Gauge/Gravity Duality
Big picture – slides
Key computations - board
TODAY
Introduction
Strings & Branes
AdS/CFT Correspondence
QCD-like dynamics lecture 2 & 3
Prof Nick Evans
Resources
D-branes – Clifford Johnson
The classic review of the AdS/CFT Correspondence is
Large N field theories, string theory and gravity.
Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz.
Phys.Rept.323:183-386,2000, hep-th/9905111
A review of holographic quark and meson physics is
Mesons in Gauge/Gravity Duals - A Review.
Johanna Erdmenger, Nick Evans, Ingo Kirsch, Ed Threlfall Eur.Phys.J.A35:81133,2008, arXiv:0711.4467 [hep-th]
Transport properties of gluon plasma
Viscosity, Black Holes, and Quantum Field Theory.
Dam T. Son, Andrei O. Starinets. Ann.Rev.Nucl.Part.Sci.57:95-118,2007.
arXiv:0704.0240 [hep-th]
These slides can be found at
www.hep.phys.soton.ac.uk/~evans/GaugeGravity/
0 Introduction
• String theory was originally an attempt to describe mesons
• QCD took over…
• String Theory became a quantum theory of gravity – a TOE
• The AdS/CFT Correspondence has re-united them
• & there are now positive links back to QCD
* new pictures of confinement
* new understanding of mass generation
* new ideas on hadronization
* novel descriptions of heavy ion physics
Smolin
& Woit
are
wrong!
Quarks come in 3
colours
8 gluons mediate the
force
Hadronization in even electron positron collisions is very
messy and uncomputable from first principles… as for
heavy ion collisions….
Chiral Lagrangian description of pions
Stringy Hints – Regge Trajectories
String theory
was originally
proposed to
explain
Stringy Hints – ‘tHooft says Large N
N2c
Nc Nf
At large N perturbation
theory simplifies and
restricts to planar
diagrams…
Dense gluons -> a string
world sheet?
The expansion parameter is
Quark production &
hence string breaking is
forbidden at large N
Waves with no energy flux off
the end of a string…
A consistent theory of photons and gravitons? The TOE era!
Actually a negative world-sheet Cosmological Constant leaves the
un-wavy string as a tachyon….
Super-Strings
To make a consistent theory add fermion partners to the X m
living on the string world-sheet
Only anomaly free in 10d
Extra world sheet waves
Same chirality
fermions or
opposite?
The dilaton vev turns out to be the string coupling constant…
The links to QCD seem utterly lost… until…
HOW SHOULD YOU THINK ABOUT THIS?
Imagine taking a field theory and mixing up the equations for
motion in x and RG flow in energy scale into a 5d theory… if
the gauge theory is strongly coupled we don’t know these mixed
equations… they turn out to be the EoM of strings in AdS!
Another view
Curved extra dimensions generate a Kaluza Klein tower
Here the glueballs of the gauge theory are this tower (at large N
an infinite number are stable).. The spectrum matches the KK
tower of an AdS radial direction…
1/2 The World of Branes
T- Duality in String Theory
Closed strings on a compact space generate two sets
of integer spaced mass states…
There is a symmetry between spectra
on two different radii….
D-branes
Only closed strings see the new direction… not open strings…
Electromagnetic
particles live on a
sub-space or
“brane”
Gravitons live in a higher
dimension “bulk”
Dirichlet
boundary
conditions
The extra scalar measures the position
of a dynamical brane in the bulk….
Including multiple D-branes in the space provides flavour indices for
the string ends points…. U(1) -> U(N) gauge symmetries
Dirac-Born-Infeld Low Energy
Effective Action
Determined by space-time symmetries….
cf a point particle (0-brane)
+
cf a 1-brane (string)
+
Brane Induced Metrics
1 AdS/CFT Correspondence
All fields in
adjoint
representation
b = 0 - this N=4
theory has no
running coupling –
it’s conformal!!
The space-time
symmetries are
boosted to SO(2,4)
Scalar Moduli Space
SO(6) global symmetry on 6 scalars… fermions transform as 4
of SO(6) due to
Potential:
Provided we pick the same generator the 6 scalar vevs can be any
value… physically inequivalent vacua…
N
Generically U(N) -> U(1)
As we decouple string modes we must leave finite values for
vevs….
Remember the scalar vevs describe the position of the D3s in the
bulk geometry….
AdS5 x
5
S
Symmetries of the Geometry
The five sphere
isometries are SO(6)…
The AdS space has an
SO(2,4) isometry group
& same moduli space…
Duality?
Note with a’ -> 0 Newton’s constant has gone to zero in the
gauge theory and the GR theory is completely disconnected…
Gravity approximate strings only at large g s N
Gravity is only weakly coupled for small g s
We’re at large ‘tHooft coupling in gauge theory…
Maldacena – are these two copies of the
same physics?
Extra 5th dimension?
Different u is looking at
gauge theory at different
RG scale…..
…. red shift in the black
hole geometry…
Kim, Romans, van
Nieuwenhuizen – 1985
Had computed the sugra
spectrum on AdS5xS5…
All sugra states match
duality conjecture… (but
do not give full operator
spectrum – string states
important too!)
The latter is the dilaton - also dual to the gauge coupling – g s = g 2
YM
Glueballs
The N=4 theory is conformal and so doesn’t have bound states…
A simple cure is to compactify the 3d space in which the N=4
theory lives to introduce a scale and break conformal invariance..
Lets just do the case where the 3+1d space becomes a 3-torus with
all radii equal….
We can now investigate the glueball spectrum…