A Paper on String-theory, AdS/cFT, and Sarkar Equation of Holography

I. INTRODUCTION

AdS space has certain specific properties that make it a natural representative for a Holographic Hamiltonian description. In this paper, I will discuss a remarkable form or structure of AdS holography, which originates from the mathematics of string theory.

This remarkable form comes from the higher-order symmetry of the theory, which is actually a powerful version of supersymmetry. However, we will ignore the mathematical aspects of the theory and emphasize the simple physical principles. The particular space I am interested in working with is not five-dimensional AdS, but rather $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ . This is actually a 10-dimensional product space constructed from two factors: first, a 5-dimensional AdS, and second, the 5-sphere or S(5). But why S(5)? Because higher-order supersymmetry operates in superstring theory. Typically, the supergravity theories that arise from string theory do not have a cosmological constant. But by bending some of the spatial directions into a compact manifold, we can obtain a lower-dimensional "Kaluza-Klein" theory, where a cosmological constant can be generated in a certain manner. Conceptually speaking, the internal 5-sphere is not important, but from a mathematical standpoint, it is quite important if we want to construct a precise statement. I want to begin my paper by giving a detailed idea of AdS geometry. For this purpose, we assume that the 5-dimensional AdS space is actually a 4-dimensional solid ball multiplied by an infinite 1-dimensional time axis. This geometry can be described in terms of some dimensionless coordinates such as t, r, $\mathrm{\Omega }$ . Here t is time, r is a radial coordinate whose value varies as 0, or between 0 and 1, meaning , and $\mathrm{\Omega }$ parameterizes the unit 3-sphere.

This geometry has uniform curvature ${\mathbf{R}}^{-2}$ , where $\mathbf{R}$ is the curvature radius. Under these circumstances, the metric becomes:

This metric can also be written in another way:

where $i=1,2,3$ . Equation (2) can be related to equation (1) in two different ways. First, at the boundary $r=1$ , equation (2) works as an approximation of equation (1). The 3-sphere can be replaced by a flat tangential plane (parameterized by $x^i$ ), and the radial coordinate can be replaced by $y$ , where $y=(1-r)$ . The second way in which equation (1) and equation (2) can be related is that equation (2) actually works as an exact metric for an incomplete patch of AdS space. A time-like geodesic reaches $y=\mathrm{\infty }$ in finite proper time; therefore, the space in equation (2) is geodesically incomplete. Its horizon lies at $y=\mathrm{\infty }$ , and thus the time coordinate in equation (1) is not equal to the time coordinate in equation (2). We can also write the metric in equation (2) in the form $z=1/y$ , giving:

According to this form, we can say that at $z=0$ we find a horizon because the time-time component of the metric vanishes there, and at $z=\mathrm{\infty }$ we find the boundary. However, to construct AdS(5)⊗S(5) space, we need to introduce 5 additional coordinates ${\omega }_5$ describing the unit 5-sphere, whose metric is:

Although the AdS boundary is at infinite proper distance from any point inside the ball, light can travel to that boundary and return within a finite time. For example, it takes a total time $t=\pi$ to go from the origin $r=0$ to the boundary $r=1$ and return!

For all practical purposes we may say that AdS space is a finite cavity with perfectly reflecting walls. The size of the cavity is denoted by R. And R is much larger than any microscopic scale such as the Planck scale or the string scale.

We shall call $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ the bulk space, and the 4-dimensional boundary the AdS boundary, where $y=0$ .

According to the holographic principle, everything inside the bulk can be described by a theory whose degrees of freedom can be identified at the boundary $y=0$ .

The holographic principle claims something more: it states that the boundary theory should not contain more than one degree of freedom per Planck area.

To understand this, let us determine the area of the boundary. From equation (1) it is clear that the metric diverges near the boundary. Later we will regulate this divergence by moving slightly away from the boundary $y=0$ . But for now, we assume that the number of degrees of freedom per unit coordinate area is infinite.

