dr jonathan shock (uct)

Strongly Coupled Field Theories and the Holographic Principle

Friday, 19 September 14

Aims of this talk

• By the end you should know:• What a holographic duality is• What the AdS/CFT correspondence is• How it helps us to understand strong

coupling phenomena• That it leads us to insights into emergent

spacetime


The problem of strong coupling

• We rely heavily on perturbative analysis in many fields.

• Strong coupling phenomena occur all around us:• Superconductivity (Gubser)

• Confinement (Maldacena, Klebanov)

• Chiral Symmetry breaking (Evans, Myers, JS)

• Rabi splitting (Yokoyama)

• Gravity in the ultra early universe (Skenderis)

• The quantum hall effect (Kraus)Friday, 19 September 14

ways to tackle strong coupling in quantum

field theories• Expand in another small parameter

• Inverse of a heavy quark mass• Inverse of a large number of colours - this

rearranges itself into a string theory!• Put the system on the lattice (truly non-

perturbative)• Processing power becomes a limiting factor• Finite chemical potential causes problems


Path Integral Ambiguities

C2(x1, x2) = !!(x1)!(x2)" =

!D!e#

!L[!]dxd!(x1)!(x2)!

D!e#!L[!]dxd

Observables lose the direct information about the degrees of freedom in the theory

The answer is just a function of spacetime points or momentum


• We can sometimes make a change of variables• Alter the apparent degrees of freedom of the

theory by integrating out and shuffling the dynamics

• May take us from a non-perturbative to a perturbative picture

Path Integral Ambiguities

C2(x1, x2) = !!(x1)!(x2)" =

!D!e#

!L[!]dxd!(x1)!(x2)!

D!e#!L[!]dxd


Examples of Dualities

Take a 3 dimensional QFT with particle excitations• Particle-Vortex duality (Burgess and Dolan; Murugan, JS et al)

arXiv:1404.5926


http://arxiv.org/abs/1404.5926

http://arxiv.org/abs/1404.5926


Find vortex solutions (depends on V: non-trivial homotopy of solution on the boundary)

p2 p 3 p

2 2 pAngular direction

p

2 p

3 p

4 p

5 p

6 pPhase of field

non-trivially wrapped solutions Æ vortices

• Particle-Vortex duality



Find a change of variables from particle degrees of freedom to vortex degrees of freedom by a process of integrating out in the path integral

Find vortex solutions (depends on V - non trivial homotopy of solution on the boundary)

• Particle-Vortex dualityTake a 3 dimensional QFT with particle excitations



Find a change of variables from particle degrees of freedom to vortex degrees of freedom by a process of integrating out in the path integral

Find vortex solutions (depends on V - non trivial homotopy of solution on the boundary)

• Particle-Vortex dualityTake a 3 dimensional QFT with particle excitations

Left with an interacting theory of vortices - may be weakly interacting when the particles are strongly interacting


Gravitational duality

• How about if we had two theories which were dual to one another but lived in completely different spaces?

• One is a strongly coupled field theory with a lot of symmetries

• The other is a weakly coupled gravitational theory in higher dimensions


holographic dualitiesTwo descriptions of (mem)branes in string theory

They will look incredibly different:

A gauge theory in four dimensions

A theory of gravity in ten dimensions

But they are holographically linked!


D-branes

An extended hypersurface:

Open strings end on them

Closed strings couple to them

They are a coherent state of closed strings which include gravitons

They come in different dimensions depending on the particular string theory - we will concentrate on D3-branes (3 spatial dimensions + 1 time direction)


We will show:There are two descriptions of D3-branes

The two descriptions are in different numbers of dimensions

Yet are dual to one another

We can use one description to answer questions about the other


Stack multiple branes in the same place...a symmetry arises from this: SU(N)

The world volume theory of a D3-brane

N D3-branes

The open strings have two ends on the branes - two

labels ... the adjoint representation of SU(N)

gluons and more


Low energy description of open strings on the brane: a well known field theory with a very special symmetry group:

The conformal group: Transformations leaving all angles intact (includes simple scalings)


4d conformal group

(SO(4,2))

2d conformal group



The theory has lots of exotic

spacetime symmetries

N=4 Super Yang Mills SU(N) gauge theory

Remember, we are describing the excitations on the D3-brane world volume. Two parameters associated with this: N and

λ=g N2





The theory has massless

vector fields



λ=g N2






vector fields

The theory has lots of internal symmetries...



