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...Parallel Strings and Wrapped M2 Branes from the ABJM

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Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
Multiple Parallel Strings and
Wrapped M2 Branes from
the ABJM Model ?
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
Tamiaki Yoneya
University of Tokyo - Komaba
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
KEK 2009 Theory Workshop
Multiple
Parallel Strings
from ABJM?
1. Introduction
Contents
1. Introduction
2. Wrapped M2 branes from ABJM
3. Parallel strings
4. Effective action for parallel strings
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
5. Discussion
Multiple
Parallel Strings
from ABJM?
1. Introduction
Revisiting the M-theory conjecture
1. Introduction
• M-theory
• ABJM
First recall the conjecture.
2. Wrapping
M2 brane
11
spacetime
dimensions
10
9
3. Parallel
strings
M Theory
Circle
Type IIA
Circle/Z2
Type IIB
Type
IIAB
Type I
Type I
Hetero
SO(32)
Hetero
E8 x E8
Hetero
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Perturbative Theories
S duality
T duality
Cirlce Compactification
Unfortunately, no substantial progress, from the end of the previous
century, on what the M theory really is.
I
Multiple
Parallel Strings
from ABJM?
Radius of the circle direction :
1. Introduction
R11 = gs `s
• M-theory
• ABJM
2. Wrapping
M2 brane
M2 brane as gs → 0
I
”longitudinal”: wrapped along the 11-th circle direction
• From
AdS4 × S 7
• Bulk vs.
boundary
⇒ (fundamental) string
I
3. Parallel
strings
”transverse” : extended along directions orthogonal to the 11-th
circle
⇒ D2 brane
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
I
Fundamental length scale of M theory = Planck scale
`P = gs1/3 `s
and
`P `s
as gs → 0.
R11
• Summary
⇓
In the weak coupling (∼ 10 dimensional) limit, M2 branes should
smoothly reduce to perturbative strings of type IIA theory.
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
We would like to discuss this question in the context of the ABJM
model, a candidate low-enegy theory for multiple M2 branes, in the
simplest possible setting.
• So far, almost all previous works have been focused on the
”transverse” configurations of M2 branes:
weak coupling limit k ∼ ∞
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
m
S 7 , which is transverse to M2 branes, into CP3
S 7 /Zk → CP3 ,
Zk → S 1 ∼ M-theory circle
2π/k = Chern-Simons coupling constant
• Summary
• Remark: case of single M2 brane
The dynamics of a single M2 brane is already quite non-trivial, and
hence the reduction to string(s) is not completely understood,
quantum-mechanically.
Sekino-TY, hep-th/0108176 , Asano-Sekino-TY, hep-th/0308024
I
I
I
wrapped M2 brane
⇓ directly
matrix-string theory
[1 + 1D SYM with coupling 1/gs (N → ∞)]
√
large N limit with gYM = 1/ gs → ∞ can be studied by using
GKPW relation in the PP (BMN)-wave limit, under the
assumption of gauge/gravity correspondence.
The result of two-point correlators shows that the effective
scaling dimension of scalar fields is
2 1
∆eff =
=
5 − p p=1
2
This is consistent with the existence of 3D CFT description of M2
branes.
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Main features of the ABJM model
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
I
Susy Chern-Simons U(N)×U(N) gauge theory in 3D with
SO(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 × S 7 ⇔ effective CFT of N M2 branes in flat 11D
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
I
But, only with N = 6 susy, manifestly.
Multiple
Parallel Strings
from ABJM?
• Notations
(following Bandres-Lipstein-Schwarz, 0807.0880)
I
bosonic fields:
(XA ,
X A)
I
fermionic fields: (ΨA ,
ΨA )
1. Introduction
(4, 4 of SU(4))
• M-theory
• ABJM
(4, 4 of SU(4), 3D 2-component spinor)
I
Chern-Simons U(N)×U(N) gauge fields: (Aµ ,
ˆ µ)
A
SABJM
k
=
2π
Z
i
h
¯ A γ µ Dµ ΨA
d x Tr − D µ X A Dµ XA + i Ψ
3
k
+SCS +
2π
Z
d 3 x (L6 + L2,2 )
L6 = potential term of O(X ),
L2,2 = X Ψcoupling terms of O(X 2 Ψ2 )
Z
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
6
k
4π
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
• Action
SCS =
2. Wrapping
M2 brane
h
i
2i
ˆ µ ∂ν A
ˆ λ − 2i A
ˆ µA
ˆνA
ˆ λ)
d 3 x µνλ Tr Aµ ∂ν Aλ + Aµ Aν Aλ −A
3
3
Multiple
Parallel Strings
from ABJM?
