Non-singlet baryons in gauge/
gravity duality
Yolanda Lozano (U. Oviedo)
Corfu, September 2012
martes 25 de septiembre de 2012
Motivation:
Non-singlet baryons are predicted in N=4 SYM by the
AdS/CFT correspondence.
Do they also exist in more realistic theories?
martes 25 de septiembre de 2012
Motivation:
Non-singlet baryons are predicted in N=4 SYM by the
AdS/CFT correspondence.
Do they also exist in more realistic theories?
How:
Analyze the holographic description in various
backgrounds with reduced supersymmetries and/or
confining
martes 25 de septiembre de 2012
Motivation:
Non-singlet baryons are predicted in N=4 SYM by the
AdS/CFT correspondence.
Do they also exist in more realistic theories?
How:
Analyze the holographic description in various
backgrounds with reduced supersymmetries and/or
confining
- Reduced SUSY: AdS5 × Y5 , Lunin-Maldacena β
deformed, Frolov multi- β deformed
- Confining: Maldacena-Nuñez
martes 25 de septiembre de 2012
Results:
- Non-singlet baryons exist in all these backgrounds
- Same number of quarks in all AdS5 × Y5 Einstein
manifolds with 5-form flux, independent of SUSY
- More restricted number of quarks in MN
- Stable against fluctuations
- Non-singlet baryons at finite ‘t Hooft coupling
martes 25 de septiembre de 2012
Results:
- Non-singlet baryons exist in all these backgrounds
- Same number of quarks in all AdS5 × Y5 Einstein
manifolds with 5-form flux, independent of SUSY
- More restricted number of quarks in MN
- Stable against fluctuations
- Non-singlet baryons at finite ‘t Hooft coupling
(Based on arXiv:1203.6817, D. Giataganas,Y.L., M. Picos, K.
Siampos, JHEP)
martes 25 de septiembre de 2012
1. The baryon vertex in AdS5 × S 5
martes 25 de septiembre de 2012
1. The baryon vertex in AdS5 × S 5
Gauge invariant coupling of N external quarks
martes 25 de septiembre de 2012
1. The baryon vertex in AdS5 × S 5
Gauge invariant coupling of N external quarks
Through AdS/CFT external quarks are regarded as endpoints
of F-strings in AdS
AdS-Boundary
x2
x3
x1
xN
... Quarks
Vertex
martes 25 de septiembre de 2012
1. The baryon vertex in AdS5 × S 5
Gauge invariant coupling of N external quarks
Through AdS/CFT external quarks are regarded as endpoints
of F-strings in AdS
AdS-Boundary
x2
x3
x1
xN
... Quarks
Vertex
Baryon vertex in the gravity side: D5-brane wrapped on the
5-sphere (Witten’98):
SCS = 2π T5
martes 25 de septiembre de 2012
�
R×S 5
P [F5 ] ∧ A = N TF 1
�
R
dtAt
1. The baryon vertex in AdS5 × S 5
Gauge invariant coupling of N external quarks
Through AdS/CFT external quarks are regarded as endpoints
of F-strings in AdS
AdS-Boundary
x2
x3
x1
xN
... Quarks
Vertex
Baryon vertex in the gravity side: D5-brane wrapped on the
5-sphere (Witten’98):
SCS = 2π T5
�
R×S 5
P [F5 ] ∧ A = N TF 1
�
R
dtAt
N charge cancelled by N F-strings ending on the 5-brane
martes 25 de septiembre de 2012
The strings stretching to the boundary behave as fermions,
giving the antisymmetry of the baryon vertex.
martes 25 de septiembre de 2012
The strings stretching to the boundary behave as fermions,
giving the antisymmetry of the baryon vertex.
However, within the gauge/gravity correspondence it is
possible to construct bound states of k quarks with k < N
(non-singlets) (Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98;
Imamura’98)
martes 25 de septiembre de 2012
The strings stretching to the boundary behave as fermions,
giving the antisymmetry of the baryon vertex.
