## Monday, July 18, 2011

### The Non-Witten Half of IAS - Princeton

In physics, the AdS/CFT correspondence (anti de Sitter/conformal field theory correspondence), sometimes called the Maldacena duality, is the conjecturedequivalence between a string theory and gravity defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whosedimension is lower by one or more. The name suggests that the first space is the product of anti de Sitter space (AdS) with some closed manifold like sphereorbifold, ornoncommutative space, and that the quantum field theory is a conformal field theory (CFT).[1]
An example is the duality between Type IIB string theory on AdS5 × S5 space (a product of five dimensional AdS space with a five dimensional sphere) and a supersymmetricN = 4 Yang–Mills gauge theory (which is a conformal field theory) on the 4-dimensional boundary of AdS5. It is the most successful realization of the holographic principle, a speculative idea about quantum gravity originally proposed by Gerard 't Hooft and improved and promoted by Leonard Susskind.
The AdS/CFT correspondence was originally proposed by Juan Maldacena in late 1997.[2] Important aspects of the correspondence were given in articles by Steven GubserIgor Klebanov and Alexander Markovich Polyakov,[3] and by Edward Witten.[4] The correspondence has also been generalized to many other (non-AdS) backgrounds and their dual (non-conformal) theories. In about five years, Maldacena's article had 3000 citations and became one of the most important conceptual breakthroughs in theoretical physics of the 1990s, providing many new lines of research into both quantum gravity and quantum chromodynamics (QCD).

## Maldacena's example

It all began with Maldacena's observation. A stack of N D3-branes in type IIB string theory has massless brane fields residing on it. With respect to the brane, they form Yang–Mills supermultiplets transforming under $\mathcal{N} = 4$ SUSY in 3+1D. The vector hypermultiplets form a gauge group $U(N)\cong SU(N)\times U(1)$. This isn't quite a conformal field theory, even though it runs to one in the infrared once gravitation and string dynamics decouple. In the infrared, the U(1) hypermultiplet decouples, but the SU(N) hypermultiplets remain interacting as the beta function is zero. The metric background is given by an extremal 3-brane black hole. The event horizon is infinitely far away; the distance to it diverges logarithmically. The near horizon geometry is approximately $AdS_5 \times S^5$ with the approximation becoming more and more exact closer to the horizon. Now, take the scaling limit as the string scale goes to zero with the string coupling kept fixed. All the string and gravitational dynamics decouple, and the U(1) hypermultiplet too. We are left with a bona fide $\mathcal{N}=4$ superconformal field theory. If we take the limit in which we are always in the near horizon region, the geometry becomes exactly $AdS_5 \times S^5$. A D3-brane has a self-dual charge under the self-dual NS 5-form flux. A stack of N of them gives rise to an integral flux of N over S5

## Conformal boundary

A suitable Weyl transformation assures that AdS has a boundary. It turns out that this boundary is a conformal field theory having one less dimension. To make things more concrete, choose a particular coordinatization, the half-space coordinatization:
$ds^2 = (kz)^{-2}\left( dz^2 + \eta_{\mu\nu} \, dx^\mu \, dx^\nu \right).$
After a Weyl transformation ω = kz, we get
$\ ds^2 = dz^2 + \eta_{\mu\nu}\,dx^\mu \,dx^\nu,$
which has the Minkowski metric as the boundary at z = 0. This is called the conformal boundary.

## Source fields

Basically, the correspondence runs as follows; if we deform the CFT by certain source fields by adding the source $\int d^dx J_{CFT}(x)\mathcal{O}(x)$, this will be dual to an AdS theory with a bulk field J with the boundary condition
$\lim_{\text{boundary}} J \omega^{\Delta-d+k} = J_{\text{CFT}} \,$
where Δ is the conformal dimension of the local operator $\mathcal{O}$ and k is the number of covariant indices of $\mathcal{O}$ minus the number of contravariant indices. Only gauge-invariant operators are allowed.
Here, we have a dual source field for every gauge-invariant local operator we have.
Using generating functionals, the relation is expressed as
$\left\langle \mathcal{T}\left\{ \exp\left(\int d^dx J_{4D}(x)\mathcal{O}(x)\right) \right\} \right\rangle_{CFT} = Z_{AdS}\left[\lim_{\text{boundary}} J \omega^{\Delta-d+k} = J_{4D}\right]$

The left hand side is the vacuum expectation value of the time-ordered exponential of the operators over the conformal field theory. The right hand side is the quantum gravity generating functional with the given conformal boundary condition. The right hand side is evaluated by finding the classical solutions to the effective action subject to the given boundary conditions.

