Magnetic strings , M5 branes , and N=4 SYM on del Pezzo surfaces:
K-strings, D-branes, and the Gauge/Gravity Correspondence · 2007-12-16 · K-strings, D-branes,...
Transcript of K-strings, D-branes, and the Gauge/Gravity Correspondence · 2007-12-16 · K-strings, D-branes,...
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
K-strings, D-branes, and the Gauge/GravityCorrespondence
Kory Stiffler
University of Iowa Department of Physics & AstronomyDiffeomorphisms & Geometry Research Group
Miami 2007December 16, 2007
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
What are k-strings?
I Describe SU(M) gauge theory configurations
I Colorless combinations of SU(M) color sources
I k-string tensions are associated with each combination
I Commonly analyzed with lattice gauge theory
I Can predict baryon formation
The Goal of this presentation
I To show how we can use the gauge/gravity correspondenceto analyze k-string configurations.
I Compare results to lattice gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) 1-string [for a review see: Shiffman hep-ph/0510098v2]
I formed from SU(M)color-anti-color source pairs
I pulled a large distance Lapart
I color flux tube in betweenpair results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) 1-string [for a review see: Shiffman hep-ph/0510098v2]
I formed from SU(M)color-anti-color source pairs
I pulled a large distance Lapart
I color flux tube in betweenpair results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) 1-string [for a review see: Shiffman hep-ph/0510098v2]
I formed from SU(M)color-anti-color source pairs
I pulled a large distance Lapart
I color flux tube in betweenpair results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) k-string[for a review see: Shiffman hep-ph/0510098v2]
k = |l −m|
I l fundamental and manti-fundamental strings
I placed a parallel distanced << L apart
I SU(M) color flux tube inbetween pairs results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) k-string[for a review see: Shiffman hep-ph/0510098v2]
k = |l −m|
I l fundamental and manti-fundamental strings
I placed a parallel distanced << L apart
I SU(M) color flux tube inbetween pairs results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) k-string[for a review see: Shiffman hep-ph/0510098v2]
k = |l −m|
I l fundamental and manti-fundamental strings
I placed a parallel distanced << L apart
I SU(M) color flux tube inbetween pairs results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(M) k-string[for a review see: Shiffman hep-ph/0510098v2]
k = |l −m|
I l fundamental and manti-fundamental strings
I placed a parallel distanced << L apart
I SU(M) color flux tube inbetween pairs results
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
K-string Tensions FromLattice MQCD[for a review see: Shiffman hep-ph/0510098v2]
I Lattice MQCD predicts asine law for this k-stringtension:
Tk ∝ sin
(kπ
M
)
I defining M-ality as:
M-ality ≡ k Mod M
I Tk vanishes when M-alityvanishes
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
K-string Tensions FromLattice MQCD[for a review see: Shiffman hep-ph/0510098v2]
I Lattice MQCD predicts asine law for this k-stringtension:
Tk ∝ sin
(kπ
M
)
I defining M-ality as:
M-ality ≡ k Mod M
I Tk vanishes when M-alityvanishes
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
K-string Tensions FromLattice MQCD[for a review see: Shiffman hep-ph/0510098v2]
I Lattice MQCD predicts asine law for this k-stringtension:
Tk ∝ sin
(kπ
M
)
I defining M-ality as:
M-ality ≡ k Mod M
I Tk vanishes when M-alityvanishes
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
A Simple Example: An SU(3) 3-string
One one side
I 3 quarks
I k = 3
I M-ality is zero
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
A Simple Example: An SU(3) 3-string
One one side
I 3 quarks
I k = 3
I M-ality is zero
I SU(3) 3-string tensionvanishes
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(3) 3-string: A Familiar Picture FromThe Standard Model
Three quark anti-quark pairspulled a large distance L >> dapart
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(3) 3-string: A Familiar Picture FromThe Standard Model
Three quark anti-quark pairspulled a large distance L >> dapart
I A Baryon and anAnti-Baryon form
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The SU(3) 3-string: A Familiar Picture FromThe Standard Model
Three quark anti-quark pairspulled a large distance L >> dapart
I A Baryon and anAnti-Baryon form
I Tension vanishes
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dp-branes and String Charge Conservation[Zwiebach A First Course in String Theory 2005]
Dp-branes gives rise to
I A new U(1) gaugeinvariant field
Fab = Bab + Fab/T0
a, b = 0 . . . p
I Such that string current isconserved
I M stacked D-branes giverise to U(1)× SU(M)gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dp-branes and String Charge Conservation[Zwiebach A First Course in String Theory 2005]
Dp-branes gives rise to
I A new U(1) gaugeinvariant field
Fab = Bab + Fab/T0
a, b = 0 . . . p
I Such that string current isconserved
I M stacked D-branes giverise to U(1)× SU(M)gauge theory
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dp-brane World Volume: A Flux Tube forK-strings
From the Dp-branes perspective:
I SU(M) gauge theory on itsworld volume
I Sourced by stringsendpoints
I This looks like the k-stringproblem
I Dp-brane acts as flux tube
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dp-brane World Volume: A Flux Tube forK-strings
From the Dp-branes perspective:
I SU(M) gauge theory on itsworld volume
I Sourced by stringsendpoints
I This looks like the k-stringproblem
I Dp-brane acts as flux tube
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The AdS/CFT and Gauge/GravityCorrespondences [Maldacena hep-th/9711200v3, hep-th/0309246v5; Klebanov hep-th/0009139v2]
M stacked D3-branes give rise to:
I N = 4 supersymmetric SU(M) gauge theory on worldvolume of the cooincident D3-branes.
