Gauge/Gravity Duality

Prof Nick Evans

Big picture – slides

Key computations - board

TODAY

Introduction

Strings & Branes

AdS/CFT Correspondence

QCD-like dynamics lecture 2 & 3

ResourcesThe classic review of the AdS/CFT Correspondence isLarge 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 isMesons in Gauge/Gravity Duals - A Review.Johanna Erdmenger, Nick Evans, Ingo Kirsch, Ed Threlfall Eur.Phys.J.A35:81-133,2008, arXiv:0711.4467 [hep-th]

Transport properties of gluon plasmaViscosity, 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/

D-branes – Clifford Johnson

• 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

0 Introduction

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 TrajectoriesString theory was originally proposed to explain

Stringy Hints – ‘tHooft says Large N

At large N perturbation theory simplifies and restricts to planar diagrams…

Dense gluons -> a string world sheet?

Quark production & hence string breaking is forbidden at large NThe expansion parameter is

N

N Nc

2

f

c

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-StringsTo make a consistent theory add fermion partners to the X 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…

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!

The links to QCD seem utterly lost… until…

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 BranesT- 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-branesOnly closed strings see the new direction… not open strings…

Gravitons live in a higher dimension “bulk”

Electromagnetic particles live on a sub-space or “brane”

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

Dirichlet boundary conditions

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

this N=4 theory has no running coupling – it’s conformal!!

The space-time symmetries are boosted to SO(2,4)

Scalar Moduli Space

Provided we pick the same generator the 6 scalar vevs can be any value… physically inequivalent vacua…

Generically U(N) -> U(1)N

SO(6) global symmetry on 6 scalars… fermions transform as 4 of SO(6) due to

Potential:

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….

AdS x S55

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 ’ -> 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 N

Gravity is only weakly coupled for small gs

s

Maldacena – are these two copies of the same physics?

We’re at large ‘tHooft coupling in gauge theory…

Extra 5th dimension?

Different u is looking at gauge theory at different RG scale…..

…. red shift in the black hole geometry…

The latter is the dilaton - also dual to the gauge coupling – g = gs YM2

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!)

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…