The stress-energy operator on the CFT side is dual to the transverse components of the metric on the AdS side. Since the stress-energy operator has a conformal weight of 4, the AdS metric ought to go as ω − 2, which is true for AdS. Also, the graviton has to be massless, just as it should.
If there is a global internal symmetry G on the CFT side, its Noether current J will be dual to the transverse components of a gauge connection for a Yang–Mills gauge theory with G as the gauge group on the AdS side. Since J has a conformal weight of 3, the dual Yang–Mills gauge boson ought to have zero bulk mass, just as it should.
A scalar operator with conformal weight Δ will be dual to a scalar bulk field with a bulk mass of $k\sqrt{\Delta(\Delta-4)}$.

## Particles

A CFT bound state of size r is dual to a bulk particle approximately localized at z=r.

## Supersymmetry

We need to match up conformal supersymmetry in 4D with AdS supersymmetry in 5D. The symmetry supergroups in both cases happen to match up, as they should. There are $8\mathcal{N}$ real SUSY generators and the bosonic part consists of the conformal AdS group Spin(4,2) times an internal group $SU(\mathcal{N})_T \times U(1)_A$. See superconformal algebra for more details.
For the case $\mathcal{N}=4$, we have 32 real SUSY generators and an internal group $SU(4)_T\times U(1)_A$. Now, $SU(4) \cong \mathrm{Spin}(6)$ and Spin(6) is the isometry group of S5 with spinorial fields. The bosonic spatial isometry group of $\mathrm{AdS}_5 \times S^5$ is $\mathrm{Spin}(4,2) \times \mathrm{Spin}(6)$.
In $\mathcal{N}=(2,0)$ 10D SUSY, we have 32 real SUSY generators. In a generic curved spacetime, some of the SUSY generators will be broken but in the special compactification of $\mathrm{AdS}_5 \times S^5$ with both factors having the same radius, we are left 32 real unbroken generators. However, the bosonic spatial isometries with 55 generators in the flat case is now broken to $\mathrm{Spin}(4,2) \times \mathrm{Spin}(6)$ with 30 generators. $\mathcal{N}=(2,0)$ also has a U(1)R symmetry and this is identified with U(1)A.
The source of the curvature lies in the nonzero value of a self-dual 5-form flux belonging to the SUGRA multiplet. The integral of this 5-flux over S5 has to be a nonzero integer (if it's zero, we have no stress-energy tensor). Because the part of the 5-flux lying in AdS5 contains a time component, it gives rise to negative curvature. The part of the 5-flux lying in S5 doesn't have a time component, and so, it gives rise to a positive curvature.
The SUGRA multiplet also contains a dilaton and axion field. They correspond to the gauge field coupling and theta angle of the dual superYang–Mills theory.

There are $4\mathcal{N}$ real SUSY generators with $\mathrm{Spin}(\mathcal{N})$ as the obligatory R-symmetry.

11D $\mathcal{N}=1$ supergravity contains 32 real SUSY generators. There is a particular compactification, $\mathrm{AdS}_4 \times S^7$, the Freund–Rubin compactification, which preserves all 32 real generators. The bosonic isometry group is reduced to $\mathrm{Spin}(3,2) \times \mathrm{Spin}(8)$. After a Kaluza–Klein decomposition over S7, we get $\mathcal{N}=8$ SUSY. A 7-form magnetic flux is present over S7. Its integral over S7 has to be integer and nonzero.

## Applications

A plethora of papers is found in the literature which uses techniques of ADS CFT to understand strongly coupled system such as RHIC and condensed matter systems.

## Other topics

Certain "higher spin gauge theories" on AdS space appear to be holographically dual to a CFT with O(N) symmetry.[5] This has been called the Klebanov–Polyakov correspondence.
The AdS/CFT correspondence should not be confused with algebraic holography or "Rehren duality"; although these are sometimes identified with AdS/CFT, string theorists agree that they are different things.[6][7][8]