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The baryon vertex with magnetic fluxBert Janssen
Universidad de Granada & CAFPE
Instituto de Fısica Teorica, U.A.M./C.S.I.C
In collaboration with: Y. Lozano (U. Oviedo) and D. Rodrıguez Gomez (U. Oviedo & U.A.M)
References: in preparation.
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Outlook1. Introduction
• Brief review of AdS/CFT
• Witten’s baryon vertex
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Outlook1. Introduction
• Brief review of AdS/CFT
• Witten’s baryon vertex
2. The baryon vertex with magnetic flux
• Construction
• Stability
• Supersymmetry
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Outlook1. Introduction
• Brief review of AdS/CFT
• Witten’s baryon vertex
2. The baryon vertex with magnetic flux
• Construction
• Stability
• Supersymmetry
3. Microscopic description
• Brief review of the dielectric effect
• Microscopic description in terms of D1-strings
• F1’s in the microscopic description
• Comment on S-duality
4. Conclusions
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1 Introduction1.1 Brief review of AdS/CFTExplicit example of gauge/gravity correspondence: [Maldacena]
Type IIB Supergravity (strings) on AdS5 × S5
∼ N = 4 Super Yang-Mills with gauge group SU(N)
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1 Introduction1.1 Brief review of AdS/CFTExplicit example of gauge/gravity correspondence: [Maldacena]
Type IIB Supergravity (strings) on AdS5 × S5
∼ N = 4 Super Yang-Mills with gauge group SU(N)
ds2 =u2
L2ηabdxadxb +
L2
u2du2 + L2dΩ2
5, G5 = 4L−1√
|gAdS| + 4L4√
|gS|
AdS 5
t
u
φ
S5
D3
SYM
SU(N) IIB sugra
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AdS5 × S5: near horizon limit of N D3-branesN = 4 SU(N) Super Yang-Mills: Gauge theory on D3 worldvolume
−→ D = 4 gauge theory captures holographically D = 10 gravity
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AdS5 × S5: near horizon limit of N D3-branesN = 4 SU(N) Super Yang-Mills: Gauge theory on D3 worldvolume
−→ D = 4 gauge theory captures holographically D = 10 gravity
−→ Dictonary between D = 4 gauge theory and D = 10 gravity:
• N =∫
S5 F5 : ] of branes = rank of SYM gauge group
• Coupling constants: g2Y MN = 4πgsN = (L/`s)
4
• SO(4, 2) × SO(6) isometry group of AdS5 × S5
∼ SO(4, 2) × SU(4)R conformal group of N = 4 SYM
• Fields in AdS5 ∼ operators in SYM
• Quark (q) in SYM ∼ string stretched between boundary and horizon
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Probe D3-brane at radial distance u
→ massive string between D3 and horizon with m ∼ u in vectorrepresentation of U(N)
→ massive quark in gauge theory
→ u ∼ ∞ ⇔ m ∼ ∞ ⇔ non-dynamical quark of SYM∼ F1 between horizon and boundary
F1
u
N D3
D3q
_q
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Quark physics in AdSnon-dynamical quark of SYM ∼ F1 between horizon and boundaryqq-pair in SYM ∼ string “hanging” from boundary
Wilson line C in SYM ∼ string worldsheet ending on C [Maldacena]
q
qq_
q_
qu0
l
F1
F1
F1 C
C
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Quark physics in AdSnon-dynamical quark of SYM ∼ F1 between horizon and boundaryqq-pair in SYM ∼ string “hanging” from boundary
Wilson line C in SYM ∼ string worldsheet ending on C [Maldacena]
q
qq_
q_
qu0
l
F1
F1
F1 C
C
Solution:
x =L
uo
∫ u/u0
1
dz
z2√
z2 − 1E =
u0
π
∫
∞
1
dz( z2
√z4 − 1
−1)
−u0
π∼
√
g2Y MN
`
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1.2 Witten’s baryon vertex
qq-pair (meson) in SYM ∼ string “hanging” from boundary
Does there exist a baryon configuration?= colourless antisymmetric bound state of N quarks
→ D5-brane wrapped around S5 with N strings extending to boundary
F1
D5q
q
q
q
q
u0
F1
F1
F1
F1
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Consider a D5-brane wrapped around S5 at fixed point u0 in AdS [Witten]
SCS = −T5
∫
R×S5
P [C(4)] ∧ F
= T5
∫
R×S5
P [G(5)] ∧ A
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Consider a D5-brane wrapped around S5 at fixed point u0 in AdS [Witten]
SCS = −T5
∫
R×S5
P [C(4)] ∧ F
= T5
∫
R×S5
P [G(5)] ∧ A
Ansatz: A = At(t)dt
= T5
∫
S5
P [G(5)]
∫
R
dtAt
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Consider a D5-brane wrapped around S5 at fixed point u0 in AdS [Witten]
SCS = −T5
∫
R×S5
P [C(4)] ∧ F
= T5
∫
R×S5
P [G(5)] ∧ A
Ansatz: A = At(t)dt
= T5
∫
S5
P [G(5)]
∫
R
dtAt
Background:∫
S5
G(5) = 4π2N
= NT1
∫
dtAt,
→ N units of string charge induced on D5 worldvolume
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Consistency of Ansatz A = At(t)dt:
S = SDBI + NT1
∫
dtAt
Equation of motion:
0 ≡ ∂L∂At
= NT1.
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Consistency of Ansatz A = At(t)dt:
S = SDBI + NT1
∫
dtAt
Equation of motion:
0 ≡ ∂L∂At
= NT1.
However, we can add N fundamental strings, ending on D5:
Stotal = SDBI + NT1
∫
dtAt + NSF1 − NT1
∫
dtAt
= SDBI + NSF1
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Consistency of Ansatz A = At(t)dt:
S = SDBI + NT1
∫
dtAt
Equation of motion:
0 ≡ ∂L∂At
= NT1.
However, we can add N fundamental strings, ending on D5:
Stotal = SDBI + NT1
∫
dtAt + NSF1 − NT1
∫
dtAt
= SDBI + NSF1
→ N F1 with same orientation (all q’s), stretched between D5 and boundary
→ Configuration is antisymmetric under interchange of any two F1’s
=⇒ Baryon vertex
[Witten]
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2 The baryon vertex with magnetic flux
2.1 Construction
Baryon vertex = D5-brane wrapped around S5
S5 is U(1) fibre bundle over CP 2
dΩ25 = (dχ − B)2 + ds2
CP 2,
B = −1
2sin2 ϕ1(dϕ4 + cos ϕ2dϕ3),
ds2CP 2 = dϕ2
1 +1
4sin2 ϕ1
(
dϕ22 + sin2 ϕ2dϕ2
3 + cos2 ϕ1
(
dϕ4 + cos ϕ2dϕ3
)2)
Fibre connection B satisfies
dB = ?(dB), dB ∧ dB ∼ √gCP 2 ∼ √
gS5
→ B non-trivial gauge field on CP 2, with non-zero instanton number
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Turn on magnetic Born-Infeld flux
F =√
2n dB
⇒∫
CP 2
F ∧ F = 8π2n.
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Turn on magnetic Born-Infeld flux
F =√
2n dB
⇒∫
CP 2
F ∧ F = 8π2n.
F is magnetic ⇒ no extra terms in Chern-Simons actionNew contributions to Born-Infeld action
SDBI = −T5
∫
d6ξu
L
√
det(
gαβ + Fαβ
)
= −T5
∫
d6ξ u√
gS5
(
L4 + 2FαβFαβ)
E = 8π3T5 u(
n +L4
8
)
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F =√
2n dB induces D1 charge in D5 worldvolume
SD5 =1
2T5
∫
R×S5
P [C(2)] ∧ F ∧ F
= nT1
∫
R×S1
P [C(2)]
= n SD1
→ n D1-branes: extended in t- and χ-directionsdissolved in D5 worldvolume
NB: n dissolved D-strings should not be confusedwith the N baryon vertex F-strings!
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F1
F1
F1
χ
F1
D5
n D1
F1
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F1
F1
F1
χ
F1
D5
n D1
F1
→ Alternative, microscopic description in terms of non-Abelian D1’s?Cfr Dielectric effect: [Emparan] [Myers]
Spherical D2-brane with dissolved D0-charge (Abelian)∼ dielectric D0’s expanding into fuzzy D2 (Non-Abelian)
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F1
F1
F1
χ
F1
D5
n D1
F1
→ Alternative, microscopic description in terms of non-Abelian D1’s?Cfr Dielectric effect: [Emparan] [Myers]
Spherical D2-brane with dissolved D0-charge (Abelian)∼ dielectric D0’s expanding into fuzzy D2 (Non-Abelian)
−→ See section 3
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2.2 Bound on n
baryon vertex with n = 0: stable under perturbations in xi
stable under perturbations in u
→ analysis of dynamics due to external F1’s[Brandhuber, Itzhaki, Sonnensshein, Yankielowicz]
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2.2 Bound on n
baryon vertex with n = 0: stable under perturbations in xi
stable under perturbations in u
→ analysis of dynamics due to external F1’s[Brandhuber, Itzhaki, Sonnensshein, Yankielowicz]
What is the influence of n 6= 0?
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2.2 Bound on n
baryon vertex with n = 0: stable under perturbations in xi
stable under perturbations in u
→ analysis of dynamics due to external F1’s[Brandhuber, Itzhaki, Sonnensshein, Yankielowicz]
What is the influence of n 6= 0?
S = SD5 − NT1
∫
dtdx
√
(u′)2 +u4
L4
Bulk eqn:u4
√
(u′)2 + u4
L4
= const
Boundary eqn:u′
0√
(u′
0)2 +
u4
0
L4
=πL4
4N
(
1 +8n
L4
)
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Equations combine into
u4
√
(u′)2 + u4
L4
= β u20L
2
withβ2 = 1 − 1
16
(
1 +8πn
N
)2
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Equations combine into
u4
√
(u′)2 + u4
L4
= β u20L
2
withβ2 = 1 − 1
16
(
1 +8πn
N
)2
Observation:u is real =⇒ β should be real
⇐⇒ 0 ≤ nN
≤ 38π
(Remember: F =√
2n dB ⇒ n > 0)
→ Upper bound on n
N
(relation to string exclusion principle?)
[Maldacena, Strominger]
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Solution: Size ` of the baryon vertex (in boundary)
` =L2
u0
∫
∞
1
dyβ
y2√
y4 − β2
NB: Size of baryon vertex is inversely proporcional to u0
Size of baryon vertex is function of n/N
l
q
q
q
q
q
D3
D5
F1u 0
18 Π
14 Π
38 Π
nN
0.1
0.2
0.3
0.4
u0L2
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Energy E of the baryon vertex
E = T1u0
∫
∞
1
dy[ y2
√
y4 − β2− 1
]
− 1
.
Energy E of the baryon vertex is:proportional to u0 (conformal invariance)proportional to
√gY MN
a function of n/N
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Energy E of the baryon vertex
E = T1u0
∫
∞
1
dy[ y2
√
y4 − β2− 1
]
− 1
.
Energy E of the baryon vertex is:proportional to u0 (conformal invariance)proportional to
√gY MN
a function of n/N
18 Π
14 Π
38 Π
nN
0.75
0.8
0.85
0.9
0.95
-E
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2.3 Supersymmetry
Witten’s baryon vertex: 12
(D5-brane) × 12
(N F1) = 14
total
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2.3 Supersymmetry
Witten’s baryon vertex: 12
(D5-brane) × 12
(N F1) = 14
total
Generalised vertex: N F1 break 12
(no influence of F =√
2ndB)D5-brane supersymmetry due to F =
√2ndB ?
D5-brane κ-symmetry:
Γnε = L−1BI
(
Γ(6) + F ∧ F Γ2
)
ε = ε
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2.3 Supersymmetry
Witten’s baryon vertex: 12
(D5-brane) × 12
(N F1) = 14
total
Generalised vertex: N F1 break 12
(no influence of F =√
2ndB)D5-brane supersymmetry due to F =
√2ndB ?
D5-brane κ-symmetry:
Γnε = L−1BI
(
Γ(6) + F ∧ F Γ2
)
ε = ε
in S5 fibre coordinates:
Γnε = L−1BI
(
Γijkl + (F ∧ F )ijkl
)
Γtχε = ε
→ Operator with Tr(Γn) = 0 and Γ2n = l1 in AdS5 × S5
[Bergshoeff, Kallosh, Ortın, Papadopoulos]
→ Generalised baryon vertex breaks preserves same susy as original
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3 The microscopical description
3.1 Brief review of the dielectric effect
n coinciding D-branes =⇒ U(1)n → U(n) gauge enhancement [Witten]
=⇒ non-Abelian action [Myers]
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3 The microscopical description
3.1 Brief review of the dielectric effect
n coinciding D-branes =⇒ U(1)n → U(n) gauge enhancement [Witten]
=⇒ non-Abelian action [Myers]
SnD1 = −T1
∫
d2ξ STr
√
∣
∣
∣det
(
P [gµν + gµi(Q−1 − δ)ijgjkgkν ]
)
detQ∣
∣
∣
+ T1
∫
d2ξ STr
P [i(iX iX)C(4) − 12(iX iX)2C(4) ∧ F
with
Qij = δi
j + i[X i, Xk]gkj
(
(iX iX)C(4))
µν= 1
2[Xλ, Xρ]C
(4)ρλµν
F = 2∂A + i[A,A] (iX iX)2C(4) = 14[Xλ, Xρ][Xν, Xµ]C
(4)µνρλ
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12[Xλ, Xρ]C
(4)ρλµν is dipole coupling
• Flat space: n D1’s expand into fuzzy D3 [Myers]
Fuzzy D3
(S x R)2
D1n
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12[Xλ, Xρ]C
(4)ρλµν is dipole coupling
• Flat space: n D1’s expand into fuzzy D3 [Myers]
Fuzzy D3
(S x R)2
D1n
• AdS5 × S5: D1’s expand into fuzzy S5
→ Fuzzy S5 is Abelian U(1) fibre over fuzzy CP 2 [B.J., Lozano, Rodr.-Gomez]
NB: Chern-Simons couplings are zero ⇒ purely gravitational effect
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CP 2 is coset manifold SU(3)/U(2)
embedded in R8 via
8∑
i=1
xixi = 1
8∑
j,k=1
dijkxjxk =1√3xi
Fuzzy CP 2 generated by SU(3) generators T i in (anti-)fundamental repres
X i =T i
√
(2n − 2)/3[X i, Xj] =
if ijk
√
(2n − 2)/3Xk
[Alexanian, Balachandran, Immirzi, Ydri]
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CP 2 is coset manifold SU(3)/U(2)
embedded in R8 via
8∑
i=1
xixi = 18
∑
j,k=1
dijkxjxk =1√3xi
Fuzzy CP 2 generated by SU(3) generators T i in (anti-)fundamental repres
X i =T i
√
(2n − 2)/3[X i, Xj] =
if ijk
√
(2n − 2)/3Xk
[Alexanian, Balachandran, Immirzi, Ydri]
Substituting in D1 action:
SnD1 = −T1
∫
dtdχ u STr
l1 +L4
4(2n − 2)l1
EnD1 = 2πuT1
(
n +nL4
8(n − 1)
)
[
ED5 = 8π2uT5
(
n +L4
8
)
, T1 = 4π2T5
]
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3.2 F1’s in microscopic descriptionD5-brane in baryon vertex is expanded D1-branesWhere are F1’s that form vertex?
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3.2 F1’s in microscopic descriptionD5-brane in baryon vertex is expanded D1-branesWhere are F1’s that form vertex?
→ Chern-Simons coupling:
SCS = T1
∫
dtdχ STr
P [(iX iX)C(4)] − P [(iX iX)2C(4)] ∧ F
= −T1
4
∫
dtdχ STr
[X i, Xj][Xk, X l]G(5)χijklAt
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3.2 F1’s in microscopic descriptionD5-brane in baryon vertex is expanded D1-branesWhere are F1’s that form vertex?
→ Chern-Simons coupling:
SCS = T1
∫
dtdχ STr
P [(iX iX)C(4)] − P [(iX iX)2C(4)] ∧ F
= −T1
4
∫
dtdχ STr
[X i, Xj][Xk, X l]G(5)χijklAt
→ G(5)χijkl = L4fm
[ijfnkl]X
mXn
=L4T1
2(n − 1)
∫
dtdχ STr
At
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3.2 F1’s in microscopic descriptionD5-brane in baryon vertex is expanded D1-branesWhere are F1’s that form vertex?
→ Chern-Simons coupling:
SCS = T1
∫
dtdχ STr
P [(iX iX)C(4)] − P [(iX iX)2C(4)] ∧ F
= −T1
4
∫
dtdχ STr
[X i, Xj][Xk, X l]G(5)χijklAt
→ G(5)χijkl = L4fm
[ijfnkl]X
mXn
=L4T1
2(n − 1)
∫
dtdχ STr
At
→ A = At(t) l1dt
=n
n − 1NT1
∫
dt At
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3.2 F1’s in microscopic descriptionD5-brane in baryon vertex is expanded D1-branesWhere are F1’s that form vertex?
→ Chern-Simons coupling:
SCS = T1
∫
dtdχ STr
P [(iX iX)C(4)] − P [(iX iX)2C(4)] ∧ F
= −T1
4
∫
dtdχ STr
[X i, Xj][Xk, X l]G(5)χijklAt
→ G(5)χijkl = L4fm
[ijfnkl]X
mXn
=L4T1
2(n − 1)
∫
dtdχ STr
At
→ A = At(t) l1dt
=n
n − 1NT1
∫
dt At
⇒ N BI charges as n → ∞, cancelled by N extrnal F1’sB. Janssen (UGR) Madrid, 18 may 2006 22
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3.3 Comment on S-duality
AdS5 × S5 is S-duality invariant (φ = 0)→ distinction between D1 and F1 artificial.
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3.3 Comment on S-duality
AdS5 × S5 is S-duality invariant (φ = 0)→ distinction between D1 and F1 artificial.
→ construct baryon vertex from dielectric F1’s? [Brecher, B.J., Lozano]
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3.3 Comment on S-duality
AdS5 × S5 is S-duality invariant (φ = 0)→ distinction between D1 and F1 artificial.
→ construct baryon vertex from dielectric F1’s? [Brecher, B.J., Lozano]
SnF1 = −T1
∫
dτdσ STr
√
∣
∣
∣det
(
P [Eµν + Eµi(Q−1 − δ)ijEjkEkν ]
)
detQ∣
∣
∣
−T1
∫
dτdσ STr
P [B(2)] + iP [(iX iX)C(4)] − 1
2P [(iX iX)2B(6)]
with
Eµν = gµν + eφC(2)µν
Qij = δi
j + ie−φ[X i, Xk]Ekj
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3.3 Comment on S-duality
AdS5 × S5 is S-duality invariant (φ = 0)→ distinction between D1 and F1 artificial.
→ construct baryon vertex from dielectric F1’s? [Brecher, B.J., Lozano]
SnF1 = −T1
∫
dτdσ STr
√
∣
∣
∣det
(
P [Eµν + Eµi(Q−1 − δ)ijEjkEkν ]
)
detQ∣
∣
∣
−T1
∫
dτdσ STr
P [B(2)] + iP [(iX iX)C(4)] − 1
2P [(iX iX)2B(6)]
with
Eµν = gµν + eφC(2)µν
Qij = δi
j + ie−φ[X i, Xk]Ekj
→ EnF1 = 2πuT1
(
n +nL4
8(n − 1)
)
N D1 strings via WV scalar
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4 Conclusions
• Witten’s baryon vertex is D5-brane wrapped around S5 in AdS5 × S5
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4 Conclusions
• Witten’s baryon vertex is D5-brane wrapped around S5 in AdS5 × S5
• S5 S1
−→ CP 2 permits to add magnetic BI flux F =√
2n dB
⇒ Generalised baryon vertex
• Upperbound on n/N , related to string exclusion principle?
