AdS/CFT
FIAS Frankfurt, HGS-HIRe Powerweek, June 27-30th 2011
by Matthias Kaminski (Princeton University)
Introduction to Gauge/Gravity Correspondence & Heavy-Ion-Applications
QFT Gravity
AdS/CFT
FIAS Frankfurt, HGS-HIRe Powerweek, June 27-30th 2011
by Matthias Kaminski (Princeton University)
Introduction to Gauge/Gravity Correspondence & Heavy-Ion-Applications
QFT Gravity
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 2
Lecture I & II: Introduction to Gauge/Gravity
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 3
Invitation: Gauge/Gravity Correpondence-Reasons to ignore the correspondence
1. String Theory may not describe our nature.
2. Gauge/Gravity Correspondence may be wrong. It is a conjecture, which is not proven in general, only in special cases. proof
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 3
Invitation: Gauge/Gravity Correpondence-Reasons to ignore the correspondence
1. String Theory may not describe our nature.
2. Gauge/Gravity Correspondence may be wrong. It is a conjecture, which is not proven in general, only in special cases.
➡ Foundations of Gauge/Gravity Correspondence may be unphysical since it arises in the context of String Theory, and in addition it may be mathematically wrong.
proof
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 4
So, why am I here at 9 a.m. on a Monday morning?
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
➡ ~10% of String Theory “are” AdS/CFT
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
➡ ~10% of String Theory “are” AdS/CFT
➡ AdS/CFT is a factor 10 less “important” than QCD
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
➡ ~10% of String Theory “are” AdS/CFT
➡ AdS/CFT is a factor 10 less “important” than QCD
➡ String Theory is almost as “important” as QCD
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
➡ ~10% of String Theory “are” AdS/CFT
➡ AdS/CFT is a factor 10 less “important” than QCD
➡ String Theory is almost as “important” as QCD
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 6
Proofs of Gauge/Gravity Correspondences-Some examples
Three-point functions of N=4 Super-Yang-Mills theory
Conformal anomaly of the same theory
RG flows away from most symmetric case
... many other symmetric instances of the correspondence
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Transport coefficients (e.g. ‘universal’ viscosity bound)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Transport coefficients (e.g. ‘universal’ viscosity bound)
Model QCD equation of state
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Transport coefficients (e.g. ‘universal’ viscosity bound)
Model QCD equation of state
Deconfinement & Break: Chiral, Conformal, SUSY
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Transport coefficients (e.g. ‘universal’ viscosity bound)
Model QCD equation of state
Deconfinement & Break: Chiral, Conformal, SUSY
Condensed matter applications (strongly corr. electrons)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Transport coefficients (e.g. ‘universal’ viscosity bound)
Model QCD equation of state
Deconfinement & Break: Chiral, Conformal, SUSY
Condensed matter applications (strongly corr. electrons)
[AdS/QCD (bottom-up approach) distinct from string constr.]
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
Compute observables in strongly coupled QFTs
Meson spectra/melting, glueball spectra
Quark energy loss, Jets
Thermodynamics/Phase diagrams
Hydrodynamics (beyond Muller-Israel-Stewart), chiral effects
Transport coefficients (e.g. ‘universal’ viscosity bound)
Model QCD equation of state
Deconfinement & Break: Chiral, Conformal, SUSY
Condensed matter applications (strongly corr. electrons)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 8
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
e.g. thermal spectral function for flavor current in a hot and dense charged plasma
e.g. phase diagrams of fundamental matter (quarks) in a hot and dense plasma carryingisospin charge
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 8
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
superfluid
e.g. thermal spectral function for flavor current in a hot and dense charged plasma
e.g. phase diagrams of fundamental matter (quarks) in a hot and dense plasma carryingisospin charge
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 9
Low shear viscosity
Theory/Model Reference
Lattice QCD
Hydro (Glauber)
Hydro (CGC)
Viscous Hydro (Glauber)
!/s
[Meyer, 2007]0.134(33)
0.19
0.11
[Drescher et al., 2007]
[Drescher et al., 2007]
[Romatschke et al.,2007]0.08 , 0.16, {0.03}
!
s!
1
4"" 0.08Gauge/Gravity: [Policastro, Son, Starinets, 2001]
✓ Correct prediction!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 10
Gauge/Gravity is a Powerful Tool
non-perturbative results, strong coupling
treat many-body systems
direct computations in real-time thermal QFT (transport)
no sign-problem at finite charge densities
methods often just require solving ODEs in classical gravity
quick numerical computations (~few seconds on a laptop)
(turn around: study strongly coupled gravity)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 11
Invitation: Gauge/Gravity Correpondence-Less pessimistic view:
1. String Theory may not describe our nature uniquely. But it is mathematically correct.
2. Gauge/Gravity Correspondence is a mathematical map that is conjectured from the correct mathematical framework of String Theory.