However, within the gauge/gravity correspondence it is
possible to construct bound states of k quarks with k < N
(non-singlets) (Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98;
Imamura’98)
In AdS5 × S 5 : 5N/8 ≤ k ≤ N
In AdS4 × CP 3 : 2N/3 ≤ k ≤ N
martes 25 de septiembre de 2012
(Y.L., Picos, Sfetsos, Siampos’11)
The strings stretching to the boundary behave as fermions,
giving the antisymmetry of the baryon vertex.
However, within the gauge/gravity correspondence it is
possible to construct bound states of k quarks with k < N
(non-singlets) (Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98;
Imamura’98)
In AdS5 × S 5 : 5N/8 ≤ k ≤ N
In AdS4 × CP 3 : 2N/3 ≤ k ≤ N
(Y.L., Picos, Sfetsos, Siampos’11)
What happens in less supersymmetric and/or confining
backgrounds?
martes 25 de septiembre de 2012
The strings stretching to the boundary behave as fermions,
giving the antisymmetry of the baryon vertex.
However, within the gauge/gravity correspondence it is
possible to construct bound states of k quarks with k < N
(non-singlets) (Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98;
Imamura’98)
In AdS5 × S 5 : 5N/8 ≤ k ≤ N
In AdS4 × CP 3 : 2N/3 ≤ k ≤ N
(Y.L., Picos, Sfetsos, Siampos’11)
What happens in less supersymmetric and/or confining
backgrounds?
And at finite ‘t Hooft coupling?
martes 25 de septiembre de 2012
2. Gauge/gravity calculation of the energy
(Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98; Imamura’98; Maldacena’98)
AdS-Boundary
... Quarks
Vertex
Consider a uniform distribution of strings on an Mp shell
Non-SUSY but we can ignore the backreaction
In the probe brane approach: S = SDp + SN F 1 :
SN F 1 = −N TF 1
with τ = t
martes 25 de septiembre de 2012
σ=r
�
dt dr
�
|detP (G)|
and the AdS direction ρ = ρ(r)
2. Gauge/gravity calculation of the energy
(Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98; Imamura’98; Maldacena’98)
AdS-Boundary
ρ(0) = ρ0
ρ(L) = ∞
... Quarks
Vertex
Consider a uniform distribution of strings on an Mp shell
Non-SUSY but we can ignore the backreaction
In the probe brane approach: S = SDp + SN F 1 :
SN F 1 = −N TF 1
with τ = t
martes 25 de septiembre de 2012
σ=r
�
dt dr
�
|detP (G)|
and the AdS direction ρ = ρ(r)
SDp = −Tp
�
R×Mp
d
p+1
ξ
�
|det P (G + 2πF − B)|
In AdS5 × Y5 :
2
2
ρ
R
ds2 = 2 dx21,3 + 2 dρ2 + R2 ds2Y5
R
ρ
4
4π
N gs
4
,
R =
Vol(Y5 )
Bulk equation of motion:
F5 = 4 R4 (1 + ∗)dVol(Y5 )
�
ρ4
ρ4
R4
Boundary equation of motion: �
martes 25 de septiembre de 2012
+ ρ� 2
=c
ρ0 �
ρ40
R4
+ ρ0 � 2
T5 R4 Vol(Y5 )
=
N TF 1
4
�
T
R
Vol(Y5 )
5
2
Define 1 − β =
N TF 1
with β ∈ [0, 1]
The two equations can be combined into:
Integrating:
�
ρ4
ρ4
R4
+ ρ� 2
Size of the configuration:
2
R
L=
ρ0
On-shell energy:
E = EDp + EN F 1
martes 25 de septiembre de 2012
= β ρ20 R2
�
∞
1
dz
z2
�
β
z4 − β2
�
��
= N TF 1 ρ0
1 − β2 +
∞
1
z2
dz �
z4 − β2
�
Binding energy:
Ebin
�
��
= N TF 1 ρ0
1 − β2 +
∞
1
�
dz �
z2
z4 − β2
�
�
−1 −1
where we have substracted the energy of the constituents
(when the brane is located in ρ0 = 0 the strings become
radial and correspond to free quarks)
