Here are the course materials (including TeXed lecture notes and problem sets) from a graduate class on the AdS/CFT correspondence at MIT in Fall 2008:

A guide to the problem sets:
Problems 4,5,6,7 of pset 3 may provide a useful way to familiarize oneself with the geometry of AdS (several of these are from Barton Zwiebach's textbook, 2d edition).
Pset 4 should be accessible after Wednesday's lecture.
Pset 5 problem 4 is relevant to the calculation of the Hawking temperature described by Gary.
Pset 6 problems 2,3,4 should be fun after Thursday's lecture.

A guide to the lecture notes:
The material in Tuesday's lecture is discussed in: lecture 3 pages 4-6, lecture 4, the beginning of lecture 7.
't hooft counting: end of 7, beginning of 8.
calculation of two-point functions in vacuum: end of lecture 11 through lecture 15(Note also the very geometric case of the large-dimension limit (where geometric optics can be used in the bulk) which is discussed at the beginning of lecture 19.)
finite temperature CFT from black holes in AdS: lecture 23

This paper (not a review article) takes seriously the idea that the energy scale in real-space RG can be treated as an extra dimension. At the very least, it should be a useful discussion of many features of the correspondence for those familiar with these new real-space RG techniques: Entanglement renormalization and holography (Brian Swingle)

Quantum criticality lectures:

References for Lecture II (Mike Hermele)
A discussion of the boson Hubbard model can be found in chapters 10 and 11 of Subir Sachdev's book.
The original papers on deconfined criticality are here and here. This paper is not a review, but contains a good deal of pedagogical discussion of algebraic spin liquids. (Note also there is an erratum.)
If you want to understand the thinking behind splitting spins into slave particles and using that construction to generate effective theories (or trial wavefunctions), this paper by Xiao-Gang Wen is essential reading.
This recent review by Sachdev covers many of the topics of this lecture, and also devotes significant attention to experimental systems. Other potentially useful reviews are here and here.

Holographic Superconductors (and Superfluids):

References for a seminar by C. P. Herzog:
In my talk, I described two simple holographic models of a superconductor (or superfluid). The first model consisted of a U(1) gauge field and a scalar, and I think was described originally by Steve Gubser in the present context in the paper http://arxiv.org/abs/0801.2977. Inspired by this work, Gary Horowitz, Sean Hartnoll, and I wrote a short paper http://arxiv.org/abs/0803.3295 and a longer follow-up http://arxiv.org/abs/0810.1563 that emphasize the superconducting aspects of this model. Pavel Kovtun, Dam Son, Amos Yarom and I have two papers that emphasize the superfluid aspects of this model: http://arxiv.org/abs/0809.4870 and http://arxiv.org/abs/0906.4810.

The second model replaced the U(1) gauge field with an SU(2) gauge field and had no scalar field. An early paper on this model is http://arxiv.org/abs/0803.3483 . In the talk, I described some analytic results from my paper http://arxiv.org/abs/0902.0409 with Silviu Pufu.

Sean Hartnoll's review article was mentioned by John McGreevy above. Another review article is a write-up of my Trieste lectures, http://arxiv.org/abs/0904.1975 .

Subir Sachdev
My talk covered the material in these papers: http://arxiv.org/abs/hep-th/0701036 (quantum critical transport of the M2 CFT and duality constraints) http://arxiv.org/abs/0706.3215 (adding chemical potential and magnetic field, leading to Nernst effect)

Cenke Xu
The whole philosophy is that, in condensed matter systems at high energy there is no gauge symmetry, but at low energy the local constraint imposed energetically will lead to gauge symmetries. This philophy is summarized as LCGGSi.e. Local Constraints Generate Gauge Symmetries. This is how we obtain all the dynamical gauge field in condensed matter systems.

In my talk I started with the simplest model which can lead to emergent gauge symmetry: the quantum dimer model, and discussed the Z4 conservation of magnetic flux in this model. This Z4 conservation is the key of deconfine quantum criticality. Then I moved on to spin liquid states with U(1) gauge field and fermionic spinon excitations, and also discussed what kind of system this formalism can be applied to. In the end I discussed how to obtain SU(N) gauge field with large N in condensed matter system, which is in general obtained from multi-orbital spin system.

Sung-Sik Lee
The papers by Schafer and Schwenzer (about the not-worseness IR singularities at higher-order in perturbation theory) that came up during the talk were: http://arXiv.org/pdf/hep-ph/0405053, see in particular the discussion around pages 3-5. http://arXiv.org/pdf/hep-ph/0512309, this is the paper with the full story, in the QCD context.

