Index Concordia

Posted in AdS/CFT, Gauge Theory, Supergravity by Index Guy on June 10, 2008

We continue our discussion (started here) of very basic notions in the study of the AdS/CFT correspondence. I have followed very closely the following review:

arXiv:0711.4467

The next topic is the Anti-de Sitter/Conformal Field Theory correspondence. We start by noticing that D3-branes may be interpreted in two different ways. First (and probably what comes to mind when one talks about a Dp-brane) D3-brane are (3+1)-d hyperplanes in (9+1)-d space on which open strings can end. In the low energy limit only masless strings contribute and open string degrees of freedom correspond to MSYM with gauge group U(N), with N corresponding to number of D3-branes superimposed.

We can factorize the gauge group U(N) into SU(N) x U(1). The abelian part (U(1)) we take to describe the motion of the center of mass of the D3-brane.

The second way to look at D3-branes is as solitonic solutions of 10-dimensional type IIB supergravity with the metric

$ds^2 = \xi^{-1/2}\eta_{ij}dx^{i}dx^{j} + \xi^{1/2}(dy^2 + y^2d\Omega_{5}^{2})$

with

$\xi = 1 + \displaystyle\frac{R^4}{y^4}$ and $R^4 = 4\pi g_{s}N\alpha^{\prime 2}$

The ‘t Hooft coupling is defined by $\lambda = g_{s}N = g^{2}_{YM}N$. In all these expressions N is again the number of D3-branes and $\alpha'$ is the inverse string tension.

The metric above has been written as the sum of two terms. The first term includes the metric $\eta_{ij}$ which has the same form as (3+1)-dimensional space. These part contains the coordinates along the D3-stack. The other part of the metric includes 6-dimensional part with coordinates $y_{M}$. These six coordinates will describe the directions that are perpendicular to the brane.

We can explore two limits to this metric. First we notice that for $y \gg R$ the metric tends to that of (9+1)-dimensional Minkowski space. In the low energy limit (short distance scales) we have $y \ll R$. In this regime we define the variable $u = R^2/y$. In terms of this new variable the metric now looks like

$ds^2 = R^2 (\displaystyle\frac{\eta_{ij}dx^{i}dx^{j}}{u^2} + \displaystyle\frac{du^2}{u^2} + d\Omega_{5}^2).$

In this form we recognize the first two terms as the metric for 5-dimensional anti-de Sitter space and the third term as the metric for the 5-sphere. Hence in this “near horizon limit” we have the space breaking up into 5-d AdS with radius $R$. We recall that anti-de Sitter space is a solution to Einstein’s equation that produces constant (negative) curvature given by

$\mathcal{R} = -\displaystyle\frac{d(d - 1)}{R^2}.$

The isometries of 5-d AdS are contained in the group SO(2,4). For the 5-sphere we have the isometries coming from SO(6).  These isometries match the maximal bosonic subgroups of SU(2,2|4).

The Maldacena Conjecture has different forms. In its strongest form it claims:

MSYM with gauge group SU(N) corresponds to string theory in $AdS_{5}\times S^5$.

Another version involves certain limits:

For $N \rightarrow \infty$ and keeping the ‘t Hooft coupling fixed, planar diagrams contribute to the field theory side; string side is restricted to semi-classical limit with $g_{s} \rightarrow 0.$

Yet one more form:

For large ‘t Hooft coupling, MSYM is mapped to SUGRA in $AdS_{5}\times S^5$, with $\alpha' \rightarrow 0.$

Because the gravity side has an extra, non-compact dimension $y$ relative to the gauge theory (i.e. 4-d gauge theory versus 5-d gravity), the correspondence is said to be “holographic”. We can look at the dilatations (a sub-set of conformal transformations) in the gauge theory side. The invariance of the metric requires

$y \longrightarrow e^{-\alpha}y$

which can naturally be interpreted a representing the renormalization group scale in the gauge theory.

Other developments we can mention are the field-operator map that has been developed that maps gauge-invariant operators of MSYM is some irreducible representation of SU(4) to SUGRA fields. Five-dimensional fields can be obtained by a Kaluza-Klein reduction of the original 10-d SUGRA fields on the 5-sphere.