λ=g N2






vector fields

The theory has lots of internal symmetries...

...which are local



λ=g N2


Now the second perspective: The D3 brane stack As a gravitational

soliton


D-branes from supergravity

Take a stack of N D3-branes in ten dimensional flat space

Massive objects: curve the surrounding spacetime:



Take a stack of N D3-branes


added on the worldsheet beyond the four embedding coordinates of the string to ensureconsistency of the theory. In the standard quantization of four dimensional stringtheory an additional field called the Liouville field arises [4], which may be interpretedas a fifth space-time dimension. Polyakov has suggested [47, 48] that such a fivedimensional string theory could be related to four dimensional gauge theories if thecouplings of the Liouville field to the other fields take some specific forms. As we willsee, the AdS/CFT correspondence realizes this idea, but with five additional dimensions(in addition to the radial coordinate on AdS which can be thought of as a generalizationof the Liouville field), leading to a standard (critical) ten dimensional string theory.

1.3 Black p-Branes

The recent insight into the connection between large N field theories and string theoryhas emerged from the study of p-branes in string theory. The p-branes were originallyfound as classical solutions to supergravity, which is the low energy limit of stringtheory. Later it was pointed out by Polchinski that D-branes give their full stringtheoretical description. Various comparisons of the two descriptions led to the discoveryof the AdS/CFT correspondence.

1.3.1 Classical Solutions

String theory has a variety of classical solutions corresponding to extended black holes[49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. Complete descriptions of all possible blackhole solutions would be beyond the scope of this review, and we will discuss here onlyillustrative examples corresponding to parallel Dp branes. For a more extensive reviewof extended objects in string theory, see [60, 61].

Let us consider type II string theory in ten dimensions, and look for a black holesolution carrying electric charge with respect to the Ramond-Ramond (R-R) (p + 1)-form Ap+1 [50, 55, 58]. In type IIA (IIB) theory, p is even (odd). The theory containsalso magnetically charged (6!p)-branes, which are electrically charged under the dualdA7!p = "dAp+1 potential. Therefore, R-R charges have to be quantized according tothe Dirac quantization condition. To find the solution, we start with the low energye!ective action in the string frame,

S =1

(2!)7l8s

!d10x

#!g

"

e!2!#R + 4($")2

$! 2

(8 ! p)!F 2

p+2

%

, (1.9)

where ls is the string length, related to the string tension (2!#")!1 as #" = l2s , and Fp+2

is the field strength of the (p + 1)-form potential, Fp+2 = dAp+1. In the self-dual caseof p = 3 we work directly with the equations of motion. We then look for a solution

16

Actionp=3

F=self-dual 5 formɸ=dilaton=const

R=Ricci scalar

g=det(metric)

ls=string length



Take a stack of N D3-branes



X

zoom into the region close to the branes

The ‘near horizon’ geometry is a direct product of a 5d anti-de-Sitter space and an S

Anti-de-Sitter space has a boundary at infinity that can be reached by light in finite time

The boundary of this space is 4d Minkowski space

The isometries are SO(4,2) and SO(6)

AdS5

5S

X

5


Here comes the magic:

There is a dictionary to translate questions between the two theories:

I can ask a question about correlation functions in one, which I can’t answer within that description

I can translate the question into the language of the other description

and answer it very easily

This is the beauty of the AdS/CFT correspondence!Friday, 19 September 14

OHxL

OHyL

AdS5x S5

M4

<OHxLOHyL>~∂fx∂fyŸ SB‚x4

Boundary values of fields in the gravity theory are sources for operators in the

gauge theory!D!e!SCFT!