I Curved background which is asymptotically(r → 0)AdS5 × S5
ds2 =
(1 +
L4
r4
)−1/2 (−dt2 + (dx1)2 + (dx2)2 + (dx3)2
)+
(1 +
L4
r4
)1/2 (dr2 + r2dΩ2
5
)I The AdS/CFT correspondence for D3-branes
Stacking D-branes in other ways can result in gauge/gravitycorrespondence
I Gauge theories more complex than SU(M)
I Curved backgrounds which are NOT necessarilyasymptotically AdS5 × S5
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The AdS/CFT and Gauge/GravityCorrespondences [Maldacena hep-th/9711200v3, hep-th/0309246v5; Klebanov hep-th/0009139v2]
M stacked D3-branes give rise to:
I N = 4 supersymmetric SU(M) gauge theory on worldvolume of the cooincident D3-branes.
I Curved background which is asymptotically(r → 0)AdS5 × S5
ds2 =
(1 +
L4
r4
)−1/2 (−dt2 + (dx1)2 + (dx2)2 + (dx3)2
)+
(1 +
L4
r4
)1/2 (dr2 + r2dΩ2
5
)I The AdS/CFT correspondence for D3-branes
Stacking D-branes in other ways can result in gauge/gravitycorrespondence
I Gauge theories more complex than SU(M)
I Curved backgrounds which are NOT necessarilyasymptotically AdS5 × S5
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The Klebanov Strassler (KS) Background[Klebanov, Strassler hep-th/0007191]
ds210 = h−1/2(τ)
[−(dX 0)2 +
3∑i=1
(dX i )2
]+ h1/2(τ)ds2
6
I Is result of warping from M D5-branes and N D3-branes
I Dual to N = 1 supersymmetric SU(N + M)× SU(N) gaugegroup
I Not asymptotically AdS5 × S5
I ds26 is the deformed conifold, R × S2 × S3
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Deformed Conifold [Klebanov, Strassler hep-th/0007191]
ds26 =
1
2ε4/3K (τ)
[1
3K 3(τ)[dτ 2 + (g5)2] + [(g3)2 + (g4)2] cosh2
(τ2
)+ [(g1)2 + (g2)2] sinh2
(τ2
)]
g1 =1√2
[−sinθ1dφ1 − cosψsinθ2dφ2 + sinψdθ2]
g2 =1√2
[dθ1 − sinψsinθ2dφ2 − cosψdθ2]
g3 =1√2
[−sinθ1dφ1 + cosψsinθ2dφ2 − sinψdθ2]
g4 =1√2
[dθ1 + sinψsinθ2dφ2 + cosψdθ2]
g5 = dψ + cosθ1dφ1 + cosθ2dφ2
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
At the Tip of the Deformed Conifold
Figure: Topology of deformed conifold[Kachru TASI 2007]
We stack M D5-branes and N D3-branes at the tip of thedeformed conifold, τ = 0
I Here, two dimensions of the D5-branes shrink to zero sizeI M D5-branes become M fractional D3-branesI SU(N + M)× SU(N) becomes SU(M) [Klebanov, Strassler hep-th/0007191;
Klebanov, Tsetylin hep-th/0002159 v2]
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The KS Background [Klebanov, Strassler hep-th/0007191]
ds210 = h−1/2(τ)
[−(dX 0)2 +
3∑i=1
(dX i )2
]
+ h1/2(τ)1
2ε4/3K (τ)
[1
3K 3(τ)[dτ 2 + (g5)2]
+ [(g3)2 + (g4)2] cosh2(τ
2
)+ [(g1)2 + (g2)2] sinh2
(τ2
)]
We will use the KS background, evaluated at τ = 0 in . . .