martes 25 de septiembre de 2012
Binding energy:
Ebin
�
��
= N TF 1 ρ0
1 − β2 +
∞
1
�
dz �
z2
z4 − β2
�
�
−1 −1
where we have substracted the energy of the constituents
(when the brane is located in ρ0 = 0 the strings become
radial and correspond to free quarks)
As a function of L :
Ebin
√
λ
= −f (β)
L
martes 25 de septiembre de 2012
with
f (β) ≥ 0
Binding energy:
Ebin
�
��
= N TF 1 ρ0
1 − β2 +
∞
1
�
dz �
z2
z4 − β2
�
�
−1 −1
where we have substracted the energy of the constituents
(when the brane is located in ρ0 = 0 the strings become
radial and correspond to free quarks)
As a function of L :
Ebin
√
λ
= −f (β)
L
with
⇒ - The configuration is stable
f (β) ≥ 0
- Ebin ∼ √
1/L dictated by conformal invariance
- Ebin ∼ λ non-trivial prediction for the non-perturbative
regime of the gauge theory
martes 25 de septiembre de 2012
Universal behavior for all AdS5 × Y5 backgrounds, independent
of SUSY
martes 25 de septiembre de 2012
Universal behavior for all AdS5 × Y5 backgrounds, independent
of SUSY
Same thing in beta deformed LM backgrounds and multi beta
deformed (Frolov) backgrounds (non-SUSY)
→ Based on conformality
martes 25 de septiembre de 2012
Universal behavior for all AdS5 × Y5 backgrounds, independent
of SUSY
Same thing in beta deformed LM backgrounds and multi beta
deformed (Frolov) backgrounds (non-SUSY)
→ Based on conformality
In AdS4 × CP : Ebin
3
√
λ
= −f (β)
L
with a different f (β)
(Y.L., Picos, Sfetsos, Siampos’11)
martes 25 de septiembre de 2012
Universal behavior for all AdS5 × Y5 backgrounds, independent
of SUSY
Same thing in beta deformed LM backgrounds and multi beta
deformed (Frolov) backgrounds (non-SUSY)
→ Based on conformality
In AdS4 × CP : Ebin
3
√
λ
= −f (β)
L
with a different f (β)
(Y.L., Picos, Sfetsos, Siampos’11)
Confining background?
Maldacena-Nuñez: Ebin ∼ L
(Loewy, Sonnenschein’01)
martes 25 de septiembre de 2012
Universal behavior for all AdS5 × Y5 backgrounds, independent
of SUSY
Same thing in beta deformed LM backgrounds and multi beta
deformed (Frolov) backgrounds (non-SUSY)
→ Based on conformality
In AdS4 × CP : Ebin
3
√
λ
= −f (β)
L
with a different f (β)
(Y.L., Picos, Sfetsos, Siampos’11)
Confining background?
Maldacena-Nuñez: Ebin ∼ L
(Loewy, Sonnenschein’01)
Non-singlets?
martes 25 de septiembre de 2012
3. Reduce the number of quarks
In AdS5 × S 5 : Baryon vertex classical solutions with number
of quarks 5N/8 ≤ k ≤ N (non-singlet)
(Brandhuber, Itzhaki, Sonnenschein,Yankielowitz’98;
Imamura’98)
Stable against fluctuations for 0.813 N ≤ k ≤ N
(Sfetsos, Siampos’08)
martes 25 de septiembre de 2012
3.1. The classical solution
The boundary equation of motion changes:
�
ρ�0
ρ40
R4
+ ρ�0 2
⇒ 5N/8 ≤ k ≤ N .
martes 25 de septiembre de 2012
T5 R4 Vol(Y5 ) N − k
=
+
≤1
k TF 1
k
3.1. The classical solution
The boundary equation of motion changes:
�
ρ�0
ρ40
R4
+ ρ�0 2
⇒ 5N/8 ≤ k ≤ N .
martes 25 de septiembre de 2012
T5 R4 Vol(Y5 ) N − k
=
+
≤1
k TF 1
k
For k = 5N/8 the baryon reduces
to free quarks
3.1. The classical solution
The boundary equation of motion changes:
�
ρ�0
ρ40
R4
+ ρ�0 2
⇒ 5N/8 ≤ k ≤ N .
T5 R4 Vol(Y5 ) N − k
=
+
≤1
k TF 1
k
For k = 5N/8 the baryon reduces
to free quarks
Same for all Y5 , independent on SUSY
martes 25 de septiembre de 2012
3.1. The classical solution
The boundary equation of motion changes:
�
ρ�0
ρ40
R4
+ ρ�0 2
⇒ 5N/8 ≤ k ≤ N .
T5 R4 Vol(Y5 ) N − k
=
+
≤1
k TF 1
k
For k = 5N/8 the baryon reduces
to free quarks
Same for all Y5 , independent on SUSY
In fact, same bound for beta and multi beta deformed
martes 25 de septiembre de 2012
3.1. The classical solution
The boundary equation of motion changes:
�
ρ�0
ρ40
R4
+ ρ�0 2
⇒ 5N/8 ≤ k ≤ N .
T5 R4 Vol(Y5 ) N − k
=
+
≤1
k TF 1
k
For k = 5N/8 the baryon reduces
to free quarks
Same for all Y5 , independent on SUSY
In fact, same bound for beta and multi beta deformed
Same analysis in Maldacena-Nuñez: 3N/4 ≤ k ≤ N
martes 25 de septiembre de 2012
3.1. The classical solution
The boundary equation of motion changes:
�
ρ�0
ρ40
R4
+ ρ�0 2
⇒ 5N/8 ≤ k ≤ N .
T5 R4 Vol(Y5 ) N − k
=
+
≤1
k TF 1
k
For k = 5N/8 the baryon reduces
to free quarks
Same for all Y5 , independent on SUSY
In fact, same bound for beta and multi beta deformed
Same analysis in Maldacena-Nuñez: 3N/4 ≤ k ≤ N
⇒ Non-singlet states in non-SUSY or confining backgrounds
martes 25 de septiembre de 2012
3.2. Stability analysis
Important in establishing the physical parameter space
(Avramis, Sfetsos, Siampos’06-08)
martes 25 de septiembre de 2012
3.2. Stability analysis
Important in establishing the physical parameter space
(Avramis, Sfetsos, Siampos’06-08)
Ansatz for the fluctuations (for the strings):
δxµ (t, ρ) = δxµ (ρ)e−iωt for xµ = r, θ, φ
martes 25 de septiembre de 2012
3.2. Stability analysis
Important in establishing the physical parameter space
(Avramis, Sfetsos, Siampos’06-08)
Ansatz for the fluctuations (for the strings):
δxµ (t, ρ) = δxµ (ρ)e−iωt for xµ = r, θ, φ
Expand the Nambu-Goto action to quadratic order and study
the zero mode problem ↔ Critical curve in the parametric
space separating the stable and unstable regions
Stability reduced to an eigenvalue problem of the general
Sturm-Liouville type
Instabilities emerge from longitudinal fluctuations of the
strings
martes 25 de septiembre de 2012
For AdS5 × Y5 and beta deformed:
Bound for the number of F-strings coming from stability:
�
N
k≥
(1 + 1 − β 2 )
1 + γc
γc = 0.538
More restrictive than the bound imposed by the existence
of a classical solution:
�
N
k ≥ (1 + 1 − β 2 )
2
For MN: Same bound for stability and existence of classical sol.
martes 25 de septiembre de 2012
For AdS5 × Y5 and beta deformed:
Bound for the number of F-strings coming from stability:
�
N
k≥
(1 + 1 − β 2 )
1 + γc
γc = 0.538
More restrictive than the bound imposed by the existence
of a classical solution:
�
N
k ≥ (1 + 1 − β 2 )
2
For MN: Same bound for stability and existence of classical sol.
⇒ Non-singlet states exist for λ >> 1
martes 25 de septiembre de 2012
For AdS5 × Y5 and beta deformed:
Bound for the number of F-strings coming from stability:
�
N
k≥
(1 + 1 − β 2 )
1 + γc
γc = 0.538
More restrictive than the bound imposed by the existence
of a classical solution:
�
N
k ≥ (1 + 1 − β 2 )
2
For MN: Same bound for stability and existence of classical sol.
⇒ Non-singlet states exist for λ >> 1
Can we reach the finite ‘t Hooft coupling region?
martes 25 de septiembre de 2012
5. Baryons at finite ‘t Hooft coupling
Generalize the baryon vertex adding a magnetic flux.
A non-trivial flux adds lower dim brane charges →
Complementary description of the baryon in terms of
D1-branes expanding by Myers dielectric effect
martes 25 de septiembre de 2012
5. Baryons at finite ‘t Hooft coupling
Generalize the baryon vertex adding a magnetic flux.
A non-trivial flux adds lower dim brane charges →
Complementary description of the baryon in terms of
D1-branes expanding by Myers dielectric effect
- Description in terms of the expanded brane (macroscopical)
valid in the sugra limit: R >> 1 ⇔ λ >> 1
- Micro when
martes 25 de septiembre de 2012
4πR2
<< ls2 ⇔ λ << n2
n
5. Baryons at finite ‘t Hooft coupling
Generalize the baryon vertex adding a magnetic flux.
A non-trivial flux adds lower dim brane charges →
Complementary description of the baryon in terms of
D1-branes expanding by Myers dielectric effect
- Description in terms of the expanded brane (macroscopical)
valid in the sugra limit: R >> 1 ⇔ λ >> 1
- Micro when
4πR2
<< ls2 ⇔ λ << n2
n
⇒ It allows to explore the region of finite λ
martes 25 de septiembre de 2012
5. Baryons at finite ‘t Hooft coupling
Generalize the baryon vertex adding a magnetic flux.
A non-trivial flux adds lower dim brane charges →
Complementary description of the baryon in terms of
D1-branes expanding by Myers dielectric effect
- Description in terms of the expanded brane (macroscopical)
valid in the sugra limit: R >> 1 ⇔ λ >> 1
- Micro when
4πR2
<< ls2 ⇔ λ << n2
n
⇒ It allows to explore the region of finite λ
- Complementary at finite n . Should agree at large n
martes 25 de septiembre de 2012
The magnetic flux modifies the dynamics:
martes 25 de septiembre de 2012
The magnetic flux modifies the dynamics:
In AdS5 × Y p,q :
2
2
ρ
R
ds2 = 2 dx21,3 + 2 dρ2 + R2 ds2Y p,q
R
ρ
4
4π
N gs
4
R =
,
Vol(Y p,q )
ds2Y p,q
1
= ds (M4 ) + ( dψ + σ)2
3
2
k µ = δψµ : Reeb vector
M4 : Local Kähler-Einstein metric
martes 25 de septiembre de 2012
The magnetic flux modifies the dynamics:
In AdS5 × Y p,q :
2
2
ρ
R
ds2 = 2 dx21,3 + 2 dρ2 + R2 ds2Y p,q
R
ρ
4
4π
N gs
4
R =
,
Vol(Y p,q )
ds2Y p,q
1
= ds (M4 ) + ( dψ + σ)2
3
2
k µ = δψµ : Reeb vector
M4 : Local Kähler-Einstein metric
1
Take F = N J with J = dσ the Kähler form:
2
4
p,q
2 2
�
T
R
Vol(Y
)
4π
N
5
2
1−β =
(1 +
)
4
N TF 1
R
β≤1⇒
martes 25 de septiembre de 2012
Nmax ↔ β = 0
Binding energy:
Ebin
�
��
= N TF 1 ρ0
1 − β2 +
∞
1
�
dz �
z2
z4 − β2
�
�
−1 −1
- Ebin negative and decreases monotonically with β
- Ebin = 0 for β = 0 (N free radial strings stretching from
ρ0 to ∞ plus a Dp-brane at ρ0 )
Ebin ! qTF1 Ρ0
!