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 11
Invitation: Gauge/Gravity Correpondence-Less pessimistic view:
1. String Theory may not describe our nature uniquely. But it is mathematically correct.
2. Gauge/Gravity Correspondence is a mathematical map that is conjectured from the correct mathematical framework of String Theory.
➡ Gauge/Gravity Correspondence may be a mathematically correct map, which relates particular quantum field theories to particular gravity theories (assuming the conjecture can be proven).
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 11
Invitation: Gauge/Gravity Correpondence-Less pessimistic view:
1. String Theory may not describe our nature uniquely. But it is mathematically correct.
2. Gauge/Gravity Correspondence is a mathematical map that is conjectured from the correct mathematical framework of String Theory.
➡ Gauge/Gravity Correspondence may be a mathematically correct map, which relates particular quantum field theories to particular gravity theories (assuming the conjecture can be proven).
➡ Gauge/Gravity may be used as a mathematical tool to map effective field theories to gravity theories, even if string theory is not describing our nature.
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
horizon
radial coordinate r
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
➡ perturbations are described by a linearized wave equation in curved space, i.e. second order linear differential equation in the radial coordinate r
horizon
radial coordinate r
”!!2" =#2"
#t2”
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
➡ perturbations are described by a linearized wave equation in curved space, i.e. second order linear differential equation in the radial coordinate r
➡ mathematics tells us there exist two solutions
horizon
radial coordinate r
”!!2" =#2"
#t2”
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
➡ perturbations are described by a linearized wave equation in curved space, i.e. second order linear differential equation in the radial coordinate r
➡ mathematics tells us there exist two solutions
➡ physics tells us only the infalling solution is realized
horizon
radial coordinate r
”!!2" =#2"
#t2”
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
➡ perturbations are described by a linearized wave equation in curved space, i.e. second order linear differential equation in the radial coordinate r
➡ mathematics tells us there exist two solutions
➡ physics tells us only the infalling solution is realized
➡ later: even the outgoing solution contains physics
horizon
radial coordinate r
”!!2" =#2"
#t2”
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
➡ perturbations are described by a linearized wave equation in curved space, i.e. second order linear differential equation in the radial coordinate r
➡ mathematics tells us there exist two solutions
➡ physics tells us only the infalling solution is realized
➡ later: even the outgoing solution contains physics
horizon
radial coordinate r
”!!2" =#2"
#t2”
reasonable
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 12
An example: Correct math, wrong physics?-Perturbations near a classical black hole:
➡ perturbations are described by a linearized wave equation in curved space, i.e. second order linear differential equation in the radial coordinate r
➡ mathematics tells us there exist two solutions
➡ physics tells us only the infalling solution is realized
➡ later: even the outgoing solution contains physics
horizon
radial coordinate rforbidden!
”!!2" =#2"
#t2”
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 13
Why Anti-deSitter, isn’t our world deSitter ?
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 14
Invitation: Gauge/Gravity Correpondence-Optimistic view:
1. Some “String Theory” may describe our nature uniquely.
2. Gauge/Gravity Correspondence may be an exact duality between two ways of describing this one unique theory.
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 14
Invitation: Gauge/Gravity Correpondence-Optimistic view:
1. Some “String Theory” may describe our nature uniquely.
2. Gauge/Gravity Correspondence may be an exact duality between two ways of describing this one unique theory.
➡ Gauge/Gravity Correspondence suggests that our present way of assigning some phenomena to gravity, and others to quantum field theories should be unified.
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 14
Invitation: Gauge/Gravity Correpondence-Optimistic view:
1. Some “String Theory” may describe our nature uniquely.
2. Gauge/Gravity Correspondence may be an exact duality between two ways of describing this one unique theory.
➡ Gauge/Gravity Correspondence suggests that our present way of assigning some phenomena to gravity, and others to quantum field theories should be unified.
➡ The full holographic dual to our world would contain both: our gravitational and standard model physics
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 14
Invitation: Gauge/Gravity Correpondence-Optimistic view:
1. Some “String Theory” may describe our nature uniquely.
2. Gauge/Gravity Correspondence may be an exact duality between two ways of describing this one unique theory.
➡ Gauge/Gravity Correspondence suggests that our present way of assigning some phenomena to gravity, and others to quantum field theories should be unified.
