K-strings, D-branes, and the Gauge/Gravity Correspondence · 2007-12-16 · K-strings, D-branes, and the Gauge/Gravity Correspondence Kory Sti er Introduction K-strings in lattice

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


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Kory Stiffler

Introduction





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


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Kory Stiffler

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Acknowledgments

What are k-strings?










Correspondence

Kory Stiffler

Introduction





Acknowledgments

What are k-strings?










Correspondence

Kory Stiffler

Introduction





Acknowledgments

What are k-strings?










Correspondence

Kory Stiffler

Introduction





Acknowledgments

What are k-strings?










Correspondence

Kory Stiffler

Introduction





Acknowledgments

What are k-strings?










Correspondence

Kory Stiffler

Introduction





Acknowledgments

What are k-strings?










Correspondence

Kory Stiffler

Introduction





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


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Kory Stiffler

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Kory Stiffler

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Kory Stiffler

Introduction





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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


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k = |l −m|





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Kory Stiffler

Introduction





Acknowledgments


k = |l −m|





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Kory Stiffler

Introduction





Acknowledgments


k = |l −m|





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Kory Stiffler

Introduction





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


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Tk ∝ sin

(kπ

M

)


M-ality ≡ k Mod M



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Introduction





Acknowledgments



Tk ∝ sin

(kπ

M

)


M-ality ≡ k Mod M



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Acknowledgments

A Simple Example: An SU(3) 3-string

One one side

I 3 quarks

I k = 3

I M-ality is zero


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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


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Acknowledgments

The SU(3) 3-string: A Familiar Picture FromThe Standard Model

Three quark anti-quark pairspulled a large distance L >> dapart


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I A Baryon and anAnti-Baryon form


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I A Baryon and anAnti-Baryon form

I Tension vanishes


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Kory Stiffler

Introduction





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


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Kory Stiffler

Introduction





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


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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


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Introduction





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


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


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Introduction





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


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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]


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Kory Stiffler

Introduction





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 . . .


Correspondence

Kory Stiffler

Introduction





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 . . .


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Kory Stiffler

Introduction





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]


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Acknowledgments



SDBI3 = − T 20

2πgs

∫d4ξ


+T 2

0

2π

∫exp(F) ∧

∑q

Cq + Sf︸︷︷︸0

Choosing Θ1 = 0 =⇒ Sf = 0.[Martucci, et al. hep-th/0504041]


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Introduction





Acknowledgments



SDBI3 = − T 20

2πgs

∫d4ξ


+T 2

0

2π

∫exp(F) ∧

∑q

Cq


D3-brane world volume parametrization

ξ =

txθφ


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Kory Stiffler

Introduction





Acknowledgments



SDBI3 = − T 20

2πgs

∫d4ξ


+T 2

0

2π

∫exp(F) ∧

∑q

Cq


D3-brane world volume reparametrization

ξ =

t = X 0

x = X 1

θ = θ1

φ = −φ2


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Kory Stiffler

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SDBI3 = − T 20

2πgs

∫d4ξ


+T 2

0

2π

∫exp(F) ∧

∑q

Cq


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


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Kory Stiffler

Introduction





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)


Correspondence

Kory Stiffler

Introduction





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)


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Kory Stiffler

Introduction





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


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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.


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Xµ = Xµ0 + λ δXµ(ξ) µ = 0 . . . 9


Θ1 = 0 + λ δΘ1




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Xµ = Xµ0 + λ δXµ(ξ) µ = 0 . . . 9


Θ1 = 0 + λ δΘ1




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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


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X 2 = λδX 2, X 3 = λδX 3







Θ1 = λδΘ1


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Acknowledgments



X 2 = λδX 2, X 3 = λδX 3







Θ1 = λδΘ1


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Introduction





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


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Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


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


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




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


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


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 +




Correspondence

Kory Stiffler

Introduction





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


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


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




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


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


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




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


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 +



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


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 +




Correspondence

Kory Stiffler

Introduction





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.


Correspondence

Kory Stiffler

Introduction





Acknowledgments

Bosons




d4ξ

[(k −M/2)sinθ


]= 0



δθm =1

2cscθ∂φΨ

δφp = −1

2cscθ∂θΨ



Correspondence

Kory Stiffler

Introduction





Acknowledgments

Bosons




d4ξ

[(k −M/2)sinθ


]= 0



δθm =1

2cscθ∂φΨ

δφp = −1

2cscθ∂θΨ



Correspondence

Kory Stiffler

Introduction





Acknowledgments

Bosons




d4ξ

[(k −M/2)sinθ


]= 0



δθm =1

2cscθ∂φΨ

δφp = −1

2cscθ∂θΨ



Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





Acknowledgments

Bosons


∇a∇aδX2 = 0

∇a∇aδX3 = 0





∇aδFab = 4π

0J ∂tΨ00


Correspondence

Kory Stiffler

Introduction





Acknowledgments

Bosons


∇a∇aδX2 = 0

∇a∇aδX3 = 0





∇aδFab = 4π

0J ∂tΨ00


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





Acknowledgments








Correspondence

Kory Stiffler

Introduction





Acknowledgments








Correspondence

Kory Stiffler

Introduction





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


Correspondence

Kory Stiffler

Introduction





Acknowledgments

Many Thanks

I The audience


my thesis advisor





I Chris Doran

I Heather Bruch

I Xiaolong Liu

I Leo Rodriguez

I Tuna Yildirim

I Da Xu


Correspondence

Kory Stiffler

Introduction





Acknowledgments

Many Thanks

I The audience


my thesis advisor





I Chris Doran

I Heather Bruch

I Xiaolong Liu

I Leo Rodriguez

I Tuna Yildirim

I Da Xu


Correspondence

Kory Stiffler

Introduction





Acknowledgments

Many Thanks

I The audience


my thesis advisor





I Chris Doran

I Heather Bruch

I Xiaolong Liu

I Leo Rodriguez

I Tuna Yildirim

I Da Xu

Documents

K-strings, D-branes, and the Gauge/Gravity Correspondence · 2007-12-16 · K-strings, D-branes, and the Gauge/Gravity Correspondence Kory Sti er Introduction K-strings in lattice