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
The KS Background [Klebanov, Strassler hep-th/0007191]
ds210 = h−1/2(τ)
[−(dX 0)2 +
3∑i=1
(dX i )2
]
+ h1/2(τ)1
2ε4/3K (τ)
[1
3K 3(τ)[dτ 2 + (g5)2]
+ [(g3)2 + (g4)2] cosh2(τ
2
)+ [(g1)2 + (g2)2] sinh2
(τ2
)]We will use the KS background, evaluated at τ = 0 in . . .
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dirac-Born-Infeld (DBI) action: Calculating thek-string Tension
Here we use D3-branes:
SDBI3 = − T 20
2πgs
∫d4ξ
√−det(gab + Fab)
+T 2
0
2π
∫exp(F) ∧
∑q
Cq + Sf
All fermionic degrees of freedom, Θ1, are in Sf .[Martucci, et al. hep-th/0504041]
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dirac-Born-Infeld (DBI) action: Calculating thek-string Tension
Here we use D3-branes:
SDBI3 = − T 20
2πgs
∫d4ξ
√−det(gab + Fab)
+T 2
0
2π
∫exp(F) ∧
∑q
Cq + Sf︸︷︷︸0
Choosing Θ1 = 0 =⇒ Sf = 0.[Martucci, et al. hep-th/0504041]
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dirac-Born-Infeld (DBI) action: Calculating thek-string Tension
Here we use D3-branes:
SDBI3 = − T 20
2πgs
∫d4ξ
√−det(gab + Fab)
+T 2
0
2π
∫exp(F) ∧
∑q
Cq
Choosing Θ1 = 0 =⇒ Sf = 0.[Martucci, et al. hep-th/0504041]
D3-brane world volume parametrization
ξ =
txθφ
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dirac-Born-Infeld (DBI) action: Calculating thek-string Tension
Here we use D3-branes:
SDBI3 = − T 20
2πgs
∫d4ξ
√−det(gab + Fab)
+T 2
0
2π
∫exp(F) ∧
∑q
Cq
Choosing Θ1 = 0 =⇒ Sf = 0.[Martucci, et al. hep-th/0504041]
D3-brane world volume reparametrization
ξ =
t = X 0
x = X 1
θ = θ1
φ = −φ2
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Dirac-Born-Infeld (DBI) action: Calculating thek-string Tension
Here we use D3-branes:
SDBI3 = − T 20
2πgs
∫d4ξ
√−det(gab + Fab)
+T 2
0
2π
∫exp(F) ∧
∑q
Cq
Choosing Θ1 = 0 =⇒ Sf = 0.[Martucci, et al. hep-th/0504041]
D3-brane world volume reparametrization and solution choices
ξ =
t = X 0
x = X 1
θ = θ1 = θ2
φ = −φ2 = φ1
X 2 = 0X 3 = 0τ = 0ψ = constant
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Hamiltonian for D3-branes in KS Background[Herzog, Klebanov hep-th/0111078; Firouzjahi, Leblond, Tye hep-th/0603161v1]
I In this Background, Fab = Fab/T0
I Choose the U(1) gauge potential A0 = 0
I Investigate solutions where Ftx is only non-vanishing U(1)gauge field
I Leaves only one conjugate variable
D =∂L∂Ftx
I Hamiltonian is a Legendre transformed DBI action
H = h(0)−1/2√
∆2 + T 20 (D − Ω)2
∆ =2T 2
0
gsh(0)1/2ε4/3K (0)cos2
(ψ
2
)Ω =
M
2π(ψ + sinψ)
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Hamiltonian for D3-branes in KS Background[Herzog, Klebanov hep-th/0111078; Firouzjahi, Leblond, Tye hep-th/0603161v1]
I In this Background, Fab = Fab/T0
I Choose the U(1) gauge potential A0 = 0
I Investigate solutions where Ftx is only non-vanishing U(1)gauge field
I Leaves only one conjugate variable
D =∂L∂Ftx
I Hamiltonian is a Legendre transformed DBI action
H = h(0)−1/2√
∆2 + T 20 (D − Ω)2
∆ =2T 2
0
gsh(0)1/2ε4/3K (0)cos2
(ψ
2
)Ω =
M
2π(ψ + sinψ)
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
K-string Tension From D3-brane Hamiltonian
Minimizing the Hamiltonian with respect to the remaining field,ψ, gives us approximately the k-string tension:
Tk ≈MT0
h(0)1/2πcos
(Dπ
M
)=
MT0
h(0)1/2πsin
(kπ
M
)where setting D = k −M/2 has reproduced the sine law, whichagain vanishes for vanishing M-ality
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fluctuations from the Classical Solutions
I Lattice QCD can calculate corrections to the k-stringtension: Luscher terms.