There is also a distilled (perhaps over-distilled) version of this story on pages 27-29 of http://arXiv.org/pdf/0709.4635, which is a big review paper that thomas schafer and krishna rajagopal are coauthors of.

Holography & Dynamical Critical Phenomena

Michael Mulligan
In my talk, I discussed recent work on gravity duals to field theories with dynamical critical exponent different from unity. I began by introducing the Lifshitz field theory in the context of quantum dimer models. I then discussed candidate gravity duals to anisotropic fixed points, and then described the computation of a certain two-point correlation function and holographic RG flow in these candidate spacetimes. I concluded with an overview of more recent work: (1) gravity duals to anisotropic field theories at finite temperature, (2) a z=2 gauge theory in 2+1 dimensions, (3) entanglement entropy in these spacetimes, and (4) string theory embeddings. Below are references for various parts of my talk.

Johanna Erdmenger
I presented a holographic superconductor for which the field theory Lagrangian in 3+1 dimensions is explicitly known, and for which there is a stringy picture of the condensation process. This model arises from embedding two coincident D7 branes into the 10-dimensional AdS black hole geometry. This amounts to adding degrees of freedom in the fundamental representation of the gauge group to the original N=4 Super Yang-Mills theory. The black hole background puts the field theory at finite temperature. Moreover, an SU(2) isospin chemical potential and the corresponding density are turned on by considering a non-trivial gauge field in the Dirac-Born-Infeld action for the two coincident D7 branes.

The ideas of adding flavor to AdS/CFT using D7 brane probes is due to Karch and Katz http://arxiv.org/abs/hep-th/0205236.
The phase diagram at finite temperature and finite baryon density in this approach is described by Kobayashi, Mateos, Matsuura, Myers and Thomson in http://arxiv.org/abs/hep-th/0611099.

Holographic Entanglement Entropy

Tadashi Takayanagi
In my talk, I gave an overview of recent developments on holographic calculations of entanglement entropy. The entanglement entropy (EE) is interesting from the viewpoint
of both quantum gravity and condensed matter physics. Historically, the EE in QFT has been first discussed to understand the microscopic origin of black hole entropy. Recently, it has been noticed that the EE can be an order parameter of quantum phase transition. In the holographic description (via e.g. AdS/CFT), the EE in a QFT turns out to be equal to the area of a certain minimal surface divided by 4G_N, which looks the same as the
Bekenstein-Hawking formula of black hole entropy. By using the holographic dual of confining gauge theories, we can confirm that the EE is a good order parameter of confining/deconfining transion. I also explained a holographic caluculation of EE in a IIB supergravity which is supposed to be dual to a Lifshitz-like anisotropic scale invariant theory (the gravity dual of Lifshitz-like theories has been explained in detail by Mulligan's talk in this workshop). Our type IIB string theory background was originally obtained from a D3-D7 brane model which gives a holographic dual of fractional quantum Hall effect.

Most of what was discussed in the talk appears (in greater detail) in this paper. An important precursor of these results is the paper of Alicea et. al. A cool picture "illustrating" the duality can be found here.

## Table of Contents

## Introductory background:

Lectures on holographic methods for condensed matter physics (Sean Hartnoll)Lectures on Holographic Superfluidity and Superconductivity (Chris Herzog)

Quantum criticality and black holes (Subir Sachdev and Markus Muller)

## General relativity:

Gravity: An Introduction to Einstein's General Relativity (James Hartle)Spacetime and Geometry (Sean Carroll)

## AdS/CFT lectures:

John McGreevy## Lecture Notes

Here are some notes based on the lectures:applied_AdS.pdf(Any helpful comments on the notes are much appreciated.)

Sean's notes are very much in the spirit of the present enterprise:

Lectures on holographic methods for condensed matter physics (Sean Hartnoll)

This paper gives a derivation of the duality with liberal use of hindsight and no use of string theory:

Gauge/gravity duality (Gary Horowitz and Joe Polchinski)

The next two review articles do less to hide the string theory lurking in the UV:

Large-N field theories, string theory and gravity (MAGOO)

TASI lectures on AdS/CFT (Juan Maldacena)

This review article emphasizes aspects of the correspondence related to supersymmetry, but has a very clear description of the computation of CFT correlation functions in vacuum (the goal of lecture 2):

Supersymmetric gauge theories and the AdS/CFT correspondence (Eric D'Hoker and Dan Freedman)

For real-time calculations:

Viscosity, black holes and quantum field theory (Dam Son and Andrei Starinets)

Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm (Nabil Iqbal and Hong Liu)

Here are the course materials (including TeXed lecture notes and problem sets) from a graduate class on the AdS/CFT correspondence at MIT in Fall 2008:

## Applied String Theory

A guide to the problem sets:Problems 4,5,6,7 of pset 3 may provide a useful way to familiarize oneself with the geometry of AdS (several of these are from Barton Zwiebach's textbook, 2d edition).