This indicates that the boundary theory will be a quantum field theory.

Another important aspect relates to the symmetry of AdS. Look again at the metric of equation (2)! It is obvious that the geometry is invariant under ordinary 4-dimensional Minkowski Poincaré transformations of coordinates $t$ and $x^i$ . Additionally, there is a dilational symmetry:

On the other hand, if we consider only the representation of AdS in equation (1), we find additional symmetry. For example, all the rotations of the sphere $\mathrm{\Omega }$ fall into the symmetry group. The full symmetry group of AdS(5) is $O(4,2)$ . Additionally, there is an O(6) symmetry corresponding to the rotations of the internal 5-sphere.

Since our goal is to argue that the holographic description of physics inside the bulk space-time is important, we must show how the symmetries of AdS act on the boundary. The 4-dimensional Poincaré symmetry acting on the boundary is straightforward. The dilational symmetry acts easily on the coordinates $t$ and $x$ .

These transformations acting on the boundary are known as conformal transformations, which preserve light-like directions on the boundary.

Indeed, the full AdS symmetry group, when acting on the boundary $y=0$ , becomes the conformal group of 4-dimensional Minkowski space.

The implication of this boundary symmetry is that the holographic boundary theory must be invariant under conformal group transformations.

From this perspective (along with the infinite coordinate density of the boundary), the holographic theory is actually a CFT or conformal quantum field theory.

As previously mentioned, $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ is a solution of 10-dimensional supergravity, which is the low-energy limit of superstring theory.

It is true that additional symmetries may exist beyond the conformal group, such as the O(6) symmetry related to the internal 5-sphere. These extra symmetries correspond to $\mathrm{N}=4$ supersymmetry.

This symmetry is easily understood in the holographic theory.

Ultimately, we arrive at the conclusion that the quantum gravity inside $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ is actually equivalent to a superconformal Lorentz-invariant QFT on the AdS boundary.

To get a measure of the number of degrees of freedom in $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ , we must construct a cutoff field theory inside the bulk.

A simple cutoff corresponds to a microscopic length equal to the 10-dimensional Planck length $l(p)$ . One way to do this is to introduce a sort of spatial lattice in the 9-dimensional space. It need not be a regular lattice; only its average spacing must be $l(p)$ .

Then we can construct a simple Hamiltonian lattice theory.

To count the degrees of freedom, we consider the area of AdS as infinite.

For that, we consider an L-surface at $r=1-\delta$ .

The 9-dimensional spatial volume inside the L-surface can be obtained easily from equation (2), and it is critically divergent:

Thus, the number of lattice sites or degrees of freedom inside the bulk becomes:

In such a theory, we can compute the maximum entropy. According to the holographic bound:

where A is the area of the L-boundary. Therefore:

In other words, when $\mathrm{R}/\mathrm{l}(\mathrm{p})$ is large, the holographic description requires a reduction factor of $\{\mathrm{l}(\mathrm{p})/\mathrm{R}\}$ in the number of degrees of freedom per unit Planck volume.

Stated differently, the holographic principle can describe all microscopic physics inside a very large AdS space using $\{\mathrm{l}(p)/\mathrm{R}\}$ degrees of freedom per Planck volume.

No matter how large R is, the theory remains able to describe microscopic bulk physics beautifully.

To obtain the holographic description of $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ , we must understand the symmetries clearly. That is why I began with a conceptual description.

Even more importantly, only a particular class of known systems possesses $N=4$ supersymmetry, namely SU(N) supersymmetric Yang-Mills theories (SYM).

The connection between gravity, or the string-theoretic generalization of $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ , and the super Yang-Mills theory on the boundary has deep conceptual aspects.

In this paper I will discuss only certain specific features.

The correspondence states that superstring theory in the bulk of $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ is equivalent to $\mathrm{N}=4$ , $3+1$ -dimensional $\mathrm{SU}(\mathrm{N})$ SYM theory on the AdS boundary.