A very important statement:

The generating functional of particular gauge-invariant operators in the CFT coincides with the generating functional for tree diagrams in SUGRA with the boundary values of SUGRA fields coinciding with sources.

No proof of the anti-de Sitter/Conformal Field Theory correspondence that takes into account its string-theoretical origin exist yet.

Finally some last remarks about confinement. We know that quarks and gluons are confined to hadrons. Let us look at the interaction energy between two heavy quarks at different separations. Heavy quarks can be introduce in AdS/CFT by placing a “probe” D3-brane at large $y$ in the AdS space. (This corresponds to the UV in the field theory.) Now the strings with endpoints at the probe D3-brane can stretch to the other N D3-branes at the origin. These stretched strings are formally very massive gauge bosons that can be associated with the breaking of the gauge symmetry by one of the six scalars adquiring some VEV. The strings can be thought of as heavy sources though massive and in fundamental representation.

Now consider two such strings representing a quark-antiquark pair. There can be two possible configurations. First, each string could just stretch from the probe to the stack at the origin. Then there will be no interactions between the two. On the other hand, the two strings may join. One may compute the energy of this configuration in pure AdS space and the result is inversely proportional to the separation legth $L$ of the quark pair. Hence, as one separates the quarks further, the string dips further into the AdS space (meaning moving away from the boundary).

When one deviates from AdS space (and from conformal field theories) the situation changes. Having a non-conformal theory introduces a sense of scale to compare things. We can introduce a “mass gap” by using a hard block (at some radius) that will block the SUGRA fields and restrict them from values of the radius (what we identified as the renormalization group scale) below some mass gap energy. When we have a hard block [Is this the same as a hard wall? That article by Polchinski and Strassler is very confusing…] the string that connects the two quarks will behave as mentioned above only for small separations. For large separation the string will eventually hit the block and lie along it. Further separation just extends the string along the block. The energy now must be proportional to the separation. This is what we called confinement.

We can also consider the case with finite temperature. [What exactly do people mean by finite temperature quantum field theory?] When the particles in the gauge theory have large kinetic energy, one expects de-confinement. Finite temperature is associated with having a black hole in the AdS space, since a black hole has all thermodynamic properties that one can associate to a thermal bath. There will be a cut-off of the space at long energies where the horizon will radially lay. The cut-off energies will correspond to energies below the (finite) temperature scale. Going back to our string linking the two heavy quarks we see that as the string moves deeper into the space it will find the horizon. Part of the string will fall into the horizon leaving a disconnected string from each quark/antiquark to the horizon. In this sense we say the quarks are screened by each other since they can now be moved independently.

Confining theories contain a discrete spectrum of bound states. We can obtain a discrete glueball [What is a glueball?] spectrum from a gravity dual with the following arguments. Glueballs are associated to the gauge field operator given by the group-index trace of $F^2$. In the AdS/CFT this operator is associated to the massless dilaton. We can look at the equation of motion for the dilaton and check for “plane-wave solutions”. These “plane-wave solutions” will consist of a 4-d phase factor and an amplitude that will depend on the radial part. Discrete spectra exist for discrete values of the glueball mass.

As mentioned above, when we deviate from AdS we loose the conformal symmetry. People talk about “deformed geometry” with a metric that looks like

$ds^2 = e^{2A(u)}\eta_{ij}dx^{i}dx^{j} + du^2.$

Pure AdS space is given by $A = \ln{R/u}$. We expect the deformation from AdS space to be “stronger” when we go to the IR (large values of the coordinate u). For confining geometries the function $A(u)$ will diverge in the IR creating a hard wall. We can obtain a Schroedinger-like equation and for this hard wall we obtain solutions similar to the infinite well, which is discrete.

Actually the infinite well is not what we want. Recall that the energy eigenvalues of the infinite well go as the square of some quantum number. This is a contradiction to the expected linear behavior (Regge). This can be argued to be due to the breakdown of the SUGRA approximation. One can find solutions that obey the Regge behavior, but these are not derivable from first principles. These models are called “soft-wall models” since the dilaton has a specific behavior that smoothens-out spacetime in the IR, instead of the hard cut-off.