Updates on K-strings from the Supersymmetric D-brane ...

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Updates on K-strings from the Supersymmetric D-brane

Perspective

Kory Stiffler Indiana University Northwest

Miami Conference 2015: 100 Years of GR 12/21/2015

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Outline

•  What is a k-string? •  Gauge/Gravity Duality •  Supersymmetric Probe D-branes

•  Bosonic Modes: Lüscher term •  Fermionic Modes: ?

•  Conclusions

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What is a k-string?

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Lüscher’s Fundamental String (Lüscher, 1981)

�  quark-antiquark pair tied together by a string

�  Forms color flux tube of energy:

•  T: tension

•  d: space-time dimension

•  L: quark-antiquark separation

2

( 2) 124dE T L OL L

πβ

− ⎛ ⎞= − + + ⎜ ⎟⎝ ⎠

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SU(N) k-strings: Tension Energy:

Tensionvanisheswhenk=0,N(N-alityk)

Exhibitedbymodelsin•  la;cegaugetheory•  analy>cHamiltonianmethods•  stringtheoryusinggauge/gravityduali>es

sin sine law

casimir lawk

kNNT

N kkN

π⎧⎪⎪

∝ ⎨−⎪

⎪⎩

2

1 , | |k kE T L O k l mL Lα

β ⎛ ⎞= + + + = −⎜ ⎟⎝ ⎠

Lüscherstring k-string:mul>pleLüscherstrings,d <<L,k= |l -m |

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N-ality k

•  N=3 •  k=3-string

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N-ality k

•  N=3 •  k=3-string •  Baryon & anti-baryon form

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N-ality k

•  N=3 •  k=3-string •  Baryon & anti-baryon form •  Tension vanishes •  Behavior called 3-ality 3 •  Generally: N-ality k

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Energy:

Lattice calculations (0802.1490,1007.4720):

String theory: Dp-brane probes of Supergravity (0909.5681, 1012.0021, 1209.5149):

These match for

SU(N) k-strings: Lüscher Term

Lüscherstring k-string:mul>pleLüscherstrings,d <<L,k= |l -m |

2


β ⎛ ⎞= + + + = −⎜ ⎟⎝ ⎠

α

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Gauge/Gravity Duality k-strings = D-branes

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Closed strings = Supergravity(SUGRA) Open strings + Dp-branes = gauge theory •  Dp-branes: membranes on which

strings can end •  p = spatial dimension of the brane.

Gauge/Gravity Duality

Open strings attached to D2-branes

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Gauge/Gravity Duality Open strings + D-branes = gauge theory

Gauge flux flows between string to D-brane:

D-brane itself IS the color flux tube

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Gauge/Gravity Duality D-branes = k-strings?

Gauge flux flows between string and D-brane:

D-brane ~ SU(N) k-strings

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Gauge/Gravity Duality Open strings + D-branes = U(N) gauge theory

1 D-brane = U(1) gauge theory •  1 distinct way to connect string to the brane •  Dual to gauge theory with adjoint

representation of dimension 1 => U(1)

3 ‘stacked’ D-branes => U(3) gauge theory •  The D-branes are stacked in the sense that they are

infinitesimally close together. •  32=9 distinct ways to connect strings to the branes •  Dual to gauge theory with adjoint representation of

dimension 32=9 => U(3)

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Gauge/Gravity Duality Open strings + D-branes = U(N) gauge theory

N ‘stacked’ D-branes => U(N) gauge theory •  N2 distinct ways to connect strings to the branes •  Dual to gauge theory with adjoint representation of

dimension N2=> U(N)

…

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Gauge/Gravity Duality Open to Closed strings: D-branes as a Supergravity Source

N stacked D-branes = U(N) gauge theory

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N stacked D-branes = U(N) gauge theory •  Naturally couple to Ramond-Ramond (R-R) flux (closed strings)

Ex:

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N stacked D-branes = U(N) gauge theory •  Naturally couple to Ramond-Ramond (R-R) flux (closed strings) •  Closed strings = supergravity (low energy effective theory)

