Wilson Lines

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Dear all,

I have recently read the effects of T-duality of both closed and open strings (Zwiebach 2009, Johnson 2003) and found it very interesting indeed. However, I found it difficult to understand the concept of Wilson lines. Could someone please explain me the idea and concept of Wilson lines for both closed and open strings? How does it work and what is the point of it?

Could someone also tell me what is S-duality (g --> 1/g)? How does it actually work?

Is there a good review paper on an introduction to Wilson lines and S-duality? If so, where can I find them?

Many thanks!
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Answers and Replies

  • #2
Since no one else has responded, try here:


I posted about these in another thread but can't find it...I had a source that discussed these as Wilson-Polykov (Sp) loops and these were closely related to Roger Penrose's idea of spin foam...

Anyway, one formulation leads to the other....if you think of lattice network, space time "appears" as a crystal like structure with integer value edge lengths as areas and nodal integer values as volumes of spacetime...

Roger Penrose mentions (page 941) that Wilson loops were developed by Rovelli, Smolin and Jacobsen in general relativity and discusses related mathematics (topology) of knots and Links in THE ROAD TO REALITY at page 943 and goes on the subsequent pages to describe his formulation and preference that spacetime should be based on discreteness rather than continuity....
spin networks is also a closely related topic...
  • #3
Ok I found some notes I kept but have not yet edited/studied.....which may help you, ..Sorry I don't know the exact source, but it was from some guys here on Physicsforums ..(edit) found it:

REFERENCE : maestro Carlo Rovelli “Loop Quantum Gravity”
Physics World, November 2003

In String Theory, the main “competitor” when it comes to quantumgravity starts from the fact that there must be some kind of fictitious background space, thus actually undoing the aspects of general relativity. All calculations can then be made with respect to this background field and in the end the background independence must “somehow” be recovered. LQG starts from a totally different approach, though. We start from the knowledge we have from General Relativity, thus no background field, and we then try to rewrite the entire Quantum Field Theory in a certain way that no background-field is needed.

How to implement this nice background-independence in QFT has already been introduced, i.e. The Wilson Loop and more generally the spin networks :

The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.

The strategy is as follows : in stead of working with one specific metric like in “ordinary” QFT, just sum up over all possible metrics. So QFT should be redefined into somekind of pathintegral over all possible geometries. A wavefunction is then expressed in terms of all these geometries and one can calculate the probability of one specific metric over another. This special LQG-adapted wavefunction must obey the Wheeler-DeWitt equation, which can be viewed at as some kind of Schrödinger-equation for the gravitational field. So just like the dynamics of the EM-field is described by the Maxwell-equations, they dynamics of the gravitational-field are dedeterminedy the above mentioned equation. Now how can we describe the motion of some object or particle in this gravitational field. Or in other words, knowing the Maxwell equations, what will be the variant of the Lorentz-force ???

This is where the loops come in. First questions one must ask is :

Why exactly them loops ?

Well, let’s steal some ideas from particle physics... In QFT we have fermionic matter-fields and bosonic force-fields. The quanta of these force-fields or the socalled force-carrier-particles that mediate forces between matter-particles. Sometimes force-carriers can also interact with eachother, like strong-force-mediating gluons for example. These force carriers also have wavelike properties and in this view they are looked as excitations of the bosonic-forcefields. For example some line in a field can start to vibrate (think of a guitar-string) and in QFT one then says that this vibration is a particle. This may sound strange but what is really meant is that the vibration has the properties of some particle with energy, speed, and so on, corresponding to that of the vibration. These lines are also known as Faraday’s lines of force. Photons are "generated" this way in QFT, where they are excitations of the EM-field. Normally these lines go from one matter-particle to another and in the absence of particles or charges they form closed lines, aka loops. Loop Quantum Gravity is the mathematical description of quantum gravity in terms of loops on a manifold. We have already shown how we can work with loops on a manifold and still be assured of background-independence and gauge-invariance for QFT. So we want to quantize the gravitational field by expressing it in terms of loops. These loops are quantum excitations of the Faraday-lines that live in the field and who represent the gravitational force. Gravitons or closed loops that arise as low-energy-excitations of the gravitational field and these particles mediate the gravitational force between objects.

It is important to realize that these loops do not live on some space-time-continuum, they are space-time !!! The loops arise as excitations of the gravitational field, which on itself constitutes “space”. Now the problem is how to incorporate the concept of space or to put it more accurately : “how do we define all these different geometries in order to be able to work with a wave function ?”

The Wheeler-DeWitt equation has solutions describing excitations of the gravitational field in terms of loops. A great step was taken when Abhay Ashtekar rewrote the General Theory of Relativity in a similar form as the Yang-Mills-Theory of QFT. The main gauge-field was not the gravitational field. No, the gravitational field was replaced by the socalled connection-field that will then be used to work with different metrics. In this model space must be regarded as some kind of fabric weaved together by loops. This fabric contains finite small space-parts that reflect the quantization of space. It is easy to see that there are no infinite small space regions, thus no space-continuum. Quantummechanics teaches us that in order to look at very small distance-scales, an very big amount of energy is needed. But since we also work in General Relativity we must take into account the fact that great amounts of energy concentrated at a very small scale gives rise to black holes that make the space-region disappear. By making the Schwardzschildradius equal to the Comptonradius we can get a number expressing the minimum size of such a space-region. The result is a number that is in the order of the Planck Length.

Now how is space constructed in LQG ? Well, the above mentioned minimal space-regions are denoted by spheres called the nodes. Nodes are connected to eachother by lines called the links.

By quantizing a physical theory, operators that calculate physical quantities will acquire a certain set of possible outcomes or values. It can be proven that in our case the area of the surface between two nodes is quantized and the corresponding quantumnumbers can be denoted along a link. These surfaces I am referring are drawn as purple triangles. In this way a three-dimensional space can be constructed.

One can also assign a quantumnumber which each node, that corresponds to the volume of the grain. Finally, a physical state is now represented as a superposition of such spin-networks.

marlon, thanks to marcus for the necessary information and corrections of this text
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