• Classical moduli space = (C4 /Zk )N /SN
I
residual gauge symmetry: (U(1)×U(1))N /SN
A
1. Introduction
A
I
X → diagonal matrices with identification X = e
I
At k = 1, R8 /SN ⇔ N M2 branes in flat space
2πi/k
X
A
Would like to study, in the case k = 1, whether we can understand
ordinary 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
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
N
kr 2
r = transverse distance scale among strings
Multiple
Parallel Strings
from ABJM?
2. Wrapped M2 brane from ABJM
Double dimensional reduction
1. Introduction
The ABJM model implicitly assumes the static gauge for world-volume
coordinates: world 3-coordinates x µ = longitudinal 3 directions of 11D
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
⇓
Wrapping along the M-circle in 11-th direction can be performed by
the “double” dimensional reduction (gs 1)
I Recover the length dimension with respect to target space by
A
A µ
(X A , ΨA , x µ ) → `−1
P (X , Ψ , x ),
ˆ µ ) → `P (Aµ , A
ˆ µ)
(Aµ , A
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
I
gauge fixing along the periodic direction
ˆ 2,
∂2 A2 = 0 = ∂2 A
I
ˆ 2 ) ≡ R −1 (B, B)
ˆ
(A2 , A
11
x2 = x2 + 2πR11 , R11 = gs `s
Z
Z
d 3 x → R11 d 2 x,
∂2 → 0 for all fields
• Summary
• Reduced 2D action (µ, ν, λ . . . ∈ (0, 1)) 5
Z
h
k
1 ˆ A
ˆ
SPS =
d 2 x Tr −D µ X A Dµ XA + 2 2 (BX
−X A B)(BXA −XA B)
2
2π`s
gs `s
Z
i
k
+ . . . + SBF +
d 2 x (L6 + L2,2 )
2π`2s
Z
k
ˆλ − iB
ˆλ
ˆ νA
ˆA
ˆνA
d 2 x νλ Tr B∂ν Aλ + iBAν Aλ − B∂
SBF =
2π
1
A
B
C
L6 =
Tr
X
X
X
X
X
X
+
·
·
·
A
B
C
3gs2 `6s
1
¯ A XB ΨC XD + · · ·
L2,2 = 4/3 Tr iABCD Ψ
gs `4s
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 stretching
along a fixed longitudinal direction in flat 10D spacetime.
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Multiple
Parallel Strings
from ABJM?
3. Parallel strings
Parallel strings from AdS4 ×S7
1. Introduction
• M-theory
• ABJM
On the bulk side, start from the M2 brane metric
2
ds11
= h−2/3 (−dt 2 +dx12 +dx22 )+h1/3 (dr 2 +dΩ27 ),
32π 2 N`6P
h = 1+
r6
Using the usual relation between 11D and 10D string-frame,
2
2
ds11
= e −2φ/3 dsstring
+ e 4φ/3 dx22
the background fields around N parallel strings
stretching along x2 is
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
2
dsstring
=h
−1
2
(−dt +
dx12 )
e φ = h−1/2 , B01 = h−1
2
+ dr +
dΩ27
Remarks:
I
BPS ⇔ −g00 = g11 = B01
The world-sheet string action is completely free
Z
1
µν
µν
2 √
Sstring = −
−γ
g
(X
)γ
+
B
(X
)
∂µ X A ∂ν X B
d
ξ
AB
AB
4π`2s
Z
X
1
2
=−
d
x
∂µ X A (x)∂ µ XA (x)
4π`2s
A=transverse
in the static (conformal) gauge ξ 0 = t = X 0 ,
is manifestly SO(8) symmetric.
I
Near-horizon limit: r (gs2 N)1/6
2
dsstring
=
ξ 1 = x1 = X1 and
(Q ∝ Ngs2 `6s )
r6
(−dt 2 + dx 2 ) + dr 2 + r 2 dΩ27
Q
⇒ scaling symmetry:
(I) : r → λ1/2 r ,
(t, x), → λ−1 (t, x),
(II) : (t, x) → ρ(t, x),
gs → ρgs
2
2
dsstring
→ λ−1 dsstring
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Bulk vs. boundary
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
The structure of ABJM moduli space seems consistent with the above
properties 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.
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Let us study general structure of the effective action for parallel strings
on the basis of the reduced action SPS 4
Multiple
Parallel Strings
from ABJM?
Scaling symmetry of SPS
I
1. Introduction
(I) : inherited from 3D conformal symmetry
1/2
r →λ
r,
(t, x), → λ
−1
• M-theory
• ABJM
2
dsstring
(t, x),
→λ
−1
2
dsstring
m
3. Parallel
strings
ˆ µ , B)
ˆ → λ(Aµ , B, A
ˆ µ , B),
ˆ
(Aµ , B, A
¯ A ) → λ(ΨA , Ψ
¯ A ),
(ΨA , Ψ
I
XA → λ1/2 XA ,
k → λ−1 k
(II) : related to 2D conformal symmetry
(reminiscent of matrix-string theory)
(t, x) → ρ(t, x),
gs → ρgs
m
¯A
(ΨA , Ψ ) → ρ
−1/2
¯A
(ΨA , Ψ ),
2. Wrapping
M2 brane
ˆ µ ) → ρ−1 (Aµ , A
ˆ µ)
(Aµ , A
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Let the (transverse) distance scale among parallel strings be r . The
scaling symmetries constrain the (bosonic part of) effective action as
(string unit : `s = 1)
Seff =
∞
X
Z
cL,q,g ,h
2
d x
k −L+1 gsq−2 r −2L+6
L=0,q=2,g =0,h=0
∂r q
r3
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
×N 2−2g −h+L−1
L = # of loops,
q = # of derivatives
g =genus,
h = # of holes
with respect to color index loops in planar expansion
⇓
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
I
perturbative loop expansion is meaningful when
N
1
kr 2
I
In the limit gs → 0, the derivative expansion is also meaningful.
In the free limit, can restrict to the lowest order q = 2.
Multiple
Parallel Strings
from ABJM?
1. Introduction
• Unfortunately, the near-horizon limit on the bulk side is not
compatible with the perturbative regime of the reduced action for
finite fixed k and for weak string coupling,
since
near horizon condition : r ⇓
1 gs N/k 3/2
⇔ r (N/k)
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
(as typical AdS/CFT correspondence !)
(gs2 N)1/6
• M-theory
• ABJM
1/2
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
• However, independently of the near-horizon condition, we can study
effective actions for our ‘would-be’ gauge theory of multiple parallel
strings, for sufficiently large r
r N
1/2
at k = 1,
N = finite
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
• 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 × CP 3 ( k → ∞) in one-loop order.
Not only that, SYM can correctly reproduce the long-distance
gravitational interactions (even 3-body forces!) among D-branes
at least up to two-loop order.
I
Note also that physical interpretation of the off-diagonal parts of
matrix coordinates X A in the case of ABJM (and also of BLG
theories) is totally unclear.
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Multiple
Parallel Strings
from ABJM?
4. Effective action for parallel strings from
the reduced action
(N + 1) → (N) + (1) decompostion
1. Introduction
• M-theory
• ABJM
Let us study one-loop effective action (L = 1) for simplest background
X A = ( 0, 0, . . . , 0, r A ),
| {z }
U(N + 1) → U(N) × U(1)
• off-diagonal fluctuating fields:
(a = 1, . . . , N , all are complex N-vectors)
two pairs of (4, 4) scalar fields
A
A
UaA , U a , VaA , V a
I
their fermion partners (2D Dirac)
pairs of 2D vector fields
ˆ µ a, A
ˆµ a
Aµ a , Aµ a , A
I
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
A
A
ΘAa , Θa , ΦAa , Φa
I
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
N
I
2. Wrapping
M2 brane
ˆ 2 a)
pairs of auxiliary scalar fields (originated from A2 a , A
ˆa , B
ˆa
Ba , B a , B
• 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
¯a ) = 0,
∂µ Aµa − ir (r · V
r
1 ¯µ
∂µ Aa + ir (¯r · Va ) = 0
r
⇓
I
emergence of usual kinetic terms for fluctuating gauge fields
I
mass terms are diagonalized with eigenvalues
4
4
4
4
for complex scalars
2
2
2
2
for Dirac fermions
(r , r , r , r )
(r , r , r , r )
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
2
r =r ·r
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
• mass ∝ r 2 → off-diagonals ∼ open-membrane bits ?
• SU(4) R-symmetry is enhanced to SO(8) for completely static
parallel strings ∂r = 0. Not trivial!
However,
I
no enhancement for non-static background ∂r 6= 0
Result of explicit computation
1−loop
• Scaling symmetries ⇒ ∆Sbosonic
∼ O (∂r )2 /r 2 ,
provided no cancellation
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
• Explicit computation :
3. Parallel
strings
k=1
Seff
=
Z
k
d 2 ξ − ∂¯r A ∂r A
2π
N (∂¯r · r )2 + (¯r · ∂r )2
5N (¯r · ∂r )(r · ∂¯r )
−
−
4π
r4
2π
r4
(ψψ)2
+O
r4
m
1-loop deformation of susy transformation law
N B
ψψ
A
I AB µ I
˜
δψ = −Γ
γ ∂µ (1 + 2 2 )r + O
kr
r3
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• 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 transverse
space 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
N
&
= gs−1/3 N 2/3 1
2
r
(gs N)1/3
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Large N non-perturbative behavior?
Multiple
Parallel Strings
from ABJM?
The scaling symmetry constrains the non-perturbative form of the
1. Introduction
• M-theory
gs = 0 effective action as
• ABJM
Z
2. Wrapping
k
k=1
M2 brane
Seff
= d 2 ξ − ∂¯r A ∂r A
2π
3. Parallel
strings
2
2
N
(∂¯r · r ) + (¯r · ∂r )
N
(¯r · ∂r )(r · ∂¯r ) •AdSFrom× S 7
4
−f1
− f2
2
4
2
• Bulk vs.
kr
r
kr
r4
boundary
Assuming that the limit r → 0 is smooth for a fixed N, it seems
reasonable to expect that
f1 (x) → c1 /x 2 ,
f2 (x) → c2 /x 2
Then in the near-horizon region at finite fixed k,
N
−1/3 2/3
f1
∼
f
g
N
→0
similarly for f2
1
s
kr 2
It is plausible that ABJM model is non-perturbatively consistent with
“ multiple parallel strings / N = 6 BF gauge theory ”
correspondence
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
5. Discussion
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
• Comment : case of BLG model
2. Wrapping
M2 brane
3. Parallel
strings
I
A4 (SO(4)) BLG model with manifest SO(8) R-symetry is
equivalent to ABJM model with gauge group SU(2)×SU(2)
but
I
I
different classical moduli space : R8 × R8 /D2k (D2k =dihedral
group of order 4k)
for k = 1, (roughly speaking) two M2 branes in the (transverse)
orbifold space R8 /Z2 .
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Multiple
Parallel Strings
from ABJM?
I
enhancement of R-symmetry to SO(8) is only kinematical
1. Introduction
(0, r A ) in ABJMk=1 for N = 2 → z I
(I = 1, 2, . . . , 8) in LBG Ak=1
4
with a particular (SO(8)-invariant) constraint
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
z ·z =0
and then
(r 2 = z · z)
(r · ∂r )(r · ∂r )
r4
↓
(z · ∂z)(z · ∂z)
r4
z=0
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
(SU(4) invariant)
(SO(8) invariant)
• Summary
Summary
Multiple
Parallel Strings
from ABJM?
1. Introduction
We have examined the consistency of ABJM (and BLG) theory with
M-theory conjecture.
I
scaling behavior matches between bulk sugra picture and gauge
theory at the boundary
I
usual non-renormaltization theorem for the kinetic term is not
valid in perturbation theory
I
suggest the existence of some nontrivial 2D non-linear sigma
model with N = 6 susy, representing perhaps some kind of
Casimir effect
I
plausibility argument for non-perturbative consistency in the large
N limit
Seems worthwhile pursue further.
I
For instance, relation between this theory and the matrix-string
theory picture of wrapped membranes.
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
Multiple
Parallel Strings
from ABJM?
What’s next?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
“Find new wisdoms through old things.”
(Confucius 551-479 BC)
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
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]
Multiple
Parallel Strings
from ABJM?
1. Introduction
• M-theory
• ABJM
2. Wrapping
M2 brane
3. Parallel
strings
Thank you!
• From
AdS4 × S 7
• Bulk vs.
boundary
4. Effective
action
• (N + 1) →
(N) + (1)
• Explicit
computation
• Large N
5. Discussion
• Summary
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