!dxO!0 = Zgravity(!boundary = !0)


Tests of the correspondence

• There is a one to one correspondence between global symmetries on both sides

• There is a one to one correspondence between field theory operators and gravitational fields (including all KK-modes)

• This is vital for the operational form of the correspondence to make sense.

Maldacena; Gubser, Klebanov, Polyakov; Witten


Tests of the correspondence

• Dynamical tests are much harder:• Strong coupling calculations are hard• We can look at the system in limits of

integrability and in these cases we get agreement between the two sides.

• There are two parameters in the theory and by taking limits we can do exact comparisons

Bianchi - (Non-)perturbative tests of the AdS/CFT correspondence

See work by Berenstein, Honda et al.Friday, 19 September 14

AdS/QCD

• N=4 Super Yang Mills is a nice playground but it’s not QCD.• Highly supersymmetric• Conformal• Large N


AdS/QCD

• We can break the supersymmetry and the conformal symmetry

• 3 is pretty close to infinity, no?


AdS/QCD

• We can add quarks by adding new objects on the gravity side:• Space filling D7 branes


AdS/QGP• If we turn on finite temperature in the

field theory we can look at the quark-gluon plasma produced at the LHC and other heavy ion experiments.

• In the gravity side the thermal nature of the field theory corresponds to adding a horizon: Look at a black hole solution in AdS space


Results from AdS/QGP

• Meson melting phase transition• Phase structure in the presence of

background fields• Superconducting ground state for QCD

at very high magnetic field and finite temperature


Superconducting QCD groundstate

R ! 1 (or R ! 0), the free energy increases. Intuitively one can understand this by making useof the properties of Abrikosov vortices that we understand from type II superconductors. Thesevortices repel. Since R ! 1 and R ! 0 correspond to elongating the rhombic lattice cell (whilekeeping the area constant) neighbouring vortices are squeezed together, and since they repel, thisis energetically unfavourable. The energy di↵erence as a function of R is plotted in figure 5. In

1 2 3 4R

-0.0170

-0.0165

-0.0160

-0.0155

-0.0150

-0.0145

DW

Figure 5. The change in free energy density as a function of R = Lx/Ly, the ratio of side lengths of a

constant area lattice cell. This plot is for the AdS Schwarzschild model, but the plot for the hard wall

model is identical up to a rescaling of the axes. When R = 1, the lattice is square and the free energy

achieves a local maximum. When R =p3 and 1/

p3, the lattice is triangular and the free energy is at a

global minimum. Note that the plot has the symmetry �⌦(R) = �⌦(1/R), which simply corresponds to

swapping the x, y-axes.

figure 6 we present the contour plot of the modulus squared of the condensate in the x, y-planefor the minimum energy solution corresponding to the triangular lattice. This is calculated bystudying the boundary expansion of the field Ex and reading o↵ the normalisable term. We could

0 2 4 6 80

1

2

3

4

5

x Bc

yB c

Figure 6. A contour plot of the modulus squared of the field theory condensate dual to Ex in the ground

state triangular lattice. Darker colours mean smaller values.

also plot the magnetisation of the ground state, which is found from the normalisable term in theboundary value expansion of @xa3y � @ya

3x. However, it takes the same form as the condensate and

the numerics indicate that it di↵ers only up to a scale.

– 15 –

Prepared for submission to JHEP MPP-2012-144

Magnetic field induced lattice ground states from

holography

Yan-Yan Bu,

a,bJohanna Erdmenger,

aJonathan P. Shock,

aMigael Strydom

a

aMax-Planck-Institut fur Physik (Werner-Heisenberg-Institut),

Forhringer Ring 6, 80805 Munchen, Germany.bState Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science,

Beijing 100190, People’s Republic of China

E-mail: (yybu,jke,jonshock,mstrydom)@mppmu.mpg.de

Abstract: We study the holographic field theory dual of a probe SU(2) Yang-Mills field in abackground (4 + 1)-dimensional asymptotically Anti-de Sitter space. We find a new ground statewhen a magnetic component of the gauge field is larger than a critical value. The ground state formsa triangular Abrikosov lattice in the spatial directions perpendicular to the magnetic field. Thelattice is composed of superconducting vortices induced by the condensation of a charged vectoroperator. We perform this calculation both at finite temperature and at zero temperature with ahard wall cuto↵ dual to a confining gauge theory. The study of this state may be of relevance toboth holographic condensed matter models as well as to heavy ion physics. The results shown hereprovide support for the proposal that such a ground state may be found in the QCD vacuum whena large magnetic field is present.