I We fluctuate around our classical string theory solution
Xµ = Xµ0 + λ δXµ(ξ) µ = 0 . . . 9
Aa = Aa0 + λ δAa(ξ) a = t, x , θ, φ
Θ1 = 0 + λ δΘ1
leading to solutions which oscillate with eigenmodes aroundthe classical solution.
I These eigenmodes give us the k-string corrections; compareto Luscher terms.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fluctuations from the Classical Solutions
I Lattice QCD can calculate corrections to the k-stringtension: Luscher terms.
I We fluctuate around our classical string theory solution
Xµ = Xµ0 + λ δXµ(ξ) µ = 0 . . . 9
Aa = Aa0 + λ δAa(ξ) a = t, x , θ, φ
Θ1 = 0 + λ δΘ1
leading to solutions which oscillate with eigenmodes aroundthe classical solution.
I These eigenmodes give us the k-string corrections; compareto Luscher terms.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fluctuations from the Classical Solutions
I Lattice QCD can calculate corrections to the k-stringtension: Luscher terms.
I We fluctuate around our classical string theory solution
Xµ = Xµ0 + λ δXµ(ξ) µ = 0 . . . 9
Aa = Aa0 + λ δAa(ξ) a = t, x , θ, φ
Θ1 = 0 + λ δΘ1
leading to solutions which oscillate with eigenmodes aroundthe classical solution.
I These eigenmodes give us the k-string corrections; compareto Luscher terms.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fluctuations from the Classical Solutions
Recalling our specific classical solutions, we acquire the shiftedbosons
X 2 = λδX 2, X 3 = λδX 3
θm = λδθm, φp = λδφp
ψ = ψ0 + λδψ, τ = λδτ
and U(1) gauge potentials
Ax = F 0tx t + λδAx
Aθ = λδAθ, Aφ = λδAφ
and the 32 component fermionic spinor
Θ1 = λδΘ1
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fluctuations from the Classical Solutions
Recalling our specific classical solutions, we acquire the shiftedbosons
X 2 = λδX 2, X 3 = λδX 3
θm = λδθm, φp = λδφp
ψ = ψ0 + λδψ, τ = λδτ
and U(1) gauge potentials
Ax = F 0tx t + λδAx
Aθ = λδAθ, Aφ = λδAφ
and the 32 component fermionic spinor
Θ1 = λδΘ1
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fluctuations from the Classical Solutions
Recalling our specific classical solutions, we acquire the shiftedbosons
X 2 = λδX 2, X 3 = λδX 3
θm = λδθm, φp = λδφp
ψ = ψ0 + λδψ, τ = λδτ
and U(1) gauge potentials
Ax = F 0tx t + λδAx
Aθ = λδAθ, Aφ = λδAφ
and the 32 component fermionic spinor
Θ1 = λδΘ1
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Shifted Action
Recall how action separated:
SDBI3 = Sb + Sf
Fluctuation from the classical solution results in:
Sb = Sb0 + λSb1 + λ2Sb2 + . . .
Sf = λ2Sf 2 + . . .
I Can analyze bosonic fluctuations separately from fermionicfluctuations
I Straightforward, but quite gruelling calculation at this point
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Shifted Action
Recall how action separated:
SDBI3 = Sb + Sf
Fluctuation from the classical solution results in:
Sb = Sb0 + λSb1 + λ2Sb2 + . . .
Sf = λ2Sf 2 + . . .
