Transcript Glueball Decay in Holographic QCD

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
2
Recent exciting progress in string theory:
Low energy actions of
multiple Membranes in M-theory
was found !
Why this is so exciting?
3
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.)
4
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 perturbative open 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.
5
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) !
6
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
7
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 curve
Thus, it is very interesting to find
low energy action of multiple M5-branes
8
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
9
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!
10
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
11
M2-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
12
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
13
D4-brane with magnetic flux or noncommutative R²
= 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
14
Strategy to construct it
N D2-branes (N →∞)
N M2-branes (N →∞)
3 dim SYM
ABJM model
S1 compactification
[ X i , X j ]  iij1NN
?
M5-brane
D4-brane
(with non-zero flux ∝ 1/Θ)
S1
(with non-zero 3-form flux)
compactification
15
Strategy to construct it
N D2-branes (N →∞)
N M2-branes (N →∞)
3 dim SYM
ABJM model
S1 compactification
[ X i , X j ]  iij1NN
We found
a classical solution!
M5-brane
D4-brane
(with non-zero flux ∝ 1/Θ)
S1
(with non-zero 3-form flux)
compactification
16
Interestingly,
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.
17
2. M2-branes and ABJM action
18
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
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
19
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 )
20
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.
We want to find a conformal action for M2-brane
21
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
,
22
ABJM action:
23
( (2+1)d N=6 ) SUSY transformation:
Gaiotto-Giobi-Yin, Hosomichi et.el, Bagger-Lambert, ST, Bandres-Lipstein-Schwarz
24
ABJM action has
12 SUSY and SU(4)xU(1) global symmetry
and
Conformal symmetry
(1)This action with U(N)xU(N) gauge group
describes N M2-branes on
c.f. BLG is SU(2)xSU(2)
(2) ABJM derived this action
as a limit of a D-brane configuration
25
(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-algebra
had been unclear for ABJM action.
26
3. ABJM to 3d YM
and M2-M5 bound state
27
Orbifold
M2-branes probing
(2+1)d ABJM theory
(Chern-Simon)
θ= 2 π / k
to R^7 x S¹
M2-branes probing R^7 x S¹
= D2-branes probing R^7
(2+1)d SuperYM theory
2πv/k
Mukhi et.al.
ABJM
Scaling limit
v → ∞, k → ∞, v / k : fixed
where v is the distance between the M2 and singularity 28
Bosonic part of ABJM
where
and
is the 3-bracket
Consider
and take a linear combination
then,
This v.e.v gives mass to gauge field
29
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
30
Potential of the ABJM action
Ansatz
(i.e. forget gauge fields and
only consider Hermite and constant part of Y¹ and Y²)
31
e.o.m.
additional Ansatz
(the solution becomes D2-D4 in the limit v →∞)
where f → 0 for v →∞
32
N M2-branes (N →∞)
N D2-branes (N →∞)
ABJM model
3 dim SYM
S1 compactification
[ X i , X j ]  iij1NN
M5-brane
(with non-zero flux)
D4-brane
S1 compactification
(with non-zero flux ∝ 1/Θ)
33
e.o.m. (infinite order nonlinear PDE)
34
Two equations for one function f(x,y).
Are these really consistent?
35
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.
36
This identity is shown from
the fundamental identity of Lie 3-algebra
and following identities
including both 2-bracket and 3-bracket:
37
a perturbative solution is
We can show that the solutions have only one parameter,
although there seem two parameters.
Another remark: solution is real
38
We claim that
the solution represents
an M5-brane with 3-from flux wrapping following space
i
i
Y  re , Y  r ' e , Y  Y  0
1
2
M2
M5
3
4
0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10
○○ ○
○○ ○ ○ ○
○
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 (?)
39
We can find
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!:
40
general expression of the solutions with Poisson bracket
The e.o.m. is approximated in the limit as
take a following ansatz:
then the solution is
41
Relation to Nambu-Poisson bracket
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
42
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.
43
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
44
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!
45
Lie 3-algebra and 3-form flux
The potential can be written by the 3-bracket:
where
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
46
4. Multiple M5-brane action
from ABJM
47
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.
48
The result is
49
Now, we assume order of the fluctuations as follows:
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):
50
Parameterization of fluctuations
For A =1,2, let us remember the D2-D4 case.
the solution (from D2 point of view) is
51
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)
52
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.
53
Finally, we have action of multiple M5-branes (with flux)
where
and using “open string metric”
and coupling constant is
which is not constant
54
5. Conclusion
55
• 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 3form 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
56
Many important things are left!
• 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
57
Fin.
58