Pset 4 should be accessible after Wednesday's lecture.

Pset 5 problem 4 is relevant to the calculation of the Hawking temperature described by Gary.

Pset 6 problems 2,3,4 should be fun after Thursday's lecture.

A guide to the lecture notes:The material in Tuesday's lecture is discussed in: lecture 3 pages 4-6, lecture 4, the beginning of lecture 7.

't hooft counting: end of 7, beginning of 8.

calculation of two-point functions in vacuum: end of lecture 11 through lecture 15(Note also the very geometric case of the large-dimension limit (where geometric optics can be used in the bulk) which is discussed at the beginning of lecture 19.)

finite temperature CFT from black holes in AdS: lecture 23

This paper (not a review article) takes seriously the idea that the energy scale in real-space RG can be treated as an extra dimension. At the very least, it should be a useful discussion of many features of the correspondence for those familiar with these new real-space RG techniques: Entanglement renormalization and holography (Brian Swingle)

## Quantum criticality lectures:

References for Lecture II (Mike Hermele)A discussion of the boson Hubbard model can be found in chapters 10 and 11 of Subir Sachdev's book.

The original papers on deconfined criticality are here and here.

This paper is not a review, but contains a good deal of pedagogical discussion of algebraic spin liquids. (Note also there is an erratum.)

If you want to understand the thinking behind splitting spins into slave particles and using that construction to generate effective theories (or trial wavefunctions), this paper by Xiao-Gang Wen is essential reading.

This recent review by Sachdev covers many of the topics of this lecture, and also devotes significant attention to experimental systems. Other potentially useful reviews are here and here.

Holographic Superconductors (and Superfluids):References for a seminar by C. P. Herzog:In my talk, I described two simple holographic models of a superconductor (or superfluid). The first model consisted of a U(1) gauge field and a scalar, and I think was described originally by Steve Gubser in the present context in the paper http://arxiv.org/abs/0801.2977. Inspired by this work, Gary Horowitz, Sean Hartnoll, and I wrote a short paper http://arxiv.org/abs/0803.3295 and a longer follow-up http://arxiv.org/abs/0810.1563 that emphasize the superconducting aspects of this model. Pavel Kovtun, Dam Son, Amos Yarom and I have two papers that emphasize the superfluid aspects of this model:

http://arxiv.org/abs/0809.4870 and http://arxiv.org/abs/0906.4810.

The second model replaced the U(1) gauge field with an SU(2) gauge field and had no scalar field. An early paper on this model is http://arxiv.org/abs/0803.3483 . In the talk, I described some analytic results from my paper http://arxiv.org/abs/0902.0409 with Silviu Pufu.

Sean Hartnoll's review article was mentioned by John McGreevy above. Another review article is a write-up of my Trieste lectures, http://arxiv.org/abs/0904.1975 .

## Fermions at criticality :

Sung-Sik LeeReferences on heavy fermions :

http://arxiv.org/abs/cond-mat/0612006 (my discussion on heavy fermion systems followed this paper)

http://arxiv.org/abs/cond-mat/0606317 (thorough review of experiments)

http://arxiv.org/abs/cond-mat/0604240 (brief and pedagogical review on theories)

General discussions on spin liquid can be found in :

http://arxiv.org/abs/0708.2115

http://arxiv.org/abs/cond-mat/0410445

2+1D spinon Fermi surface with U(1) gauge field :

http://arxiv.org/abs/cond-mat/9303037 (suggested that the theory is stable in the large N limit)

http://arxiv.org/abs/cond-mat/9406024 (similar conclusion with the above paper based on the computation of some higher order diagrams)

http://arxiv.org/abs/0804.3800 (non-perturbative proof of instanton suppression)

http://arxiv.org/abs/0905.4532 (large N limit; genus expansion; stability in a single patch theory)

File of the talk :Fermions_at_criticality.pdf## Quantum critical transport:

Subir SachdevMy talk covered the material in these papers:

http://arxiv.org/abs/hep-th/0701036 (quantum critical transport of the M2 CFT and duality constraints)

http://arxiv.org/abs/0706.3215 (adding chemical potential and magnetic field, leading to Nernst effect)

For a pre-AdS/CFT view on this topic see

http://arxiv.org/abs/cond-mat/9705206

The Boltzmann equation approach to non-zero chemical potential and/or magnetic field is discussed in

http://arxiv.org/abs/0805.1413 for graphene and in http://arxiv.org/abs/0811.0269

for the field theory of the Bose Hubbard model (by Bhaseen, Green, and Sondhi).