I Can analyze bosonic fluctuations separately from fermionicfluctuations
I Straightforward, but quite gruelling calculation at this point
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
BosonsAnalyzing the corrections to the bosonic action up to secondorder
Sb = Sb0 + λSb1 + λ2Sb2
gives us λSb1 + λ2Sb2 =
1
1536 Π T02 M -4 Λ
2 Cos@Λ ∆Θm@x, Θ, Φ, tDD Csch@Λ ∆Τ@x, Θ, Φ, tDD Sin@ΘD
H-Λ Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tD + Sinh@Λ ∆Τ@x, Θ, Φ, tDD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDLL
8 Λ I-∆AΘH0,0,1,0L@x, Θ, Φ, tD + ∆AΦ
H0,1,0,0L@x, Θ, Φ, tDM +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
HSinh@Λ ∆Τ@x, Θ, Φ, tDD - Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLIΛ Cos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD ∆ΘmH0,1,0,0L@x, Θ, Φ, tD + Λ ∆ΦpH0,0,1,0L@x, Θ, Φ, tD ICos@ΘD
Sin@Λ ∆Θm@x, Θ, Φ, tDD + Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM +
Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD I1 - Λ2
∆ΘmH0,0,1,0L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDMM
I∆ΦpH0,0,0,1L@x, Θ, Φ, tD ∆ΘmH1,0,0,0L@x, Θ, Φ, tD - ∆ΘmH0,0,0,1L@x, Θ, Φ, tD∆ΦpH1,0,0,0L@x, Θ, Φ, tDM +
4 Λ2 Cos@Λ ∆Θm@x, Θ, Φ, tDD Csch@Λ ∆Τ@x, Θ, Φ, tDD Sin@ΘD
H-Λ Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tD + Sinh@Λ ∆Τ@x, Θ, Φ, tDD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDLL
8 Λ I∆AΘH0,0,1,0L@x, Θ, Φ, tD - ∆AΦ
H0,1,0,0L@x, Θ, Φ, tDM +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
H-Sinh@Λ ∆Τ@x, Θ, Φ, tDD + Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLIΛ Cos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD ∆ΘmH0,1,0,0L@x, Θ, Φ, tD + Λ ∆ΦpH0,0,1,0L@x, Θ, Φ, tD ICos@ΘD
Sin@Λ ∆Θm@x, Θ, Φ, tDD + Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM +
Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD I1 - Λ2
∆ΘmH0,0,1,0L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDMM
I∆ΦpH0,0,0,1L@x, Θ, Φ, tD ∆ΘmH1,0,0,0L@x, Θ, Φ, tD - ∆ΘmH0,0,0,1L@x, Θ, Φ, tD∆ΦpH1,0,0,0L@x, Θ, Φ, tDM -
Λ2
-8 ∆AΘH0,0,0,1L@x, Θ, Φ, tD +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
HSinh@Λ ∆Τ@x, Θ, Φ, tDD - Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLI∆ΦpH0,0,0,1L@x, Θ, Φ, tD ICos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD +
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM -
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,0,0,1L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDM
I-2 Csch@Λ ∆Τ@x, Θ, Φ, tDD H-Λ Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tD +
Sinh@Λ ∆Τ@x, Θ, Φ, tDD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDLLICos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD + Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΦpH0,0,1,0L@x, Θ, Φ, tDM∆ΘmH1,0,0,0L@x, Θ, Φ, tD - ICos@2 ΘD + Cos@2 Λ ∆Θm@x, Θ, Φ, tDD - 2 Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD
Sin@ΘD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDL ∆ΘmH0,0,1,0L@x, Θ, Φ, tD + Λ Csch@Λ ∆Τ@x, Θ, Φ, tDD∆Τ@x, Θ, Φ, tD IHCos@2 ΘD - Cos@2 Λ ∆Θm@x, Θ, Φ, tDDL Cos@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD +
2 Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆ΘmH0,0,1,0L@x, Θ, Φ, tDMM
∆ΦpH1,0,0,0L@x, Θ, Φ, tDM + Λ2 8 ∆AΘ
H0,0,0,1L@x, Θ, Φ, tD +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
H-Sinh@Λ ∆Τ@x, Θ, Φ, tDD + Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLI∆ΦpH0,0,0,1L@x, Θ, Φ, tD ICos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD +
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM -
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,0,0,1L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDM
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
BosonsAnalyzing the corrections to the bosonic action up to secondorder
Sb = Sb0 + λSb1 + λ2Sb2
gives us λSb1 + λ2Sb2 =
1
1536 Π T02 M -4 Λ
2 Cos@Λ ∆Θm@x, Θ, Φ, tDD Csch@Λ ∆Τ@x, Θ, Φ, tDD Sin@ΘD
H-Λ Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tD + Sinh@Λ ∆Τ@x, Θ, Φ, tDD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDLL
8 Λ I-∆AΘH0,0,1,0L@x, Θ, Φ, tD + ∆AΦ
H0,1,0,0L@x, Θ, Φ, tDM +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
HSinh@Λ ∆Τ@x, Θ, Φ, tDD - Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLIΛ Cos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD ∆ΘmH0,1,0,0L@x, Θ, Φ, tD + Λ ∆ΦpH0,0,1,0L@x, Θ, Φ, tD ICos@ΘD
Sin@Λ ∆Θm@x, Θ, Φ, tDD + Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM +
Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD I1 - Λ2
∆ΘmH0,0,1,0L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDMM
I∆ΦpH0,0,0,1L@x, Θ, Φ, tD ∆ΘmH1,0,0,0L@x, Θ, Φ, tD - ∆ΘmH0,0,0,1L@x, Θ, Φ, tD∆ΦpH1,0,0,0L@x, Θ, Φ, tDM +
4 Λ2 Cos@Λ ∆Θm@x, Θ, Φ, tDD Csch@Λ ∆Τ@x, Θ, Φ, tDD Sin@ΘD
H-Λ Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tD + Sinh@Λ ∆Τ@x, Θ, Φ, tDD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDLL
8 Λ I∆AΘH0,0,1,0L@x, Θ, Φ, tD - ∆AΦ
H0,1,0,0L@x, Θ, Φ, tDM +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
H-Sinh@Λ ∆Τ@x, Θ, Φ, tDD + Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLIΛ Cos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD ∆ΘmH0,1,0,0L@x, Θ, Φ, tD + Λ ∆ΦpH0,0,1,0L@x, Θ, Φ, tD ICos@ΘD
Sin@Λ ∆Θm@x, Θ, Φ, tDD + Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM +
Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD I1 - Λ2
∆ΘmH0,0,1,0L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDMM
I∆ΦpH0,0,0,1L@x, Θ, Φ, tD ∆ΘmH1,0,0,0L@x, Θ, Φ, tD - ∆ΘmH0,0,0,1L@x, Θ, Φ, tD∆ΦpH1,0,0,0L@x, Θ, Φ, tDM -
Λ2
-8 ∆AΘH0,0,0,1L@x, Θ, Φ, tD +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
HSinh@Λ ∆Τ@x, Θ, Φ, tDD - Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLI∆ΦpH0,0,0,1L@x, Θ, Φ, tD ICos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD +
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM -
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,0,0,1L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDM
I-2 Csch@Λ ∆Τ@x, Θ, Φ, tDD H-Λ Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tD +
Sinh@Λ ∆Τ@x, Θ, Φ, tDD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDLLICos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD + Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΦpH0,0,1,0L@x, Θ, Φ, tDM∆ΘmH1,0,0,0L@x, Θ, Φ, tD - ICos@2 ΘD + Cos@2 Λ ∆Θm@x, Θ, Φ, tDD - 2 Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD
Sin@ΘD HΨ0 + Λ ∆Ψ@x, Θ, Φ, tDL ∆ΘmH0,0,1,0L@x, Θ, Φ, tD + Λ Csch@Λ ∆Τ@x, Θ, Φ, tDD∆Τ@x, Θ, Φ, tD IHCos@2 ΘD - Cos@2 Λ ∆Θm@x, Θ, Φ, tDDL Cos@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD +
2 Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD Sin@Ψ0 + Λ ∆Ψ@x, Θ, Φ, tDD ∆ΘmH0,0,1,0L@x, Θ, Φ, tDMM
∆ΦpH1,0,0,0L@x, Θ, Φ, tDM + Λ2 8 ∆AΘ
H0,0,0,1L@x, Θ, Φ, tD +
1
Π
gs M SechB1
2Λ ∆Τ@x, Θ, Φ, tDF
2
H-Sinh@Λ ∆Τ@x, Θ, Φ, tDD + Λ Cosh@Λ ∆Τ@x, Θ, Φ, tDD ∆Τ@x, Θ, Φ, tDLI∆ΦpH0,0,0,1L@x, Θ, Φ, tD ICos@ΘD Sin@Λ ∆Θm@x, Θ, Φ, tDD +
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,1,0,0L@x, Θ, Φ, tDM -
Λ Cos@Λ ∆Θm@x, Θ, Φ, tDD Sin@ΘD ∆ΘmH0,0,0,1L@x, Θ, Φ, tD ∆ΦpH0,1,0,0L@x, Θ, Φ, tDM
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
Here’s where the gruelling part comes into play.