In this paper SYM always refers to the specific supersymmetric gauge theory. Here N denotes the amount of supersymmetry, and N denotes the dimension of the Yang-Mills gauge group.

It is clear that SYM is conformally invariant and therefore does not have any additional dimensional parameters. The theory is defined on the boundary using dimensionless coordinates $t,\mathrm{\Omega }$ or $t,x$ . Even the momenta on the boundary are dimensionless. Therefore it is appropriate to say that all SYM quantities are dimensionless.

On the other hand, bulk gravity variables such as mass, length, temperature carry their usual dimensions.

To convert SYM variables to bulk variables, we can use a conversion factor R. Thus if E_SYM and M represent the energies of SYM and the bulk theory respectively, then:

Similarly, a bulk time interval equals the t-interval multiplied by R.

But one question may seem confusing: since $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ is a 10-dimensional spacetime, one might think that the boundary is $(8+1)$ -dimensional.

However, it is actually $(3+1)$ -dimensional, and the reason is precise.

To see this, let us consider a Weyl rescaling. The metric [equation (1)] is rescaled by a factor ${\mathrm{R}}^2/(1-{\mathrm{r}}^2{)}^2$ so that the rescaled metric becomes finite on the boundary.

The new metric becomes:

Observe that the size of the 5-sphere shrinks to zero as $r\to 1$ at the boundary. Thus, the boundary geometry is actually $(3+1)$ -dimensional.

Let us return to the connection between bulk and boundary theories.

The 10-dimensional bulk theory has two dimensionless parameters:

i. The curvature radius of AdS measured in string units $\mathrm{R}/\mathrm{l}(\mathrm{s})$ . Alternatively, $\mathrm{R}$ can be measured in 10-dimensional Planck units, giving the relationship:

ii. $g$ is a dimensionless coupling constant. The string coupling constant and the string length scale are mathematically related to the 10-dimensional Planck length and the Newton constant:

The gauge theory has only two constants:

(a) The rank of the gauge group N.
(b) The gauge coupling ${\mathbf{g}}_{\gamma \mathrm{m}}$

Of course, the two bulk parameters R and g can be determined by N and ${\mathbf{g}}_{\gamma \mathrm{m}}$

Thus, we can write the 10-dimensional Newton constant as:

There are two different limits which are quite interesting.

The AdS/CFT connection is useful for understanding the behavior of gauge theory in the 't Hooft limit, defined as:

In the bulk string perspective:

Thus, the 't Hooft limit corresponds to the classical string theory limit inside a fixed and large AdS space.

This limit is classical supergravity.

But this remarkable limit is completely different from that of the holographic principle. We consider a situation where the AdS radius increases but the parameters describing microscopic bulk physics remain fixed:

From the gauge theory viewpoint:

Our goal is to show that the number of quantum degrees of freedom in the gauge theory satisfies the holographic behavior of equation (8)!

In the metrics of equation (1) and equation (2), the proper area of any finite coordinate patch diverges as it approaches the AdS boundary. Therefore, the number of degrees of freedom associated with that patch also diverges. This is consistent with continuum QFT such as SYM, where any finite 3-dimensional region contains infinitely many modes.

For a refined counting, we must regulate both the area of the AdS boundary and the UV degrees of freedom in SYM.

These two regulators are actually two aspects of the same physical idea.

We have already considered an L-boundary inside $r=1-\delta$ , where $y=\delta$ (which acts as an IR regulator!).

According to the IR/UV relation, the IR regulator in the bulk corresponds to a UV regulator in SYM. In many cases, this is similar to string behavior (when studied in short time intervals). To understand this IR/UV relation in AdS, we must understand the connection between D-branes and $\mathrm{AdS}(5)\otimes \mathrm{S}(5)$ .

D-branes are objects that appear in superstring theory. They are stable "impurities" of various dimensionalities within the vacuum.

A Dp-brane is a p-dimensional object.

