The Infrared Ultraviolet Connection

The proper area of any finite coordinate patch tends to ∞ as the boundary of AdS is approached. Thus we expect that the number of degrees of freedom associated with such a patch should diverge. This is consistent with the fact that a continuum quantum field theory such as SYM has an infinity of modes in any finite three-dimensional patch. In order to do a more refined counting[9] we need to regulate both the area of the AdS boundary and the number of ultraviolet degrees of freedom in the SYM.As we will see, these apparently different regulators are really two sides of the same coin. We have already discussed infrared (IR) regulating the area of AdS by introducing a surrogate boundary L at r = 1− δ or similarly at y = δ.

That the IR regulator of the bulk theory is equivalent to an ultraviolet (UV) regulator in the SYM theory is called the IR/UV connection[9].It is in many ways similar to the behavior of strings as we study them at progressively shorter time scales. We will find the interesting behavior that a string appears to grow as we average its properties over smaller and smaller time scales. To understand the relation between this phenomenon and the IR/UV connection in AdS we need to discuss the relation between AdS(5) ⊗ S(5) and D-branes.

D-branes are objects which occur in superstring theory. They are stable “impurities” of various dimensionality that can appear in the vacuum. A Dp-brane is a p-dimensional object. We are especially interested in D3-branes.Suc h objects fill 3 dimensions of space and also time. Their properties are widely studied in string theory and we will only quote the results that we need. The most important property of D3-branes is that they are embedded in a 10-dimensional space. Let us assume that they fill time and the 3 spatial coordinates xⁱ. Let the other 6 coordinates be called z^m and let z ≡. We will place a “stack” of N D3-branes at z = 0.

Now a single D-brane has local degrees of freedom. For example the location in z may fluctuate. Th us we can think of the z location as a scalar field living on the D-brane. In addition there are modes of the brane which are described by vector fields with components in the t, x direction as well as fermionic modes needed for supersymmetry. Our main concern will be with the z(x, t) fluctuations whose action is known from string theory calculations to be a that of conventional 3+1 dimensional scalar field theory.

D-branes can also be juxtaposed to form stacks of D-branes. A stack of N D-branes has a mass and D-brane charge which grow with N. The mass and charge are sources of bulk fields such as the gravitational field. What makes the D-brane stack interesting to us is that the geometry sourced by the stack is exactly that of AdS(5)⊗S(5).In fact the geometry is closely related to that of a D-brane stack.

Specifically the geometry sourced by the D-branes is a particular solution of the super gravity equations of motion:

 (1.1)

where

 (1.2)

and c is a numerical constant. If we consider the limit in which cg_s N/z⁴ >> 1 then we can replace F(z) by the simpler expression

 (1.3)

Furthermore the theory of the fluctuations of the stack is N = 4 SYM. All of the fields in this theory form a single supermultiplet belonging to the adjoint (N × N matrix) representation of SU (N).

In this lecture we give an argument for the IR/UV connection based on the quantum fluctuations of the positions of the D3-branes which are nominally located at the origin of the coordinate z. The location of a point on a 3-brane is defined by six coordinates z,ω₅.We may also choose the six coordinates to be Cartesian coordinates (z¹, ..., z⁶).The original coordinate z is defined by

 (1.4)

As we indicated, the coordinates zm are represented in the SYM theory by six scalar fields on the world volume of the branes. If the six scalar fields φⁿ are canonically normalized, then the precise connection between the z' s and φ' s is

 (1.5)

Strictly speaking equation 1.5 does not make sense because the fields φ in SU (N) are N × N matrices, where we identify the N eigenvalues of the matrices in equation 1.1 to be the coordinates z^m of the N D3- branes[10].The geometry is non-commutative and only configurations in which the six matrix valued fields commute have a classical interpretation. However the radial coordinate z =  can be defined by

 (1.6)

A question which is often asked is: where are the D3-branes located in the AdS space? The usual answer is that they are at the horizon z = 0. However our experiences with similar questions will warn us that the answer may be more subtle. What we will find there is that the way information is localized in space depends on what frequency range it is probed with. High frequency or short time probes see the string widely spread in space while low frequency probes see a well localized string.

To answer the corresponding question about D3-branes we need to study the quantum fluctuations of their position. The fields φ are scalar quantum fields whose scaling dimensions are known to be exactly (length)^-1. From this it follows that any of the N² components of φ satisfies

 (1.7)

where δ is the ultraviolet regulator of the field theory. It follows from equation 1.3 that the average value of z satisfies

 (1.8)

or

 (1.9)

In terms of the coordinate y which vanishes at the boundary of AdS

 < y >²∼ δ².   (1.10)

Here it is seen that the location of the brane is given by the ultraviolet cutoff of the field theory on the boundary. Evidently low frequency probes see the branes at z = 0 but as the frequency of the probe increases the brane appears to move toward the boundary at z = ∞.The precise connection between the UV SYM cutoff and the bulk theory IR cutoff is given by equation 1.6.