introduction to gauge/gravity correspondence &...
TRANSCRIPT
![Page 1: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/1.jpg)
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
![Page 2: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/2.jpg)
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
![Page 3: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/3.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 2
Lecture I & II: Introduction to Gauge/Gravity
![Page 4: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/4.jpg)
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
![Page 5: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/5.jpg)
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
![Page 6: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/6.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 4
So, why am I here at 9 a.m. on a Monday morning?
![Page 7: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/7.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
![Page 8: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/8.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
![Page 9: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/9.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
![Page 10: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/10.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 5
Invitation:Gauge/Gravity Correspondences-Strange calculations with publications
![Page 11: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/11.jpg)
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
![Page 12: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/12.jpg)
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
![Page 13: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/13.jpg)
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
![Page 14: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/14.jpg)
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
![Page 15: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/15.jpg)
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
![Page 16: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/16.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 7
Evidence for Gauge/Gravity-Reasonable example results from Gauge/Gravity!
![Page 17: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/17.jpg)
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
![Page 18: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/18.jpg)
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
![Page 19: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/19.jpg)
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
![Page 20: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/20.jpg)
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
![Page 21: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/21.jpg)
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
![Page 22: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/22.jpg)
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)
![Page 23: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/23.jpg)
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
![Page 24: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/24.jpg)
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
![Page 25: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/25.jpg)
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)
![Page 26: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/26.jpg)
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.]
![Page 27: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/27.jpg)
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)
![Page 28: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/28.jpg)
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
![Page 29: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/29.jpg)
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
![Page 30: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/30.jpg)
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!
![Page 31: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/31.jpg)
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)
![Page 32: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/32.jpg)
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.
![Page 33: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/33.jpg)
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).
![Page 34: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/34.jpg)
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.
![Page 35: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/35.jpg)
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
![Page 36: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/36.jpg)
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”
![Page 37: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/37.jpg)
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”
![Page 38: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/38.jpg)
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”
![Page 39: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/39.jpg)
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”
![Page 40: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/40.jpg)
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
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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”
![Page 42: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/42.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 13
Why Anti-deSitter, isn’t our world deSitter ?
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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.
![Page 44: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/44.jpg)
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.
![Page 45: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/45.jpg)
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
![Page 46: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/46.jpg)
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
![Page 47: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/47.jpg)
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:
![Page 48: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/48.jpg)
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
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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
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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
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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
![Page 52: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/52.jpg)
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
![Page 53: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/53.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 20
Lecture III: Thermal Spectral Functions
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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
![Page 55: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/55.jpg)
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
![Page 56: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/56.jpg)
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
![Page 57: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/57.jpg)
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
![Page 58: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/58.jpg)
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
![Page 59: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/59.jpg)
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
![Page 60: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/60.jpg)
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
![Page 61: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/61.jpg)
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
![Page 62: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/62.jpg)
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
![Page 63: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/63.jpg)
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
![Page 64: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/64.jpg)
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
![Page 65: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/65.jpg)
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)
![Page 66: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/66.jpg)
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)
![Page 67: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/67.jpg)
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!
![Page 68: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/68.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 23
Lecture IV: Beyond Hydrodynamics
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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
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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
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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
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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
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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]#
![Page 74: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/74.jpg)
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)
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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
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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)
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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
"
+#$µ
Tµ! =!
3(4uµu! + gµ!) + !µ!
!µ
=1
2"µ!"#
u!#"u#
IV. Chiral transport Relativistic fluids with one conserved charge, with an anomaly (chiral or parity)
!µjµ= CEµBµ!µT
µ!= F
!"J"
![Page 78: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/78.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 26
[Erdmenger, Haack, M.K., Yarom 0809.2488]
Conservation equations
Constitutive equations
-First order hydrodynamics
jµ = nuµ! !T (gµ! + uµu!)"!
! µ
T
"
+#$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
![Page 79: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/79.jpg)
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
"
+#$µ
Tµ! =!
3(4uµu! + gµ!) + !µ!
!µ
=1
2"µ!"#
u!#"u#
New vorticity term arises!related to triangle anomaly
[Son,Surowka 0906.5044]
! = C
!
µ2!
2
3
µ3n
" + P
"
!µjµ= !
1
8C"µ!"#Fµ!F"#
IV. Chiral transport Relativistic fluids with one conserved charge, with an anomaly (chiral or parity)
!µjµ= CEµBµ!µT
µ!= F
!"J"
from a gravity dual
![Page 80: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/80.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 26
[Erdmenger, Haack, M.K., Yarom 0809.2488]
Conservation equations
Constitutive equations
-First order hydrodynamics
jµ = nuµ! !T (gµ! + uµu!)"!
! µ
T
"
+#$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!
![Page 81: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/81.jpg)
Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 27
Remarks & ideas
!
!
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Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page
New coefficient at first order hydrodynamics (~viscosity)
27
Remarks & ideas
!
!
![Page 83: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/83.jpg)
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
!
!
![Page 84: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/84.jpg)
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
!
!
![Page 85: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/85.jpg)
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
!
!
![Page 86: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/86.jpg)
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
!
!
![Page 87: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/87.jpg)
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
!
!
![Page 88: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/88.jpg)
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]
![Page 89: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/89.jpg)
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
![Page 90: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/90.jpg)
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
![Page 91: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/91.jpg)
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:
!(")
![Page 92: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/92.jpg)
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:
!(")
![Page 93: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/93.jpg)
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
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-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
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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
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Matthias Kaminski Introduction to Gauge/Gravity & Heavy-Ion-Applications Page 33
Lecture V: Phase Transitions
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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
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D3-branesNc
N = 4 SYM with SU(Nc)dual to
Meson melting transition
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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
![Page 100: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/100.jpg)
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
![Page 101: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/101.jpg)
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
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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
![Page 103: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/103.jpg)
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
![Page 104: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/104.jpg)
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
![Page 105: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/105.jpg)
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
![Page 106: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/106.jpg)
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
![Page 107: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/107.jpg)
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
![Page 108: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/108.jpg)
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
![Page 109: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/109.jpg)
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
![Page 110: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/110.jpg)
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)
![Page 111: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/111.jpg)
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
![Page 112: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/112.jpg)
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.
![Page 113: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/113.jpg)
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
![Page 114: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/114.jpg)
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
![Page 115: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/115.jpg)
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
![Page 116: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/116.jpg)
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
![Page 117: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/117.jpg)
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
![Page 118: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/118.jpg)
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
![Page 119: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/119.jpg)
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
![Page 120: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/120.jpg)
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
![Page 121: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/121.jpg)
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
![Page 122: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/122.jpg)
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
![Page 123: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/123.jpg)
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
![Page 124: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/124.jpg)
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)
![Page 125: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/125.jpg)
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)
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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!
![Page 127: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/127.jpg)
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]
![Page 128: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/128.jpg)
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]
![Page 129: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/129.jpg)
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
![Page 130: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/130.jpg)
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
![Page 131: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/131.jpg)
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]
![Page 132: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/132.jpg)
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
![Page 133: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/133.jpg)
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
![Page 134: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/134.jpg)
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
![Page 135: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/135.jpg)
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
![Page 136: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/136.jpg)
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+ . . .
![Page 137: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/137.jpg)
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)
![Page 138: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/138.jpg)
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)
![Page 139: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/139.jpg)
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)
![Page 140: Introduction to Gauge/Gravity Correspondence & …mkaminski.people.ua.edu/.../intro_outro_lecturesfias.pdfGauge/Gravity Correspondence may be a mathematically correct map, which relates](https://reader034.vdocuments.mx/reader034/viewer/2022042417/5f33807764e367142f4594f7/html5/thumbnails/140.jpg)
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