We are particularly interested in the D3-brane.

These objects fill 3 spatial coordinates ${\mathbf{x}}^i$ and a time coordinate.

The remaining 6 coordinates we denote by ${\mathbf{z}}^{\mathrm{n}}$ , where $\mathrm{z}=\sqrt{({\mathrm{z}}^{\mathrm{n}}\cdot {\mathrm{z}}^{\mathrm{n}})}$ .

We now place N D3-branes stacked at $z=0$

A single D3-brane has some local degrees of freedom—for example, its location in the $z$ coordinates may fluctuate.

Thus, the $z$ -location can be regarded as a scalar field living on the D3-brane. Additionally, there are modes on the brane associated with directions $t$ , $x$ , which can be described by vector fields, along with fermionic modes required for supersymmetry.

The point is that the action for the fluctuations $\mathbf{z}(\mathbf{x},t)$ can be obtained from string theory calculations, producing an ordinary $3+1$ -dimensional scalar field theory.

The mass of a stack of N D3-branes is M, and as N increases the charge of the D3-brane also increases.

This mass and charge act as sources for bulk fields such as the gravitational field. What makes the D3-brane stack interesting is that it resembles the geometry of AdS(5)⊗S(5). Indeed, the geometry defined by equations (3) and (4) is closely related to the geometry emerging from the D3-brane stack.

In particular, the geometry arising from D3-branes is a specific solution of supergravity equations:

where:

Here $c$ is a numerical constant.

If we consider the limit in which , then we can replace $F(z)$ by:

Then it becomes easy to see that the D-brane metric is essentially the same as equations (3) and (4).

Moreover, the fluctuation theory of the D-brane stack is the $N=4$ SYM theory. All fields in this theory live within a single supermultiplet in the adjoint $(N\times N)$ representation of $\mathrm{SU}(N)$ .

In this paper, I will attempt to provide an argument for the IR/UV connection based on the quantum fluctuations of the D3-brane position.

This is localized near $z=0$ in equation (3).

The location of a point on a D3-brane is defined by six coordinates $\mathbf{z},{\omega }_5$ .

We assume that the six coordinates are Cartesian $({\mathbf{z}}^1,\dots ,{\mathbf{z}}^6)$

The actual or original coordinate $z$ is defined by:

As indicated, the coordinates $z^n$ can be represented by six scalar fields on the brane's worldvolume in the SYM theory.

Thus, if the six scalar fields ${\mathrm{\Phi }}^i$ are canonically normalized, the relation between $z$ and $\mathrm{\Phi }$ is:

Strictly speaking, equation (22) does not have a literal meaning because $\mathrm{\Phi }$ fields in $\mathrm{SU}(\mathbf{N})$ are $\mathrm{N}\times \mathrm{N}$ matrices. The geometry therefore becomes non-commutative, and only configurations in which the six matrix-valued fields commute have a definite classical interpretation. Furthermore, the radial coordinate $z=\sqrt{(z^n\cdot z^n)}$ is defined as:

A common question is: where is the D3-brane located inside AdS space?

The usual answer is the horizon, where $z=0$ .

But the correct answer is subtler. The localization of information depends on the frequency range in which it exists.

At high frequency or short time scales, the string appears stretched. At low frequencies it is well localized.

To provide an analogous answer for the D3-brane, we must study the quantum fluctuations of its position.

Of course, $\mathrm{\Phi }$ is a scalar quantum field whose scaling dimension is $(\mathrm{length}{)}^{-1}$ . In this case, $\mathrm{\Phi }$ satisfies for any of its ${\mathbf{N}}^2$ components:

Where, $\delta$ is the ultra-violet regulator of field theory. According to equation (20), the average value of $z$ will be:

Or, using equation (12):

In terms of the y coordinate, which vanishes at the AdS-boundary, we can write:

We know that just below the boundary, the ultra-violet cut-off is located. And any low-frequency detector at $z=0$ will be able to see the branes, but as $z$ increases, all the way up to $z=\mathrm{\infty }$ , we will see higher-frequency branes. Thus it becomes clear that the SYM UV-cutoff of the boundary acts as the bulk IR-cutoff, which is evident from equation (23).