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4 Conclusions
• Witten’s baryon vertex is D5-brane wrapped around S5 in AdS5 × S5
• S5 S1
−→ CP 2 permits to add magnetic BI flux F =√
2n dB
⇒ Generalised baryon vertex
• Upperbound on n/N , related to string exclusion principle?
• Generalised baryon vertex 1/2 supersymmetric
• F =√
2n dB introduces n D1-branes in D5 worldvolume
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4 Conclusions
• Witten’s baryon vertex is D5-brane wrapped around S5 in AdS5 × S5
• S5 S1
−→ CP 2 permits to add magnetic BI flux F =√
2n dB
⇒ Generalised baryon vertex
• Upperbound on n/N , related to string exclusion principle?
• Generalised baryon vertex 1/2 supersymmetric
• F =√
2n dB introduces n D1-branes in D5 worldvolume
• Microscopic description: D1-branes expanding into S5
⇒ agreement for n 1
(Witten’s baryon vertex not included)
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OutlookFuzzy odd-spheres as Abelian fibre bundles over fuzzy bases
[B.J., Lozano, Rodr.-Gomez]
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OutlookFuzzy odd-spheres as Abelian fibre bundles over fuzzy bases
[B.J., Lozano, Rodr.-Gomez]
• W → Sn−2 in Sn ⊂ AdSm × Sn: genuine giant gravitons
• W → Sm−2 in AdSm ⊂ AdSm × Sn: dual giant gravitons
B. Janssen (UGR) Madrid, 18 may 2006 25
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OutlookFuzzy odd-spheres as Abelian fibre bundles over fuzzy bases
[B.J., Lozano, Rodr.-Gomez]
• W → Sn−2 in Sn ⊂ AdSm × Sn: genuine giant gravitons
• W → Sm−2 in AdSm ⊂ AdSm × Sn: dual giant gravitons
• W → S3 in AdS3 × S3 × T 4: black hole [Rodr.-Gomez]
• D1 → S5 in AdS5 × S5: baryon vertex
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OutlookFuzzy odd-spheres as Abelian fibre bundles over fuzzy bases
[B.J., Lozano, Rodr.-Gomez]
• W → Sn−2 in Sn ⊂ AdSm × Sn: genuine giant gravitons
• W → Sm−2 in AdSm ⊂ AdSm × Sn: dual giant gravitons
• W → S3 in AdS3 × S3 × T 4: black hole [Rodr.-Gomez]
• D1 → S5 in AdS5 × S5: baryon vertex
• W → S5 in AdS5 × S5: new giant graviton! [in preparation]
• D1 → S3 in S5 ⊂ AdS5 × S5
• D1 → S3 in AdS5 ⊂ AdS5 × S5
B. Janssen (UGR) Madrid, 18 may 2006 25
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OutlookFuzzy odd-spheres as Abelian fibre bundles over fuzzy bases
[B.J., Lozano, Rodr.-Gomez]
• W → Sn−2 in Sn ⊂ AdSm × Sn: genuine giant gravitons
• W → Sm−2 in AdSm ⊂ AdSm × Sn: dual giant gravitons
• W → S3 in AdS3 × S3 × T 4: black hole [Rodr.-Gomez]
• D1 → S5 in AdS5 × S5: baryon vertex
• W → S5 in AdS5 × S5: new giant graviton! [in preparation]
• D1 → S3 in S5 ⊂ AdS5 × S5
• D1 → S3 in AdS5 ⊂ AdS5 × S5
• W → S3 in AdS7 ⊂ AdS7 × S4
• W → S3 in S7 ⊂ AdS4 × S7: S3 is non-contractable!
• ...
B. Janssen (UGR) Madrid, 18 may 2006 25