Keywords: AdS/CFT correspondence, Abrikosov lattice, magnetic field, superconductivity

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AdS/CMT

• How about strong coupling phenomena in condensed matter systems?

• Unconventional superconductors remain a mystery. We can create a model of superconductors.

• We can model cuprate superconductors with some good accuracy!


Emergent Spacetime

• Use the AdS/CFT correspondence to understand spacetime as an emergent phenomena from strongly coupled field theories.

Preprint typeset in JHEP style - PAPER VERSION WITS-CTP-048, UCT-CGG-251109

Emergent Spacetime

Robert de Mello Koch

1,2and Je↵ Murugan

2,3

1National Institute for Theoretical Physics,

Department of Physics and Centre for Theoretical Physics,

University of the Witwatersrand,

Wits, 2050,

South Africa

2National Institute for Theoretical Physics,

Stellenbosch,

South Africa

3Cosmology and Gravity Group,

Department of Mathematics and Applied Mathematics,

University of Cape Town,

Private Bag, Rondebosch, 7700,

South Africa

E-mail: [email protected],[email protected]

Abstract: We give an introductory account of the AdS/CFT correspondence in the12 -BPS sector of N = 4 super Yang-Mills theory. Six of the dimensions of the stringtheory are emergent in the Yang-Mills theory. In this article we suggest how thesedimensions and local physics in these dimensions emerge. The discussion is aimed atnon-experts.

Keywords: AdS/CFT correspondence, super Yang-Mills theory, holography.

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See also the work of Berenstein, Takayanagi, Balasubramanian, Ramgoolam,...


Information Geometry

• Can we study the instanton moduli space of field theories and find the Fisher metric via an information geometric framework to find the emergent geometry in a natural way?

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Instantons, the Information Metric,

and the AdS/CFT Correspondence

Matthias Blau, K.S. Narain, George Thompson

The Abdus Salam ICTP

Strada Costiera 11

34014 Trieste, Italy

mblau/narain/[email protected]

Abstract

We describe some remarkable properties of the so-called Information Metric on

instanton moduli space. This Metric is manifestly gauge and conformally invariant

and coincides with the Euclidean AdS5-metric on the one-instanton SU(2) moduli

space for the standard metric on R4. We propose that for an arbitrary boundary

metric the AdS/CFT bulk space-time is the instanton moduli space equipped with

the Information Metric.

To test this proposal, we examine the variation of the instanton moduli and the

Information Metric for first-order perturbations of the boundary metric and obtain

three non-trivial and somewhat surprising results: (1) The perturbed Information

Metric is Einstein. (2) The perturbed instanton density is the corresponding mass-

less boundary-to-bulk scalar propagator. (3) The regularized boundary-to-bulk

geodesic distance is proportional to the logarithm of the perturbed instanton den-

sity. The Hamilton-Jacobi equation implied by (3) equips the moduli space with a

rich geometrical structure which we explore.

These results tentatively suggest a picture in which the one-instanton sector of

SU(2) Yang-Mills theory (rather than some large-N limit) is in some sense holo-

graphically dual to bulk gravity.

1

See upcoming work by Murugan and J.S


Some reasonable criticisms

• The AdS/CFT correspondence is not proven

• The results we have found are not that close to the real world

• We aren’t able to go much further than large N


The Future

• A more accurate picture of unconventional superconductors

• More insight into the deformations we can make to the correspondence

• A deeper understanding into emergent spacetime

• A formal proof of the correspondence


Thank you for your attention!

Acknowledgements: Sam Harris, I Ning Huang and collaborators!


dr jonathan shock (uct)

Education