0.5
1.0
!0.1
1.5
2.0
2.5
3.0 L
2
Nmax ∼
√
λ
!0.2
!0.3
!0.4
!0.5
!0.6
martes 25 de septiembre de 2012
Ebin
√
λ
= −f (β)
L
Non-singlets:
The bound for the number of quarks is modified ( (k, N )
parameter space bounded by the values for which the
baryon vertex reduces to free quarks):
5
N2
k ≥ N + 2 Vol(Y p,q )
8
8π
martes 25 de septiembre de 2012
Non-singlets:
The bound for the number of quarks is modified ( (k, N )
parameter space bounded by the values for which the
baryon vertex reduces to free quarks):
5
N2
k ≥ N + 2 Vol(Y p,q )
8
8π
kmin is maximum for the 5-sphere, the most SUSY case
16 3
3
5
π = Vol(S ) >
π = Vol(T 1,1 ) > Vol(Y 2,1 ) ≈ 0.29π 3
27
Y p,q with largest volume
martes 25 de septiembre de 2012
Non-singlets:
The bound for the number of quarks is modified ( (k, N )
parameter space bounded by the values for which the
baryon vertex reduces to free quarks):
5
N2
k ≥ N + 2 Vol(Y p,q )
8
8π
kmin is maximum for the 5-sphere, the most SUSY case
16 3
3
5
π = Vol(S ) >
π = Vol(T 1,1 ) > Vol(Y 2,1 ) ≈ 0.29π 3
27
Y p,q with largest volume
In the probe brane approx all configurations are non-SUSY.
martes 25 de septiembre de 2012
Non-singlets:
The bound for the number of quarks is modified ( (k, N )
parameter space bounded by the values for which the
baryon vertex reduces to free quarks):
5
N2
k ≥ N + 2 Vol(Y p,q )
8
8π
kmin is maximum for the 5-sphere, the most SUSY case
16 3
3
5
π = Vol(S ) >
π = Vol(T 1,1 ) > Vol(Y 2,1 ) ≈ 0.29π 3
27
Y p,q with largest volume
In the probe brane approx all configurations are non-SUSY.
Spike solutions non-SUSY for T 1,1 , Y p,q and MN
(Areán, Crooks, Ramallo’04; Canoura, Edelstein, Pando Zayas, Ramallo,Vaman’05;
Imamura’04)
martes 25 de septiembre de 2012
Important to obtain information at finite ‘t Hooft coupling.
martes 25 de septiembre de 2012
Important to obtain information at finite ‘t Hooft coupling.
Micro description:
D1’s expanding into a fuzzy M4 . Only known for the S 5
1,1
(Janssen,Y.L., Rodriguez-Gomez’06) and the T
martes 25 de septiembre de 2012
The fuzzy T 1,1
The T 1,1 is a U(1) bundle over S 2 × S 2 ⇒ Take the D1’s
2
2
× Sfuzzy
wrapped on the U(1) fibre and expanding into a Sfuzzy
martes 25 de septiembre de 2012
The fuzzy T 1,1
The T 1,1 is a U(1) bundle over S 2 × S 2 ⇒ Take the D1’s
2
2
× Sfuzzy
wrapped on the U(1) fibre and expanding into a Sfuzzy
Substituting in Myers action for D1-branes:
SDBI = −
Qij
�
=
δji
�
�
�
�
�
STr e−φ |det P [Eµν + Eµi (Q−1 − δ)ij E jk Ekν ] detQ|
i
+
[X i , X k ]Ekj
2π
martes 25 de septiembre de 2012
The fuzzy T 1,1
The T 1,1 is a U(1) bundle over S 2 × S 2 ⇒ Take the D1’s
2
2
× Sfuzzy
wrapped on the U(1) fibre and expanding into a Sfuzzy
Substituting in Myers action for D1-branes:
SDBI = −
�
�
�
�
�
STr e−φ |det P [Eµν + Eµi (Q−1 − δ)ij E jk Ekν ] detQ|
i
= +
[X i , X k ]Ekj
2π
N ρ0 (m + 1)2 �
36π 2 m(m + 2) �
=
1+
8π m(m + 2)
R4
Qij
EnD1
�
δji
martes 25 de septiembre de 2012
The fuzzy T 1,1
The T 1,1 is a U(1) bundle over S 2 × S 2 ⇒ Take the D1’s
2
2
× Sfuzzy
wrapped on the U(1) fibre and expanding into a Sfuzzy
Substituting in Myers action for D1-branes:
SDBI = −
�
�
�
�
�
�
STr e−φ |det P [Eµν + Eµi (Q−1 − δ)ij E jk Ekν ] detQ|
i
= +
[X i , X k ]Ekj
2π
N ρ0 (m + 1)2 �
36π 2 m(m + 2) �
EnD1 =
1+
8π m(m + 2)
R4
1
i
J i for each S 2
where X = �
m(m + 2)
Qij
δji
martes 25 de septiembre de 2012
The fuzzy T 1,1
The T 1,1 is a U(1) bundle over S 2 × S 2 ⇒ Take the D1’s
2
2
× Sfuzzy
wrapped on the U(1) fibre and expanding into a Sfuzzy
Substituting in Myers action for D1-branes:
SDBI = −
�
�
�
�
�
�
STr e−φ |det P [Eµν + Eµi (Q−1 − δ)ij E jk Ekν ] detQ|
i
= +
[X i , X k ]Ekj
2π
N ρ0 (m + 1)2 �
36π 2 m(m + 2) �
EnD1 =
1+
8π m(m + 2)
R4
1
i
J i for each S 2
where X = �
m(m + 2)
Qij
δji
martes 25 de septiembre de 2012
Agreement with
macro descrip.
in large m
The F1 in the micro description
SCS =
�
includes
SCS
1
=
π
� � i
�
�
�
STr P e 2π (iX iX )
Cq e−B2 e2πF
q
�
�
�
dt STr (iX iX )2 ik F5 At
martes 25 de septiembre de 2012
The F1 in the micro description
SCS =
�
includes
SCS
1
=
π
� � i
�
�
�
STr P e 2π (iX iX )
Cq e−B2 e2πF
q
�
�
�
dt STr (iX iX )2 ik F5 At
2
(m + 1)
=N
m(m + 2)
martes 25 de septiembre de 2012
�
dt At
The F1 in the micro description
SCS =
�
includes
SCS
1
=
π
� � i
�
�
�
STr P e 2π (iX iX )
Cq e−B2 e2πF
q
�
�
�
dt STr (iX iX )2 ik F5 At
2
(m + 1)
=N
m(m + 2)
martes 25 de septiembre de 2012
�
dt At
→ N
�
dt At
The F1 in the micro description
SCS =
�
includes
SCS
1
=
π
� � i
�
�
�
STr P e 2π (iX iX )
Cq e−B2 e2πF
q
�
�
�
dt STr (iX iX )2 ik F5 At
2
(m + 1)
=N
m(m + 2)
�
dt At
→ N
�
dt At
Similar micro analysis in MN in terms of D1’s expanding into
a fuzzy S 2
martes 25 de septiembre de 2012
6. Conclusions
- Non-singlet baryons are predicted by gauge/gravity duality
in less supersymmetric and/or confining backgrounds
- They are stable against fluctuations
- At finite ‘t Hooft coupling:
Microscopical description of the vertex
Complete this analysis with α� corrections to the NG action
of the strings (or microscopic spike), α� corrections to the
background
martes 25 de septiembre de 2012
6. Conclusions
- Non-singlet baryons are predicted by gauge/gravity duality
in less supersymmetric and/or confining backgrounds
- They are stable against fluctuations
- At finite ‘t Hooft coupling:
Microscopical description of the vertex
Complete this analysis with α� corrections to the NG action
of the strings (or microscopic spike), α� corrections to the
background
Thanks!
martes 25 de septiembre de 2012