➡ The full holographic dual to our world would contain both: our gravitational and standard model physics
Gravity
Gauge
Gauge
Gravity
our world
holographic dual
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 15
So, why am I here at 9 a.m. on a Monday morning?
1. Gain a geometric (a dual) understanding of strongly coupled dynamics.
2. Studying examples of gauge/gravity dualities, we learn about the particular quantum field theory and the particular gravity, but possibly also about quantum gravity in general!
Two answers:
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
Lecture I: Introduction to Gauge/Gravity & Applications I
Lecture II: Introduction to Gauge/Gravity & Applications II
Lecture III: Thermal Spectral Functions
Lecture IV: Beyond Hydrodynamics
Lecture V: Phase Transitions
Lectures
16
Tuesday-Thursday 9:00 - 10:30, FIAS ground floor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
Exercise I: AdS Coordinates & Brane Embeddings
Exercise II: Thermal Green’s Functions & Viscosity
Exercise III: More than Hydrodynamics from Gravity
Exercise IV: Superfluid Phase Transition & Conductivity
Exercises
17
Monday-Thursday 13:30 - 17:00, on the roof of FIAS
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
Participants of the powerweek will be able to
- carry out computations in classical (super)gravity which are state-of-the-art in gauge/gravity research
- translate gravity results into gauge theory expressions (at least for the subset of examples presented)
- judge the relevance of gauge/gravity to their own work
Goals
18
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
Invitation: Gauge/gravity correspondence
19
-Why does it work?Stack of Nc D3-branes (coincident)
in 10 dimensions
near-horizon geometryAdS x S5
5
D3 branes in 10d
duality
Two distinct ways to describe this stack:
4-dimensional worldvolume theory on the D3-branes
(e.g. Super-Yang-Mills)N = 4
(e.g. Supergravity)gauge side
gravity side
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
Invitation: Gauge/gravity correspondence
19
-Why does it work?
Geometric setup: Strings/Branes
Find solutionconfiguration
Field Theory result
Stack of Nc D3-branes (coincident)in 10 dimensions
Add/change geometric objects on ‘gravity side’:
-How does it work?
near-horizon geometryAdS x S5
5
D3 branes in 10d
duality
Two distinct ways to describe this stack:
4-dimensional worldvolume theory on the D3-branes
(e.g. Super-Yang-Mills)N = 4
(e.g. Supergravity)gauge side
gravity side
Example: Schwarzschild radius corresponds to temperature
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 20
Lecture III: Thermal Spectral Functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 21
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
# ,
[Erdmenger, M.K., Rust 0710.0334]Example: Gauge field fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
21
Equation of motion:
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
[Erdmenger, M.K., Rust 0710.0334]Example: Gauge field fluctuations
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
21
Equation of motion:
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
!µFµ!
= 0
`Curved’ Maxwell equations:
!µ
!
!
"GGµ!
G"#
F!#
"
= 0
!µ
!
!
"GGµ!
G"#
![!A#]
"
= 0
[Erdmenger, M.K., Rust 0710.0334]Example: Gauge field fluctuations
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
21
Equation of motion:
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
[Erdmenger, M.K., Rust 0710.0334]Example: Gauge field fluctuations
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
21
Equation of motion:
Effective action:
Boundary conditions:
Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
A = (! ! !H)!iw[1 +iw
2(! ! !H) + . . . ]
[Erdmenger, M.K., Rust 0710.0334]Example: Gauge field fluctuations
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
21
Equation of motion:
Effective action:
Boundary conditions:
Fµ! = ![µA!]
Translation to Gauge Theory by duality:
SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
Aµ ! Jµ
! "# $
G
,
A = (! ! !H)!iw[1 +iw
2(! ! !H) + . . . ]
AdS/CFT
(source)
[Erdmenger, M.K., Rust 0710.0334]Example: Gauge field fluctuations
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
21
Equation of motion:
Effective action:
Boundary conditions:
Gauge Correlator:
Fµ! = ![µA!]
Translation to Gauge Theory by duality:
SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
Aµ ! Jµ
! "# $
G
,
A = (! ! !H)!iw[1 +iw
2(! ! !H) + . . . ]
Gret =NfNcT
2
8lim
!!!bdy
!
!3"!A(!)
A(!)
"
AdS/CFT
(source)
[Erdmenger, M.K., Rust 0710.0334]
[Son et al.’02]
Example: Gauge field fluctuations
Fluctuations
shooting from horizon
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0.0 0.5 1.0 1.5 2.0 2.5!6
!4
!2
0
2
4
(,0
)!