I After a lot of work, things actually simplify to a nice form.
I The linear terms from Sb1 give us a simple flux constraintthrough the D3-branes:∫
d4ξ
[(k −M/2)sinθ
4π∂tδAx + c(ψ0)(∂φδθm − ∂θδφp)
]= 0
I The second order equations are consistent with introductionof a composite field Ψ such that
δψ = Ψ + cotθ∂θΨ
δθm =1
2cscθ∂φΨ
δφp = −1
2cscθ∂θΨ
which serves to simplify analysis.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
Here’s where the gruelling part comes into play.
I After a lot of work, things actually simplify to a nice form.
I The linear terms from Sb1 give us a simple flux constraintthrough the D3-branes:∫
d4ξ
[(k −M/2)sinθ
4π∂tδAx + c(ψ0)(∂φδθm − ∂θδφp)
]= 0
I The second order equations are consistent with introductionof a composite field Ψ such that
δψ = Ψ + cotθ∂θΨ
δθm =1
2cscθ∂φΨ
δφp = −1
2cscθ∂θΨ
which serves to simplify analysis.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
Here’s where the gruelling part comes into play.
I After a lot of work, things actually simplify to a nice form.
I The linear terms from Sb1 give us a simple flux constraintthrough the D3-branes:∫
d4ξ
[(k −M/2)sinθ
4π∂tδAx + c(ψ0)(∂φδθm − ∂θδφp)
]= 0
I The second order equations are consistent with introductionof a composite field Ψ such that
δψ = Ψ + cotθ∂θΨ
δθm =1
2cscθ∂φΨ
δφp = −1
2cscθ∂θΨ
which serves to simplify analysis.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
Here’s where the gruelling part comes into play.
I After a lot of work, things actually simplify to a nice form.
I The linear terms from Sb1 give us a simple flux constraintthrough the D3-branes:∫
d4ξ
[(k −M/2)sinθ
4π∂tδAx + c(ψ0)(∂φδθm − ∂θδφp)
]= 0
I The second order equations are consistent with introductionof a composite field Ψ such that
δψ = Ψ + cotθ∂θΨ
δθm =1
2cscθ∂φΨ
δφp = −1
2cscθ∂θΨ
which serves to simplify analysis.
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
With this simplification, the boson fields are described by twomassless scalars:
∇a∇aδX2 = 0
∇a∇aδX3 = 0
two massive scalars, one with an Electric source:
∇a∇aδτ −m2τ = 0
∇a∇aδΨ−m2Ψ = qδFtx
and a U(1) Maxwell equation:
∇aδFab = 4π
0J ∂tΨ00
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
With this simplification, the boson fields are described by twomassless scalars:
∇a∇aδX2 = 0
∇a∇aδX3 = 0
two massive scalars, one with an Electric source:
∇a∇aδτ −m2τ = 0
∇a∇aδΨ−m2Ψ = qδFtx
and a U(1) Maxwell equation:
∇aδFab = 4π
0J ∂tΨ00
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosons
With this simplification, the boson fields are described by twomassless scalars:
∇a∇aδX2 = 0
∇a∇aδX3 = 0
two massive scalars, one with an Electric source:
∇a∇aδτ −m2τ = 0
∇a∇aδΨ−m2Ψ = qδFtx
and a U(1) Maxwell equation:
∇aδFab = 4π
0J ∂tΨ00
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Bosonic Eigenmodes
All fields take a harmonic form:
e i(px−ωt)Y (lm)(θ, φ)
and the solutions yield eight bosonic eigenmodes:
ω = ±√
gxxR/2l(l + 1) + p2
ω = ±√
gxxR/2l(l + 1) + p2 + gxxm2τ
ω = ±√
c1/2±√
c2gxx/2
where
c1 = 2p2 − 4g2xxJπq + gxxR(l(l + 1)− 1)
c2 = 16Jπq(−p2 + g2xxJπq)− 8gxxJ(l(l + 1)− 1)πqR + R2
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
FermionsThe fermionic action looks like
Sf 2 =T 2
0
4πgs
∫d4ξeΦ
√− det(M0) δΘ1[
(M−1
0
)abΓb∂a
+ M1 + M2 + M3]δΘ1
where
M0 = g + F
M1 =(M−1
0
)abΓb
1
4Ω µν
a Γµν
M2 =
(∨Γ
)−1 (M−1
0
)abΓb
1
8eΦ 1
3!FµνρΓµνρΓa
M3 = − 1
4(3!)FµνρΓµνρ
∨Γ =
1
2Γ(0)
√− det(g)√
−det(g + F))ΓabFab
Γ0 =1
4!
