dr jonathan shock (uct)
TRANSCRIPT
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
Friday, 19 September 14
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
Friday, 19 September 14
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
Friday, 19 September 14
• 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
Friday, 19 September 14
Examples of Dualities
Take a 3 dimensional QFT with particle excitations• Particle-Vortex duality (Burgess and Dolan; Murugan, JS et al)
arXiv:1404.5926
Friday, 19 September 14
Examples of Dualities
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
Friday, 19 September 14
Examples of Dualities
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
Friday, 19 September 14
Examples of Dualities
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
Friday, 19 September 14
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
Friday, 19 September 14
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!
Friday, 19 September 14
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!
Friday, 19 September 14
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)
Friday, 19 September 14
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
Friday, 19 September 14
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
Friday, 19 September 14
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)
The world volume theory of a D3-brane
4d conformal group
(SO(4,2))
2d conformal group
Friday, 19 September 14
The world volume theory of a D3-brane
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
Friday, 19 September 14
The world volume theory of a D3-brane
The theory has lots of exotic
spacetime symmetries
The theory has massless
vector fields
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
Friday, 19 September 14
The world volume theory of a D3-brane
The theory has lots of exotic
spacetime symmetries
The theory has massless
vector fields
The theory has lots of internal 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
Friday, 19 September 14
The world volume theory of a D3-brane
The theory has lots of exotic
spacetime symmetries
The theory has massless
vector fields
The theory has lots of internal symmetries...
...which are local
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
Friday, 19 September 14
Now the second perspective: The D3 brane stack As a gravitational
soliton
Friday, 19 September 14
D-branes from supergravity
Take a stack of N D3-branes in ten dimensional flat space
Massive objects: curve the surrounding spacetime:
Friday, 19 September 14
D-branes from supergravity
Take a stack of N D3-branes
Massive objects: curve the surrounding spacetime:
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
Friday, 19 September 14
D-branes from supergravity
Take a stack of N D3-branes
Massive objects: curve the surrounding spacetime:
Friday, 19 September 14
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
Friday, 19 September 14
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)
Friday, 19 September 14
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
Friday, 19 September 14
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
Friday, 19 September 14
AdS/QCD
• We can break the supersymmetry and the conformal symmetry
• 3 is pretty close to infinity, no?
Friday, 19 September 14
AdS/QCD
• We can add quarks by adding new objects on the gravity side:• Space filling D7 branes
Friday, 19 September 14
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
Friday, 19 September 14
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
Friday, 19 September 14
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!
Friday, 19 September 14
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,...
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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
Friday, 19 September 14
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
Friday, 19 September 14
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
Friday, 19 September 14
Thank you for your attention!
Acknowledgements: Sam Harris, I Ning Huang and collaborators!
Friday, 19 September 14