File of the talk: http://qpt.physics.harvard.edu/talks/kitp_adscmt.pdf

## Non-Fermi liquids & emergent quantum criticality from gravity

Hong LiuMy talk was based on the following papers

http://arxiv.org/abs/0907.2694 (with Faulkner, McGreevy and Vegh)

http://arxiv.org/abs/0903.2477 (with McGreevy and Vegh)

Calculating Fermionic retarded Green functions from gravity was discussed in detail in

http://arxiv.org/abs/0903.2596 (with Nabil Iqbal)

See also

http://arxiv.org/abs/0809.3402 (by Sung-Sik Lee)

http://arxiv.org/abs/0904.1993 (by Cubrovic, Zaanen and Schalm)

PDF file of my talk .

## Nonrelativistic holography I

Dam Thanh SonMy talk followed mostly my paper with Yusuke Nishida 0706.3746, and 0804.3972. See also 0804.4053 by Balasubramanian and McGreevy.

Two recent reviews of cold atom physics are:

Braaten and Hammer, cond-mat/0410417 (few-body aspects)

Giorgini, Pitaevskii and Stringari, Rev. Mod. Phys. 80, 1215 (2008) (mostly many-body aspects)

PDF file of my talk: kitp09-son.pdf

Emergent gauge fields in CMTCenke XuThe whole philosophy is that, in condensed matter systems at high energy there is no gauge symmetry, but at low energy the local constraint imposed energetically will lead to gauge symmetries. This philophy is summarized as

LCGGSi.e.Local Constraints Generate Gauge Symmetries. This is how we obtain all thedynamicalgauge field in condensed matter systems.In my talk I started with the simplest model which can lead to emergent gauge symmetry: the quantum dimer model, and discussed the Z4 conservation of magnetic flux in this model. This Z4 conservation is the key of deconfine quantum criticality. Then I moved on to spin liquid states with U(1) gauge field and fermionic spinon excitations, and also discussed what kind of system this formalism can be applied to. In the end I discussed how to obtain SU(N) gauge field with large N in condensed matter system, which is in general obtained from multi-orbital spin system.

References about nonabelian gauge field with multi-orbital spins:

arXiv:0906.3727

arXiv:0906.3718

References about SU(2) gauge symmetry of spin-1/2 system (different from above):

cond-mat/0107071

References about SU(N) global symmetry in cold atom system:

arXiv:0905.2610

References about graviton gauge symmetry in condensed matter models:

arXiv:cond-mat/0602443

arXiv:cond-mat/0609595

References about spin liquid formalism in cuprates:

cond-mat/0410445

References about bosonic spinon formalism in cuprates:

arXiv:0804.1794

arXiv:0901.0005

References about spin liquid in organic material:

cond-mat/0607015

Critical Spin Liquid with Spinon Fermi SurfaceSung-Sik LeeThe papers by Schafer and Schwenzer (about the not-worseness IR singularities at higher-order in perturbation theory) that came up during the talk were:

http://arXiv.org/pdf/hep-ph/0405053, see in particular the discussion around pages 3-5.

http://arXiv.org/pdf/hep-ph/0512309, this is the paper with the full story, in the QCD context.

There is also a distilled (perhaps over-distilled) version of this story on pages 27-29 of http://arXiv.org/pdf/0709.4635, which is a big review paper that thomas schafer and krishna rajagopal are coauthors of.