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1) SUGRA dual to supersymmetric SU(N) theory 2) Probe with D-brane wrapped with flux 3) Calculate for several years Result: The Probe brane exhibits behavior of a k-string

Gauge/Gravity Duality Supergravity instructions for a k-string

~

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What has been done SUGRA Background Dimension of dual

gauge theory Dp-brane probe

Klebanov-Strassler (KS) d=3+1 p=3

Cvetic, Gibbons, Lu, and Pope (CGLP)

d=2+1 p=4

Maldacena-Nunez (MN) d=3+1 p=3, p=5

Maldacena-Nastase (MNa) d=2+1 p=3, p=5

2


β ⎛ ⎞= + + + = −⎜ ⎟⎝ ⎠

Dp-brane bosonic Energy:

sin sine law

casimir lawk

kNNT

N kkN

π⎧⎪⎪

∝ ⎨−⎪

⎪⎩

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Supersymmetric Probe Branes

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Building a k-string

ρ

1)  MN/MNa SUGRA background: dual to SU(N) theory

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Building a k-string

Dilaton:


ρ

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ρ

Building a k-string

Dilaton:


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Building a k-string

Dilaton:


ρ

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Building a k-string 1)  MN/MNa SUGRA background: dual to SU(N) theory

ρ

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Building a k-string 1)  MN/MNa SUGRA background: dual to SU(N) theory

ρ

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Building a k-string 1)  MN SUGRA background: dual to SU(N) in 3+1

X=(t,x,y,z,ρ,θ1,θ2,φ1,φ2,ψ)

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Building a k-string 1)  MNa SUGRA background: dual to SU(N) in 2+1

X=(t,x,y,ψ1,ρ,θ1,θ2,φ1,φ2,ψ2)

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Building a k-string 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action

X(ξ),Θ(ξ)

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Energy 3) Probe brane energy ~ k-string energy

~

Ek =Tk L+αL+β +O 1

L2⎛

⎝⎜

⎞

⎠⎟

X(ξ),Θ(ξ)

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Classical Energy 3) Probe brane energy ~ k-string energy

~


L2⎛

⎝⎜

⎞

⎠⎟

X(ξ),Θ(ξ)

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~



L2⎛

⎝⎜

⎞

⎠⎟

X(ξ)

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X=(t,x,0,0,0,θ,θ,φ,φ,ψ0)


L2⎛

⎝⎜

⎞

⎠⎟

p = 3

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L2⎛

⎝⎜

⎞

⎠⎟

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L2⎛

⎝⎜

⎞

⎠⎟

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L2⎛

⎝⎜

⎞

⎠⎟

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3) Probe brane energy ~ k-string energy

~

Classical Energy

Tk∝N sinkπN

X=(t,x,0,0,0,θ,θ,φ,φ,ψ0)


L2⎛

⎝⎜

⎞

⎠⎟

D3-brane probe of MN/MNa Background:

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~


Bosonic One-Loop Energy

Luscher term

X+δx


L2⎛

⎝⎜

⎞

⎠⎟

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X+δx

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Regularize

X+δx

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L2⎛

⎝⎜

⎞

⎠⎟

D3-brane one-loop bosonic energy in MN/MNa

k-string energy

~

X+δx

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Fermionic One-Loop Energy 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action

X,Θ

=?

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X,Θ

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X,Θ

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X,Θ

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X,Θ

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Fermionic One-Loop Energy 3) Probe brane fermionic energy ~ ?

~ ???

X,Θ

=?

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Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated in MN/MNa.

ForKlebanov-Strasslerbackgroundfound:

=?

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Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated in MN/MNa.

ForKlebanov-Strasslerbackgroundfound:

=?

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Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated.

=?

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Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated.

=?

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Fermionic One-Loop Energy 3) Probe brane fermionic energy ~ ?

~ ???

X,Θ

=?

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Summary •  K-strings: color flux tubes •  Branes

•  Source supergravity •  Endowed with gauge theories

•  k-strings/D-branes: gauge/gravity duals •  Classical solution: tension •  Bosonic fluctuations

•  Regularize •  Lüscher term universality

•  Fermionic fluctuations •  Regularize? •  ?

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