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.

enter image description here

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?

enter image description here

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?

  • $\begingroup$ As you mentioned, for the computation of the potential between quarks and anti-quarks, we have an open string hanging from the two on the boundary of the AdS. See the Figure 1 and Figure 2 in ocw.mit.edu/courses/physics/… , where the construction is explained much more clearly. The quark and the anti-quark are independent of the hanging open string. $\endgroup$ – Bruce Lee Jul 15 '17 at 23:18

Here when you talk about quarks in the CFT, they are not made of a more fundamental string. The CFT, as its name says, is a field theory, not a string theory, and the quarks are excitations of quantum fields. The Wilson loop can be seen as the world-line history of such a quark (this can be seen more easily in Euclidean signature, where there is not the problem of having to go backwards in time; In Lorentzian signature, you have the interpretation as the world-line of a pair quark-antiquark).

Now in the bulk, the Wilson line is interpreted as the boundary of a string world-sheet, where the string indeed is an open string (otherwise the worldsheet would not have a boundary) whose endpoints are interpreted in the CFT as the quark and the anti-quark.

  • $\begingroup$ Thank you very much. This was also my interpretation. Anyway, in the bulk we have a type IIB superstring which is a theory of closed strings right? why i am considering also open strings then? $\endgroup$ – apt45 Jul 7 '17 at 14:14
  • $\begingroup$ Type IIB string is indeed a theory of closed strings, but when you say that, you say in fact that there are independent left- and right-moving oscillators on the string. This does not prevent from having open strings, if the endpoints lie on D-branes. $\endgroup$ – Antoine Jul 10 '17 at 15:37

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