0
d = 0.25
!0 = 0.1!0 = 0.5!0 = 0.7!0 = 0.8
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
Lower temperature
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]
[Ka
rch
, O
’Ba
nn
on ‘0
7]
Gauge theory results: spectral functions
Meson melting transition!(fundamental matter deconf trans)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0
2 4 6 8 10
1000
2000
3000
4000
XY
XY
Y X
Y X
E3E3
E3E3n = 0n = 0n = 0
n = 1n = 1n = 1
! 0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Finite isospin densityPhase diagram
Finite baryon density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]
[Ka
rch
, O
’Ba
nn
on ‘0
7]
Gauge theory results: spectral functions
Meson melting transition!(fundamental matter deconf trans)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 22
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0
2 4 6 8 10
1000
2000
3000
4000
XY
XY
Y X
Y X
E3E3
E3E3n = 0n = 0n = 0
n = 1n = 1n = 1
! 0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Finite isospin densityPhase diagram
Finite baryon density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]
Analytically: [PhD thesis ’08]
[Ka
rch
, O
’Ba
nn
on ‘0
7]
Gauge theory results: spectral functions
Meson melting transition!(fundamental matter deconf trans)
SU(2) triplet splitting!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 23
Lecture IV: Beyond Hydrodynamics
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 24
Thermal spectral function contains all information about
diffusion and quasiparticle resonances in QG-plasma.
R
R(!, q) = !2 ImGret(!, q)
IV. Thermal Spectral Function
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 24
Thermal spectral function contains all information about
diffusion and quasiparticle resonances in QG-plasma.
R
R(!, q) = !2 ImGret(!, q)
IV. Thermal Spectral Function
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 24
w
R
Thermal spectral function contains all information about
diffusion and quasiparticle resonances in QG-plasma.
R
R(!, q) = !2 ImGret(!, q)QGP
IV. Thermal Spectral Function
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 24
w
R
Thermal spectral function contains all information about
diffusion and quasiparticle resonances in QG-plasma.
R
R(!, q) = !2 ImGret(!, q)QGP
0
2 4 6 8 10
50
!50
100
150
200
250
w
R(w
,0)!
R0
IV. Thermal Spectral Function
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 24
w
R
Thermal spectral function contains all information about
diffusion and quasiparticle resonances in QG-plasma.
R
R(!, q) = !2 ImGret(!, q)QGP
0
2 4 6 8 10
50
!50
100
150
200
250
w
R(w
,0)!
R0
IV. Thermal Spectral Function
Transport coefficients using Kubo formulae, e.g.! ! lim
!!0
1
""[J t, J t]#
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 24
w
R
Thermal spectral function contains all information about
diffusion and quasiparticle resonances in QG-plasma.
R
R(!, q) = !2 ImGret(!, q)QGP
0
2 4 6 8 10
50
!50
100
150
200
250
w
R(w
,0)!
R0
IV. Thermal Spectral Function
Transport coefficients using Kubo formulae, e.g.! ! lim
!!0
1
""[J t, J t]#
(cf. today’s exercise)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 25
LLLLL
RR
RR R
Chiral vortex effect Heavy-ion-collision
ll
IV. Chiral transport effects in QGP
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 25
LLLLL
RR
RR R
Chiral vortex effect Heavy-ion-collision
ll
IV. Chiral transport effects in QGP
(similar: chiral magnetic effect)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 26
[Erdmenger, Haack, M.K., Yarom 0809.2488]
Conservation equations
Constitutive equations
-First order hydrodynamics
jµ = nuµ! !T (gµ! + uµu!)"!
! µ
T
"
+#$µ
Tµ! =!
3(4uµu! + gµ!) + !µ!
!µ
=1
2"µ!"#
u!#"u#
IV. Chiral transport Relativistic fluids with one conserved charge, with an anomaly (chiral or parity)
!µjµ= CEµBµ!µT
µ!= F
!"J"
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 26
[Erdmenger, Haack, M.K., Yarom 0809.2488]
Conservation equations
Constitutive equations
-First order hydrodynamics
jµ = nuµ! !T (gµ! + uµu!)"!
! µ
T
"
+#$µ
Tµ! =!
3(4uµu! + gµ!) + !µ!
