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Wilson loop in AdS/CFT

+ 2 like - 0 dislike
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It is well known that Wilson loop is a quite hard observable to compute. In the case in which the QFT is dual to a gravitation theory in AdS space, we can use holography to compute the Wilson loop, which is given by

$$
W(\mathcal{C}) = <\mathcal{P}\,e^{\oint_\mathcal{C} A_\mu dx^\mu}>
$$
where $\mathcal{P}$ is the path-ordered product.

The contour $\mathcal{C}$ describes the worldline of the quark-antiquark pair on the field theory side which is created at an initial time and then annihilated.

The Wilson loop on the contour measures the potential energy of the pair and signals if the theory is confining. 
A very massive quark in the field theory side can be seen as a open string in the bulk with one end-point attached to N D3 branes and the other one to another D3 brane which is sent to infinity (the boundary of AdS) to make the mass very large (infinity), see the next figure.

https://i.stack.imgur.com/Zg9L3.png

However, in many reviews, it's mentioned that the contour $\mathcal{C}$ can be seen as the boundary of a minimal surface worldsheet of a string. As in the following figure, is this string attaching to the quark and anti-quark? If yes, is it an open string?

I am confused whether the string is attaching both the quark and the anti-quark or the quarks themselves are the strings. If they are the strings, how do you close the worldsheet?

asked Jul 7 in Theoretical Physics by apt45 (50 points) [ no revision ]

2 Answers

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Indeed, on the gravity side the Wilson loop is computed by evaluating an area of the minimal surface ending on the boundary at the Wilson loop.

On the gauge theory side there are no strings, because it is just SYM theory, so the quarks and anti-quarks are point particles.

answered Jul 7 by Andrey Feldman (600 points) [ revision history ]

Sorry but this does not answer my question...

@apt45 Hm... On the gauge side of the duality there are no strings, because it is a theory of point particles -- SYM theory. On the gravity side, the object that corresponds to the Wilson loop of the SYM is the surface of the worldsheet of the open string ending on the boundary. Is it OK now?

I know you don't have strings on the gauge theory side. In the gravity side instead, you can attach a string to the quarks living at the boundary and make this worldsheet 

@apt45 There are no quarks living on the boundary on the gravity side. There is just an open strings going to conformal boundary of AdS, which doesn't support any dynamical degrees of freedom.

+ 1 like - 0 dislike

I think the string is attaching to the D-brane and the two end points of the open string are quark and anti-quark, respectively. In the figure, the world-sheet is still a world-sheet of open string (take the vertical line as the time direction, say). It looks different from normal world-sheets for open strings because now the trace of the end points is a loop. Note that, the world-sheet for a closed loop can not be attached to a D-brane.  
 

answered Jul 7 by Wein Eld (195 points) [ revision history ]

This is also strange, because in the gravity side we have type II B superstring on $AdS_5 \times S_5$ which is a theory of only closed strings... don't?

@apt45 Yes, in the gravity side, we have type 2B superstring on $AdS_5\times S_5$ which is a theory of only closed strings. But in your post, we are considering the gauge theory side, aren't we? In the both sides of AdS/CFT, we always have a D3-brane in 10 dimensional space-time. When the D3-brane is considered in the perturbative regime, it is described by the fields on the D3-brane which are induced by the massless excitations of open strings. However, when this exactly the same system is in the non-perturbative regime, the D3-brane is described as a solitonic object of superstring/suprgravity theory which inducing the tyep 2B superstring theory in the near-horizon region. That is, the duality is built on one and the same system but within different regimes. In any regime, we have the bulk and the D3-brane simultaneously. The conjecture now is, in any regime, the system can be euqally be described by either of these two theories which are actually the same theory. Hence, when the t' Hooft coupling is strong, we do not know how to solve the gauge theory but we shall still have the picture of open strings ending on the brane.

@apt45 Type IIB string theory is not a theory of only closed strings (as it is known since 1995). It is a string theory with chiral supersymmetries (which differs it from Type IIA) with identical left- and right-moving SCFTs (which differs it from Heterotic strings). There are odd-dimensional branes in the theory (in particular, the 3-brane which is a "base" of canonical example of AdS/CFT), on which open strings can end.

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