εabcd√(− det(g))
Γabcd
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fermionic Eigenvalues
I From variation of the fermionic action, acquire a Dirac-typeequation:
[(M−1
0
)abΓb∂a + M1 + M2 + M3]δΘ1 = 0
I Solving this yields the eight eigenmodes:
ω =(∓8√
v2 ∓ v3 ± 32z41 Λ1)
√1− a2
1h0
32z41 h
1/40
+ v4
I This is the current stage of our research
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Fermionic Eigenvalues
I From variation of the fermionic action, acquire a Dirac-typeequation:
[(M−1
0
)abΓb∂a + M1 + M2 + M3]δΘ1 = 0
I Solving this yields the eight eigenmodes:
ω =(∓8√
v2 ∓ v3 ± 32z41 Λ1)
√1− a2
1h0
32z41 h
1/40
+ v4
I This is the current stage of our research
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Future Calculations: Corrections to K-stringtension
I Eigenmodes from bosonic sector
I Eigenvalues from the fermionic sector
I Calculate the corrections to the k-string tension
I Compare to lattice QCD luscher terms
I Find quantitative evidence of gauge/gravity correspondence
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Future Calculations: Corrections to K-stringtension
I Eigenmodes from bosonic sector
I Eigenvalues from the fermionic sector
I Calculate the corrections to the k-string tension
I Compare to lattice QCD luscher terms
I Find quantitative evidence of gauge/gravity correspondence
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Future Calculations: Corrections to K-stringtension
I Eigenmodes from bosonic sector
I Eigenvalues from the fermionic sector
I Calculate the corrections to the k-string tension
I Compare to lattice QCD luscher terms
I Find quantitative evidence of gauge/gravity correspondence
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Many Thanks
I The audience
I Prof. Thomas Curtright and the U
my thesis advisor
I Prof. Vincent Rodgers
our collaborator from the University of Michigan
I Leopoldo Pando-Zayas
and the rest of the University of Iowa D&G group
I Chris Doran
I Heather Bruch
I Xiaolong Liu
I Leo Rodriguez
I Tuna Yildirim
I Da Xu
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Many Thanks
I The audience
I Prof. Thomas Curtright and the U
my thesis advisor
I Prof. Vincent Rodgers
our collaborator from the University of Michigan
I Leopoldo Pando-Zayas
and the rest of the University of Iowa D&G group
I Chris Doran
I Heather Bruch
I Xiaolong Liu
I Leo Rodriguez
I Tuna Yildirim
I Da Xu
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Many Thanks
I The audience
I Prof. Thomas Curtright and the U
my thesis advisor
I Prof. Vincent Rodgers
our collaborator from the University of Michigan
I Leopoldo Pando-Zayas
and the rest of the University of Iowa D&G group
I Chris Doran
I Heather Bruch
I Xiaolong Liu
I Leo Rodriguez
I Tuna Yildirim
I Da Xu
K-strings, D-branes, andthe Gauge/Gravity
Correspondence
Kory Stiffler
Introduction
K-strings in latticeMQCD
K-strings fromGauge/GravityCorrespondence
Lowest Order K-stringTension fromGauge/GravityCorrespondence
New Results:Correctionsto the Lowest OrderK-string Tension
Acknowledgments
Many Thanks
I The audience
I Prof. Thomas Curtright and the U
my thesis advisor
I Prof. Vincent Rodgers
our collaborator from the University of Michigan
I Leopoldo Pando-Zayas
and the rest of the University of Iowa D&G group
I Chris Doran
I Heather Bruch
I Xiaolong Liu
I Leo Rodriguez
I Tuna Yildirim
I Da Xu