!µ
=1
2"µ!"#
u!#"u#
IV. Chiral transport Relativistic fluids with one conserved charge, with an anomaly (chiral or parity)
!µjµ= CEµBµ!µT
µ!= F
!"J"
from a gravity dual
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 26
[Erdmenger, Haack, M.K., Yarom 0809.2488]
Conservation equations
Constitutive equations
-First order hydrodynamics
jµ = nuµ! !T (gµ! + uµu!)"!
! µ
T
"
+#$µ
Tµ! =!
3(4uµu! + gµ!) + !µ!
!µ
=1
2"µ!"#
u!#"u#
New vorticity term arises!related to triangle anomaly
[Son,Surowka 0906.5044]
! = C
!
µ2!
2
3
µ3n
" + P
"
!µjµ= !
1
8C"µ!"#Fµ!F"#
IV. Chiral transport Relativistic fluids with one conserved charge, with an anomaly (chiral or parity)
!µjµ= CEµBµ!µT
µ!= F
!"J"
from a gravity dual
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 26
[Erdmenger, Haack, M.K., Yarom 0809.2488]
Conservation equations
Constitutive equations
-First order hydrodynamics
jµ = nuµ! !T (gµ! + uµu!)"!
! µ
T
"
+#$µ
Tµ! =!
3(4uµu! + gµ!) + !µ!
!µ
=1
2"µ!"#
u!#"u#
New vorticity term arises!related to triangle anomaly
[Son,Surowka 0906.5044]
! = C
!
µ2!
2
3
µ3n
" + P
"
!µjµ= !
1
8C"µ!"#Fµ!F"#
IV. Chiral transport Relativistic fluids with one conserved charge, with an anomaly (chiral or parity)
!µjµ= CEµBµ!µT
µ!= F
!"J"
from a gravity dual
Fixed by anomaly coefficient!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
completely determined by C and equation of state
27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
completely determined by C and equation of state
3 ways to compute :
27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
completely determined by C and equation of state
3 ways to compute :
conformal symmetry
27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
completely determined by C and equation of state
3 ways to compute :
conformal symmetry
positivity of entropy current (chiral anomaly)
27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
completely determined by C and equation of state
3 ways to compute :
conformal symmetry
positivity of entropy current (chiral anomaly)
directly in specific holographic model (microscopic)
27
Remarks & ideas
!
!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
completely determined by C and equation of state
3 ways to compute :
conformal symmetry
positivity of entropy current (chiral anomaly)
directly in specific holographic model (microscopic)
27
Remarks & ideas
!
!
Relativistic hydrodynamics needs to be completed.
Not in non-relativistic setups, so repeat for 2+1 dimensional QFT, with condensed matter applications in mind (parity anomaly?)
Effects measured in heavy-ion-collisions?[Kharzeev, Son]
[1106.xxxx]
[Baier et al, Minwalla et al 2008]
[Nicolis & Son,1103.2137]
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
IV. Concepts -the dictionary
28
A = A(0)+
A(2)
!2+ . . .
non-normalizable(source)
normalizable(vev)
(! ! ")AdS
QFT!!bdy
gauge theory(strong)
gravity theory(weak)
Near AdS-boundary
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
IV. Concepts -the dictionary
28
Dictionary
operator field
GEOMETRY(radial coord. )
QFT FEATURE(energy scale)
Jµ Aµ
A = A(0)+
A(2)
!2+ . . .
non-normalizable(source)
normalizable(vev)
(! ! ")
!
vev
source A(0)
A(2) (charge)
(chem. pot.)
(gauge)
AdS
QFT!!bdy
gauge theory(strong)
gravity theory(weak)
Near AdS-boundary
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 29
IV. Quasinormal modes
Special frequencies: !n ! C
Gret =NfNcT
2
8lim
!!!bdy
!
!3"!A(!)
A(!)
"
lim!!!bdy
|A(!n)|2 = 0
!0!1
!2
(quasinormal)
e!i!r
= e!iRe{!}r
eIm{!}r
;
e.g. [Berti et al. ‘09]
Example:
!(")
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 29
IV. Quasinormal modes
Special frequencies: !n ! C
Gret =NfNcT
2
8lim
!!!bdy
!
!3"!A(!)
A(!)
"
lim!!!bdy
|A(!n)|2 = 0
!0!1
!2
(quasinormal)
Gravity: Gauge theory:
quasinormal frequencies
poles of correlator(energy, damping, stabilityof mesonic excitations)
e!i!r
= e!iRe{!}r
eIm{!}r
;
e.g. [Berti et al. ‘09]
Example:
!(")
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
-4 -2 0 2 4Re !
-4
-3
-2
-1
0
Im!
Simple example:Eigenfrequencies / normal
modesof the quantum mechanical
harmonic oscillator (no damping)
!n =1
2+ n
IV. Quasi Normal Modes (QNMs)
quasinormal frequencies
30
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
-4 -2 0 2 4Re !
-4
-3
-2
-1
0
Im!
Gret !
1
i! " Dq2
IV. Quasi Normal Modes (QNMs)
31
• QNMs are the quasi-eigenmodes of gauge field
• Dual QFT: lowest QNM identified with hydrodynamic diffusion pole (not propagating)
• Higher QN modes: gravity field waves propagate through curved b.h. background while decaying (dual gauge currents analogously)
Example: Poles of charge current correlator
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 32
!5 0 5
!3
!2
!1
0
!500
0
500
1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.50.50
0.55
0.60
0.65
0.70
0.75
0.80
IV. Quasi Normal Modes (QNMs)
Instabilities
Trajectories (dial k)Complex frequency plane
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 33
Lecture V: Phase Transitions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 34
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
m
!0
d = 0
d = 0.002
d = 0.1
d = 0.25
The mass parameter depending on the parameter .m !0
m = lim!!!bdy
! "(!) =2mquark!
#T
!0 = !(")!
!
!!!H
!(") =m
"+
c
"3+ . . .
A0 = µ !
1
!2
d
2"#!+ . . .
Near-boundary expansions:
L(!) = ! "(!)
Other relations:
! =
"
"H
,
Result to 3.2 d) from exercise III
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
N = 4 SYM with SU(Nc)dual to
Meson melting transition
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
Meson melting transition
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
(black) N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
Meson melting transition
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
(black) N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
Meson melting transition
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
7,8,9
4,5,6,7
!A!
(black)
! U(Nf )
N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
~quark
mass
Meson melting transition
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
7,8,9
4,5,6,7
!A!
(black)
Chemical potential: Aµ = !µ0A0 + Aµ
[Myers et al., hep-th/0611099] (cf. therm. FT)
! U(Nf )
[Mateos et al. 0709.1225]
N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
~quark
mass
Meson melting transition
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
7,8,9
4,5,6,7
!A!
(black)
Chemical potential: Aµ = !µ0A0 + Aµ
[Myers et al., hep-th/0611099] (cf. therm. FT)
! U(Nf )
[Mateos et al. 0709.1225]
N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
~quark
mass
Meson melting transition
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
7,8,9
4,5,6,7
!A!
(black)
Chemical potential: Aµ = !µ0A0 + Aµ
[Myers et al., hep-th/0611099] (cf. therm. FT)
! U(Nf )
Mesons
[Mateos et al. 0709.1225]
N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
~quark
mass
Meson melting transition
L(!)
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 35
D3-branesNc
D7-branesNf
7,8,9
4,5,6,7
!A!
(black)
Chemical potential: Aµ = !µ0A0 + Aµ
[Myers et al., hep-th/0611099] (cf. therm. FT)
! U(Nf )
Mesons
[Mateos et al. 0709.1225]
N = 4 SYM with SU(Nc)dual to
dual to [Karch,Katz hep-th/0205236]
~quark
mass
Meson melting transition
L(!)0 1 2 3 4 5 6 7 8 9
D3 x x x xD7 x x x x x x x x
N = 2 SU(Nf ) flavor
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 36
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
# ,
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 36
“Background”: Brane embedding & gauge field
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
# ,
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 36
“Background”: Brane embedding & gauge field
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
# ,
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
r
L
00
1
1
2
2
3
3
4
4
5
5!
A0
0.5
1
1
1.5
2.5
2
2 3 4 5
Solutions to Euler-Lagrange equations give “profiles” in
radial direction
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 37
r
L
00
1
1
2
2
3
3
4
4
5
5
!
A0
0.5
1
1
1.5
2.5
2
2 3 4 5!
A0/1
0!4
0.5
1
1
1.5
2.5
2
2 3 4 5
r
L
0
0.5
1
1
1.5
2
2 3 4 5
d = 0.25 d =10!4
4
Flavor brane embeddings (D7-branes)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 37
r
L
00
1
1
2
2
3
3
4
4
5
5
!
A0
0.5
1
1
1.5
2.5
2
2 3 4 5!
A0/1
0!4
0.5
1
1
1.5
2.5
2
2 3 4 5
r
L
0
0.5
1
1
1.5
2
2 3 4 5
d = 0.25 d =10!4
4
Flavor brane embeddings (D7-branes)
Numerical method: shooting from horizon
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 37
r
L
00
1
1
2
2
3
3
4
4
5
5
!
A0
0.5
1
1
1.5
2.5
2
2 3 4 5!
A0/1
0!4
0.5
1
1
1.5
2.5
2
2 3 4 5
r
L
0
0.5
1
1
1.5
2
2 3 4 5
d = 0.25 d =10!4
4
Flavor brane embeddings (D7-branes)
Numerical method: shooting from horizon
Quark mass fixed
dynamically!
... together with gauge
field.
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 38
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
# ,
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
38
Equation of motion:
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
38
Equation of motion:
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
!µFµ!
= 0
`Curved’ Maxwell equations:
!µ
!
!
"GGµ!
G"#
F!#
"
= 0
!µ
!
!
"GGµ!
G"#
![!A#]
"
= 0
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
38
Equation of motion:
Effective action: Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
38
Equation of motion:
Effective action:
Boundary conditions:
Fµ! = ![µA!]SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
! "# $
G
,
A = (! ! !H)!iw[1 +iw
2(! ! !H) + . . . ]
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
38
Equation of motion:
Effective action:
Boundary conditions:
Fµ! = ![µA!]
Translation to Gauge Theory by duality:
SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
Aµ ! Jµ
! "# $
G
,
A = (! ! !H)!iw[1 +iw
2(! ! !H) + . . . ]
AdS/CFT
(source)
[Erdmenger, M.K., Rust 0710.0334]Gravity solution & translation
Fluctuations
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
0 = A!! +!![
!
|det G|G22G44]!
|det G|G22G44A! !
G00
G44"H
2#2A
38
Equation of motion:
Effective action:
Boundary conditions:
Gauge Correlator:
Fµ! = ![µA!]
Translation to Gauge Theory by duality:
SD7 =
!
d8x
"
#
#
#det{[g + F ] + F}
#
#
#
Aµ ! Jµ
! "# $
G
,
A = (! ! !H)!iw[1 +iw
2(! ! !H) + . . . ]
Gret =NfNcT
2
8lim
!!!bdy
!
!3"!A(!)
A(!)
"
AdS/CFT
(source)
[Erdmenger, M.K., Rust 0710.0334]
[Son et al.’02]
Gravity solution & translation
Fluctuations
shooting from horizon
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0.0 0.5 1.0 1.5 2.0 2.5!6
!4
!2
0
2
4
(,0
)!
0
d = 0.25
!0 = 0.1!0 = 0.5!0 = 0.7!0 = 0.8
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
Lower temperature
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
! = !(d , ")
L(!) = ! "(!)
!0 = !(")!
!
!!!H!
mquark
T
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]Gauge theory results: spectral functions
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Phase diagram
Finite baryon density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]
[Ka
rch
, O
’Ba
nn
on ‘0
7]
Gauge theory results: spectral functions
Meson melting transition!(fundamental matter deconf trans)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0
2 4 6 8 10
1000
2000
3000
4000
XY
XY
Y X
Y X
E3E3
E3E3n = 0n = 0n = 0
n = 1n = 1n = 1
! 0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Finite isospin densityPhase diagram
Finite baryon density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]
[Ka
rch
, O
’Ba
nn
on ‘0
7]
Gauge theory results: spectral functions
Meson melting transition!(fundamental matter deconf trans)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 39
0 5 10 15 20 25 30 350
20000
40000
60000
80000
100000
120000
140000
(,0
)
d = 0.25
!0 = 0.999
n = 0 n = 1 n = 2 n = 3
0
2 4 6 8 10
1000
2000
3000
4000
XY
XY
Y X
Y X
E3E3
E3E3n = 0n = 0n = 0
n = 1n = 1n = 1
! 0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
Finite isospin densityPhase diagram
Finite baryon density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5!50
0
50
100
150
200
(,0
)!
0
d = 0.25
!0 = 0.8
!0 = 0.94
!0 = 0.962
[Erdmenger, M.K., Rust 0710.0334]
Analytically: [PhD thesis ’08]
[Ka
rch
, O
’Ba
nn
on ‘0
7]
Gauge theory results: spectral functions
Meson melting transition!(fundamental matter deconf trans)
SU(2) triplet splitting!