## Holography & Dynamical Critical Phenomena

Michael MulliganIn my talk, I discussed recent work on gravity duals to field theories with dynamical critical exponent different from unity. I began by introducing the Lifshitz field theory in the context of quantum dimer models. I then discussed candidate gravity duals to anisotropic fixed points, and then described the computation of a certain two-point correlation function and holographic RG flow in these candidate spacetimes. I concluded with an overview of more recent work: (1) gravity duals to anisotropic field theories at finite temperature, (2) a z=2 gauge theory in 2+1 dimensions, (3) entanglement entropy in these spacetimes, and (4) string theory embeddings. Below are references for various parts of my talk.

dimer models and the lifshitz field theory:

http://arxiv.org/abs/cond-mat/ 0311466

http://arxiv.org/abs/0809.3050

http://arxiv.org/abs/cond-mat/ 0311353

anisotropic black holes (without asymptotic Schrodinger symmetry):

http://arxiv.org/abs/0812.5088

http://arxiv.org/abs/0905.1136

http://arxiv.org/abs/0905.3183

z=2 gauge theories:

http://arxiv.org/abs/cond-mat/ 0408257

http://arxiv.org/abs/0811.2217

http://arxiv.org/abs/0906.3261

entanglement entropy:

http://arxiv.org/abs/hep-th/ 0405152

http://arxiv.org/abs/0906.1569

http://arxiv.org/abs/0904.4477

http://arxiv.org/abs/0905.0932

string embedding of anisotropic spacetime

http://arxiv.org/abs/0905.0688

Flavor Superconductors from String TheoryJohanna ErdmengerI presented a holographic superconductor for which the field theory Lagrangian in 3+1 dimensions is explicitly known, and for which there is a stringy picture of the condensation process. This model arises from embedding two coincident D7 branes into the 10-dimensional AdS black hole geometry. This amounts to adding degrees of freedom in the fundamental representation of the gauge group to the original N=4 Super Yang-Mills theory. The black hole background puts the field theory at finite temperature. Moreover, an SU(2) isospin chemical potential and the corresponding density are turned on by considering a non-trivial gauge field in the Dirac-Born-Infeld action for the two coincident D7 branes.

The talk followed the recent papers http://arxiv.org/abs/0810.2316 and http://arxiv.org/abs/0903.1864. More on isospin densities in AdS/CFT with flavor may be found in

http://arxiv.org/abs/0807.2663.

The ideas of adding flavor to AdS/CFT using D7 brane probes is due to Karch and Katz http://arxiv.org/abs/hep-th/0205236.

The phase diagram at finite temperature and finite baryon density in this approach is described by Kobayashi, Mateos, Matsuura, Myers and Thomson in http://arxiv.org/abs/hep-th/0611099.

## Holographic Entanglement Entropy

Tadashi TakayanagiIn my talk, I gave an overview of recent developments on holographic calculations of entanglement entropy. The entanglement entropy (EE) is interesting from the viewpoint

of both quantum gravity and condensed matter physics. Historically, the EE in QFT has been first discussed to understand the microscopic origin of black hole entropy. Recently, it has been noticed that the EE can be an order parameter of quantum phase transition. In the holographic description (via e.g. AdS/CFT), the EE in a QFT turns out to be equal to the area of a certain minimal surface divided by 4G_N, which looks the same as the

Bekenstein-Hawking formula of black hole entropy. By using the holographic dual of confining gauge theories, we can confirm that the EE is a good order parameter of confining/deconfining transion. I also explained a holographic caluculation of EE in a IIB supergravity which is supposed to be dual to a Lifshitz-like anisotropic scale invariant theory (the gravity dual of Lifshitz-like theories has been explained in detail by Mulligan's talk in this workshop). Our type IIB string theory background was originally obtained from a D3-D7 brane model which gives a holographic dual of fractional quantum Hall effect.

The original papers of holographic entanglement entropy are

http://arxiv.org/abs/hep-th/0603001

http://arxiv.org/abs/hep-th/0605073 and a recent overview can be found in

http://arxiv.org/abs/0905.0932

Excellent review articles of entanglement entropy in QFT

are recently written by Calabrese and Cardy (for CFTs):

http://arxiv.org/abs/0905.4013and by Casini and Huerta (for free QFT):

http://arxiv.org/abs/0905.2562

Its application to confinement/deconfinement transition can be found in

http://arxiv.org/abs/hep-th/0611035

A holographic proof of strong subadditivity has been done in

http://arxiv.org/abs/0704.3719

A holographic calculation has been extended to time-depent backgrounds in

http://arxiv.org/abs/0705.0016

The type IIB string theory dual of a Lifshitz-like theory

has been constructed in

http://arxiv.org/abs/0905.0688 whose type IIB background has been motivated by a string dual of fractional quantum Hall effect based on a D3-D7 system in http://arxiv.org/abs/0901.0924

Fermionic dual of O(4) vector modelMost of what was discussed in the talk appears (in greater detail) in this paper. An important precursor of these results is the paper of Alicea et. al. A cool picture "illustrating" the duality can be found here.