Now let us compute how many degrees of freedom are required to describe the region . The UV/IR connection implies that the theory can be described or analyzed with an ultra-violet regulator having $\delta$ cut-off. Let us consider a patch of the boundary that has unit coordinate area! Within that patch, there are $1/{\delta }^3$ cutoff cells whose size is $\delta$ . For each cell, the fields in the cut-off theory remain constant. This means that in each cell we obtain ${\mathbf{N}}^2$ degrees of freedom, which are associated with the $\mathrm{N}\otimes \mathrm{N}$

components of the adjoint representation of $\mathrm{U}(\mathrm{N})$ . Thus the number of degrees of freedom per unit area becomes:

Thus, per unit area, the number of degrees of freedom will be:

Finally, using equation (13):

This result is quite remarkable, because it is precisely what the holographic principle demands.

The high-frequency quantum fluctuations of the D3-brane's location can never be seen by a low frequency detector. In short, this is ensured by the renormalization group of the SYM-description acting on the brane. The renormalization group is what ensures that our body or any macroscopic structure cannot be easily destroyed or damaged by constant high-frequency vacuum fluctuations. However, we cannot be protected in the same way from classical fluctuations. A significant example is the thermal fluctuation of fields at high temperature. Any detector, whether low-energy or high-energy, can sense the thermal fluctuations of a brane's location. Let us now return to equation (24), but instead of using equation (25), we will use the thermal field fluctuations of $\mathrm{\Phi }$ . For each of the ${\mathbf{N}}^2$ components, the thermal fluctuation takes the form:

And equations (26) and (27) are replaced by:

That means the coordinate-distance $\mathrm{z}\sim 1/{\mathrm{T}}_{\mathrm{s}\gamma \mathrm{m}}$ strongly feels the thermal fluctuation. But if we write the expression for $\mathbf{r}$ , then it becomes:

Interestingly, this relation is equivalent to the horizon location of the AdS-black hole.

Now let us come to the main idea, which I have termed the "Sarkar-Approximation". From the AdS boundary at $\delta$ cut-off, I obtain the UV-regulator. At $r=1-\delta$ , I am making an approximation where but not zero! And at the AdS-boundary, $\delta =0$ , hence $r=1$ . I am also assuming that this $\delta$ cut-off is very close to the "Planck length!" Anyway, within this $\delta$ cut-off, a very large number of degrees of freedom operate. I can think of them as particles or modes, as I have explained earlier! Moreover, since $\delta$ is close to the Planck length, it acts as a "quantum mechanical" domain. The degrees of freedom present inside it satisfy Heisenberg's "position-momentum" uncertainty principle. Since $r=1-\delta$ is close to 1, we will represent the position uncertainty of a specific degree of freedom by $\mathrm{\Delta }r(\text{Ads:boundary})$ and the momentum uncertainty by $\mathrm{\Delta }p(\text{Ads:boundary})$ . Without going into such complicated terminology, we will simply replace them by $\mathrm{\Delta }\mathbf{r}(\mathrm{Ads}:\mathrm{boundary})\sim \mathbf{r}$ and $\mathrm{\Delta }\mathfrak{p}(\mathrm{Ads}:\mathrm{boundary})\sim \mathfrak{p}$ . Therefore:

$\mathrm{\Delta }\mathrm{r}(\mathrm{Ads}:\mathrm{boundary})\bullet \mathrm{\Delta }\mathrm{p}(\mathrm{Ads}:\mathrm{boundary})\sim \hslash$

Earlier, from equation (34) we obtained:

${\mathrm{T}}_{s\gamma \mathrm{m}}$ is the temperature of the "supersymmetric Yang-Mills" theory effective on the boundary. In some sense, the three quantities $\mathrm{p}\approx {\mathrm{M}}_{s\gamma \mathrm{m}}\approx {\mathrm{E}}_{s\gamma \mathrm{m}}$ become equal! Another important point is that $\hslash =c=G=1$ ; we are writing these three natural constants in standard-unit notation. Therefore equation (35) becomes:

The entity ${\mathrm{E}}_{\gamma }$ in equation (37) has a special value, it equals:

c is a constant coming from the supersymmetric Yang-Mills theory. It is not the speed of light! Substituting the value of ${\mathrm{E}}_{s\gamma \mathrm{m}}$ from equation (38) into equation (37), we obtain:

Another point is that $({\mathrm{N}}^2/{\mathrm{R}}^8)\sim {\mathrm{N}}^2=\mathrm{N}\mathrm{\_}\mathrm{dof}$ represents the number of degrees of freedom per unit area. Thus equation (39) becomes:

\begin{array}{l} \int $\mathrm{d}\mathrm{N}${$\text{dof}$} / $\mathrm{N}${$\text{dof}$} = \int \frac{dT_{s\gamma m}}{(1 - T_{s\gamma m})} - 3\int \frac{dT_{s\gamma m}}{T_{s\gamma m}} \ \ln ($\mathrm{N}${$\text{dof}$}) = $\int [{\mathrm{d}\mathrm{T}}_{\mathrm{s}\gamma \mathrm{m}}/(1-{\mathrm{T}}_{\mathrm{s}\gamma \mathrm{m}})]$ - 3 \int $\mathrm{d}\mathrm{T}$ _ {$\mathrm{s}$ \gamma $\mathrm{m}$} / $\mathrm{T}$ _ {$\mathrm{s}$ \gamma $\mathrm{m}$} \ \ln ($\mathrm{N}${$\text{dof}$}) = \ln (1 - $\mathrm{T}$ _ {s \gamma $\mathrm{m}$}) - 3 \ln ($\mathrm{T}$ _ {s \gamma $\mathrm{m}$}) + $\mathrm{i}$ \ \ln ($\mathrm{N}${$\text{dof}$}) = \ln \left[ \left(1 - $\mathrm{T}$ _ {$\mathrm{s}$ \gamma $\mathrm{m}$}\right) / $\mathrm{T}$ ^ {3} _ {$\mathrm{s}$ \gamma $\mathrm{m}$} \right] + $\mathrm{i}$ \ N{$\text{dof}$} = e ^ {i} \times \left[ \left(1 - T _ {s \gamma m}\right) / T ^ {3} _ {s \gamma m} \right] \tag {40} \ \end{array}

From the mathematical structure of equation (40), we can analyze a special limit. In the case of the AdS-boundary, if ${\mathrm{T}}_{s{\gamma }_{\mathrm{m}}}\to +\mathrm{\infty }$ , then $\mathrm{N}\mathrm{\_}\mathrm{dof}\to 0$ begins to occur. But to keep $\mathrm{N}\mathrm{\_}\mathrm{dof}$ unchanged, ${\mathrm{e}}^{\mathrm{i}}\to +\mathrm{\infty }$ or $\mathrm{i}\to +\mathrm{\infty }$ must hold. On the other hand, if we consider the bulk, then if ${\mathrm{T}}_{s{\gamma }_{\mathrm{m}}}\to 0$ occurs there, $\mathrm{N}\mathrm{\_}\mathrm{dof}\to +\mathrm{\infty }$ begins. But to keep $\mathrm{N}\mathrm{\_}\mathrm{dof}$ unchanged, we must have ${\mathrm{e}}^{\mathrm{i}}\to 0$ or $\mathrm{i}\to -\mathrm{\infty }$ . Thus in both cases, $\mathrm{N}\mathrm{\_}\mathrm{dof}$ remains fixed or unchanged. Now, according to the formulation of entropy: