toward m5-branes from abjm action based on going project with seiji terashima (yitp, kyoto u. )...
TRANSCRIPT
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
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)(
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)4,3,2,1(
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AdjA
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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
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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
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bos †A
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Ansatz (the solution becomes D2-D4 in the limit v → ∞)
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11/18
0)()(
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e.o.m.
,,, 22
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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
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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
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)0(
~~~~~~2
~)
~~()
~~(
~Tr
1
rrrddrdk
v
dxdyk
v
YYYYYYYYYYYYk
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k
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~
~
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2nd Evidence: Matching of TensionTension of the M5-brane from ABJM action
0
0~†A
A
AY
YY
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v
x
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2
22
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1
)()(
16/18
Volume factorTension
1~
,0~
1 01266
0125
Fxdk
vxdFSM ~~
Tension of the M5-brane world volume action
kvFBFF
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FF
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345
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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
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Pasti, Sorokin, Tonin (‘97)
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
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!!