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 40
Flavor brane embeddings (T=0, d=0)
Analytical solution for embeddings gives induced metric
Background[Kruczenski et al, hep-th/0304032]
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 40
Flavor brane embeddings (T=0, d=0)
Supersymmetric vector meson mass formula
Analytical solution for embeddings gives induced metric
Analytic solutions with spherical harmonics ( )Fluctuations
Background
l
counts nodes of solution in radial AdS directionn
[Kruczenski et al, hep-th/0304032]
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 41
[Erdmenger, M.K., Kerner, Rust 0807.2663]
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
New (lowest) mesonic excitation
Stability
Baryonic phases (reminder)
High isospin densities: instabilities
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 41
[Erdmenger, M.K., Kerner, Rust 0807.2663]
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
µq/m
qµ
q/m
q
T/MT/M
d = 0d = 0.00315
d = 4
d = 0.25
New (lowest) mesonic excitation
Stability
New phase
High isospin densities: instabilities
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 42
Building a holographic superfluid
GravityField Theory
introduce normalizble mode no non-normalizable mode condensate hovers over horizon black hole, charge
charged condensate (vev) no source condensate of charge carriers finite temperature, chem. pot.
horizon
condensate
gravity
electromag.
curvatureAdS-boundary
What do we need?
Is this stable?
[Gubser, Pufu 0805.2960][Gubser 0801.2977]
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 43
Get some intuition
GravityField Theory
strings (D3-D5) give FT charges cannot put infinitely many second brane is important
(probe limit)
charged particles condense at large enough chemical potential
µisopin ! Mq
-2 -1 0 1 2!
0
1
2
3
4
5
V!!"
L ! D!!D!! ! (Mq2" µisospin
2)!2
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 43
Get some intuition
GravityField Theory
strings (D3-D5) give FT charges cannot put infinitely many second brane is important
(probe limit)
charged particles condense at large enough chemical potential
µisopin ! Mq
-2 -1 0 1 2!
-0.5
0
0.5
1
V!!"
L ! D!!D!! ! (Mq2" µisospin
2)!2
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 43
Get some intuition
GravityField Theory
strings (D3-D5) give FT charges cannot put infinitely many second brane is important
(probe limit)
charged particles condense at large enough chemical potential
µisopin ! Mq
We need a non-Abelian structure!
-2 -1 0 1 2!
-0.5
0
0.5
1
V!!"
L ! D!!D!! ! (Mq2" µisospin
2)!2
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 44
[Erdmenger, M.K., Kerner, Rust 0807.2663]
A3
0 = µ +d
!2+ . . .
Vector meson superfluid General idea
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 44
[Ammon, Erdmenger, M.K., Kerner 0810.2316]
[Erdmenger, M.K., Kerner, Rust 0807.2663]
superc
onducting
A3
0 = µ +d
!2+ . . .
Vector meson superfluid General idea
[Gubser, Pufu 0805.2960]
A3
0 = µ +d30
!2+ . . .
A1
3 =d13
!2+ . . .
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 44
[Ammon, Erdmenger, M.K., Kerner 0810.2316]
[Erdmenger, M.K., Kerner, Rust 0807.2663]
superc
onducting
A3
0 = µ +d
!2+ . . .
Vector meson superfluid General idea
[Gubser, Pufu 0805.2960]
A3
0 = µ +d30
!2+ . . .
A1
3 =d13
!2+ . . .
spont. breaks U(1)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 45
Phenomenology of a superfluid/superconductor
second order phase transition mean-field theory critical
exponents energy gap in the conductivity
(“Cooper pair” binding) Meissner-Ochsenfeld effect,
condensate destroyed at large B
Finite magnetic field:
0.15 0.2 0.25 0.3T
0
0.5
1
1.5
2
2.5
3
3.5
B
H33
d30
T
d30
1/3
superconducting (SC)
normal phase (NC)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 45
Phenomenology of a superfluid/superconductor
second order phase transition mean-field theory critical
exponents energy gap in the conductivity
(“Cooper pair” binding) Meissner-Ochsenfeld effect,
condensate destroyed at large B
This looks like a superfluid/conductor!
Finite magnetic field:
0.15 0.2 0.25 0.3T
0
0.5
1
1.5
2
2.5
3
3.5
B
H33
d30
T
d30
1/3
superconducting (SC)
normal phase (NC)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 46
Stringy pairing picture
E3
!
E2
3
charged horizon (3-7 strings generate )
!
x3
horizon
boundary
!
condensate
gravity
electromag.
curvature
bulk charge(7-7 strings)
7-7 strings generate i.e. they break the U(1) and are thus dual to Cooper pairs.
A1
3
A3
0
B1
!3 = F1
!3 = !!A1
3
E3
! = F3
!0 = !!A3
0
E2
3 = F2
30 = A1
3A3
0