

(BX A−X AB)(BXA−XAB)

+ . . .]

+ SBF +k

2π`2s

∫d2x (L6 + L2,2)

SBF =k

2π

∫d2x ενλTr

(B∂νAλ + iBAνAλ − B∂νAλ − i BAνAλ

)L6 =

1

3g2s `

6s

Tr(X AXAX BXBX C XC + · · ·

)L2,2 =

1

g4/3s `4s

Tr(iεABCDΨAXB ΨC XD + · · ·

)I Naively, this system flows, in the extreme IR limit, to the strong

coupling regime [1/gs →∞]=[weak string coupling].

I Moduli-space approximation seems good for |p| 1/R11, 1/RP

I At k = 1, should correspond to multiple parallel strings stretchingalong a fixed longitudinal direction in flat 10D spacetime.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

3. Parallel strings

Parallel strings from AdS4×S7

On the bulk side, start from the M2 brane metric

ds211 = h−2/3(−dt2 +dx2

1 +dx22 )+h1/3(dr2 +dΩ2

7), h = 1+32π2N`6P

r6

Using the usual relation between 11D and 10D string-frame,

ds211 = e−2φ/3ds2

string + e4φ/3dx22

the background fields around N parallel stringsstretching along x2 is

ds2string = h−1(−dt2 + dx2

1 ) + dr2 + dΩ27

eφ = h−1/2, B01 = h−1


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Remarks:

I BPS ⇔ −g00 = g11 = B01

The world-sheet string action is completely free

Sstring = − 1

4π`2s

∫d2ξ√−γ(gAB (X )γµν + BAB (X )εµν

)∂µX

A∂νXB

= − 1

4π`2s

∫d2x

∑A=transverse

∂µXA(x)∂µXA(x)

in the static (conformal) gauge ξ0 = t = X 0, ξ1 = x1 = X1 andis manifestly SO(8) symmetric.

I Near-horizon limit: r (g2s N)1/6 (Q ∝ Ng2

s `6s )

ds2string =

r6

Q(−dt2 + dx2) + dr2 + r2dΩ2

7

⇒ scaling symmetry:

(I) : r → λ1/2r , (t, x),→ λ−1(t, x), ds2string → λ−1ds2

string

(II) : (t, x)→ ρ(t, x), gs → ρgs


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Bulk vs. boundary

The structure of ABJM moduli space seems consistent with the aboveproperties on the bulk side, at least classically.

• Question: What about the quantum corrections ?

I enhancement of R symmetry?

I cancellation of all interactions?

The question is essentially non-perturbative in its nature.

Let us study general structure of the effective action for parallel stringson the basis of the reduced action SPS 4


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Scaling symmetry of SPS

I (I) : inherited from 3D conformal symmetry

r → λ1/2r , (t, x),→ λ−1(t, x), ds2string → λ−1ds2

string

m

(Aµ,B, Aµ, B)→ λ(Aµ,B, Aµ, B), XA → λ1/2XA,

(ΨA, ΨA)→ λ(ΨA, Ψ

A), k → λ−1k

I (II) : related to 2D conformal symmetry(reminiscent of matrix-string theory)

(t, x)→ ρ(t, x), gs → ρgs

m

(ΨA, ΨA)→ ρ−1/2(ΨA, Ψ

A), (Aµ, Aµ)→ ρ−1(Aµ, Aµ)


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Let the (transverse) distance scale among parallel strings be r . Thescaling symmetries constrain the (bosonic part of) effective action as(string unit : `s = 1)

Seff =∞∑

L=0,q=2,g=0,h=0

cL,q,g ,h

∫d2x k−L+1gq−2

s r−2L+6(∂r

r3

)q

×N2−2g−h+L−1

L = # of loops, q = # of derivativesg =genus, h = # of holes

with respect to color index loops in planar expansion

⇓

I perturbative loop expansion is meaningful when

N

kr2 1

I In the limit gs → 0, the derivative expansion is also meaningful.In the free limit, can restrict to the lowest order q = 2.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• Unfortunately, the near-horizon limit on the bulk side is notcompatible with the perturbative regime of the reduced action forfinite fixed k and for weak string coupling,

(as typical AdS/CFT correspondence !)since

near horizon condition : r (g2s N)1/6 ⇔ r (N/k)1/2

⇓

1 gsN/k3/2


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• However, independently of the near-horizon condition, we can studyeffective actions for our ‘would-be’ gauge theory of multiple parallelstrings, for sufficiently large r

r N1/2 at k = 1, N = finite

• Relevant question :susy ‘non-renormalization theorem’ for kinetic terms, valid or not?

I In the case of D-brane susy Yang-Mills theories,non-renormalization theorems are at work.

• loop corrections start from v 4/r 7−p

• seems to be case also for AdS4 ×CP3 ( k →∞) in one-loop order.

Not only that, SYM can correctly reproduce the long-distancegravitational interactions (even 3-body forces!) among D-branesat least up to two-loop order.

I Note also that physical interpretation of the off-diagonal parts ofmatrix coordinates X A in the case of ABJM (and also of BLGtheories) is totally unclear.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

4. Effective action for parallel strings fromthe reduced action

(N + 1)→ (N) + (1) decompostion

Let us study one-loop effective action (L = 1) for simplest background

X A = ( 0, 0, . . . , 0︸︷︷︸N

, rA), U(N + 1)→ U(N)×U(1)

• off-diagonal fluctuating fields:(a = 1, . . . ,N , all are complex N-vectors)

I two pairs of (4, 4) scalar fields

UAa ,U

A

a ,VAa ,V

A

a

I their fermion partners (2D Dirac)

ΘAa ,Θ

A

a ,ΦAa ,Φ

A

a

I pairs of 2D vector fields

Aµ a,Aµ a, Aµ a, Aµ a

I pairs of auxiliary scalar fields (originated from A2 a, A2 a)

Ba,Ba, Ba, Ba


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

• Owing to the presence of the vacuum expectation value for X A, . . .

I Can integrate out the auxiliary fields B, . . .

I Can choose the followng special background-field gauge

1

r∂µA

µa − ir(r · Va) = 0,

1

r∂µA

µa + ir(r · Va) = 0

⇓I emergence of usual kinetic terms for fluctuating gauge fields

I mass terms are diagonalized with eigenvalues

(r4, r4, r4, r4) for complex scalars r2 = r · r(r2, r2, r2, r2) for Dirac fermions

• mass ∝ r2 → off-diagonals ∼ open-membrane bits ?

• SU(4) R-symmetry is enhanced to SO(8) for completely staticparallel strings ∂r = 0. Not trivial!

However,

I no enhancement for non-static background ∂r 6= 0


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Result of explicit computation

• Scaling symmetries ⇒ ∆S1−loopbosonic ∼ O

((∂r)2/r2

),

provided no cancellation

• Explicit computation :

Sk=1eff =

∫d2ξ

(− k

2π∂ rA∂rA

− N

4π

(∂ r · r)2 + (r · ∂r)2

r4− 5N

2π

(r · ∂r)(r · ∂ r)

r4

)+ O

((ψψ)2

r4

)m

1-loop deformation of susy transformation law

δψA = −ΓI ABγµεI∂µ(1 + 2N

kr2)rB + O

(ψψ

r3

)


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

No ‘non-renormalization theorem’ for the kinetic term,in contrast to the case of D-branes.

I Physical interpretaion ?non-trivial kinetic term ⇔ flat transverse metric ?

( 6= ordinary gravitational force)

• Some kind of “Casimir energy”, suggesting that the transversespace is not flat even for k = 1.

I Mathematical characterization ?N = 6 susy 2D non-linear sigma model

However, there is no direct contradiction with the possible

“multiple parallel strings / N = 6 BF gauge theory ”correspondence

which requires

N

r2&

N

(gsN)1/3= g−1/3

s N2/3 1


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Large N non-perturbative behavior?

The scaling symmetry constrains the non-perturbative form of thegs = 0 effective action as

Sk=1eff =

∫d2ξ

[− k

2π∂ rA∂rA

−f1

(N

kr2

)(∂ r · r)2 + (r · ∂r)2

r4− f2

(N

kr2

)(r · ∂r)(r · ∂ r)

r4

]Assuming that the limit r → 0 is smooth for a fixed N, it seemsreasonable to expect that

f1(x)→ c1/x2, f2(x)→ c2/x

2

Then in the near-horizon region at finite fixed k ,

f1

(N

kr2

)∼ f1

(g−1/3

s N2/3)→ 0 similarly for f2

It is plausible that ABJM model is non-perturbatively consistent with

“ multiple parallel strings / N = 6 BF gauge theory ”correspondence


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

5. Discussion

• Comment : case of BLG model

I A4 (SO(4)) BLG model with manifest SO(8) R-symetry isequivalent to ABJM model with gauge group SU(2)×SU(2)

but

I different classical moduli space : R8× R8/D2k (D2k =dihedralgroup of order 4k)

I for k = 1, (roughly speaking) two M2 branes in the (transverse)orbifold space R8/Z2.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

I enhancement of R-symmetry to SO(8) is only kinematical

(0, rA) in ABJMk=1 for N = 2→ z I (I = 1, 2, . . . , 8) in LBG Ak=14

with a particular (SO(8)-invariant) constraint

z · z = 0 (r2 = z · z) z=0

and then(r · ∂r)(r · ∂r)

r4(SU(4) invariant)

↓(z · ∂z)(z · ∂z)

r4(SO(8) invariant)


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Summary

We have examined the consistency of ABJM (and BLG) theory withM-theory conjecture.

I scaling behavior matches between bulk sugra picture and gaugetheory at the boundary

I usual non-renormaltization theorem for the kinetic term is notvalid in perturbation theory

I suggest the existence of some nontrivial 2D non-linear sigmamodel with N = 6 susy, representing perhaps some kind ofCasimir effect

I plausibility argument for non-perturbative consistency in the largeN limit

Seems worthwhile pursue further.

I For instance, relation between this theory and the matrix-stringtheory picture of wrapped membranes.


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

What’s next?

“Find new wisdoms through old things.”(Confucius 551-479 BC)

For my own approach, see my talk(s) in KEK workshop(s) last year.http://hep1.c.u-tokyo.ac.jp/ tam/jp.html

also arXiv:0804:0297[hep-th], arXiv:0706.0642[hep-th]


1. Introduction

• M-theory

• ABJM

2. WrappingM2 brane

3. Parallelstrings

• FromAdS4 × S7


4. Effectiveaction

• (N + 1) →(N) + (1)


• Large N

5. Discussion

• Summary

Thank you!

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MultipleParallelStrings and WrappedM2 Branes...

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