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I have been told that the world-sheet described by a closed string is a world-sheet without boundaries. On the contrary, the world-sheet described by an open string has boundaries. I do see why the open case has boundaries but I don't understand the closed case.

I get that the endpoints of the string now are closed so these don't give boundaries, but yet, the surface described by a closed string moving from a given time from another time has boundaries right? Imagine for example a string who happens to be a circle at one time and propagates defining a cylinder. This cylinder does have boundaries. So, what am I not seeing?

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Since the worldsheet theory is conformal, you are allowed to "shrink the boundaries to a point". So the usual viewpoint is that the worldsheet are boundary-less with certain points on them corresponding to the former boundaries.

The cylinder, for instance, becomes a twice-punctured sphere - the punctures are the places where one inserts the vertex operators which correspond to certain string states "on the boundary" by the CFT state-operator correspondence.

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  • $\begingroup$ so, for example, if I have an integral on the worldsheet, and using Stokes' theorem I write it as an integral over the boundary, is it zero? if so, do you know a rigorous proof? $\endgroup$ – Yossarian Sep 25 '15 at 10:35
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    $\begingroup$ @Scardenalli it is not zero: boundaries have operator insertions which are analogical to poles from the complex variable function theory. $\endgroup$ – Prof. Legolasov Nov 24 '15 at 1:51
  • $\begingroup$ @ACuriousMind It seems that you are overlooking a very important detail here. The conformal map that ''explode'' the punctures to ''boundaries'', also push these ''boundaries'' to infinity, so they are not quite boundaries of the worldsheet (the worldsheet becomes not compact). Conversely, a boundary that is located at finite distance will be always a D-brane or an end point of a open string. $\endgroup$ – Nogueira Aug 21 '18 at 3:29
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In string theory, the S-matrix of closed strings are obtained by summing over closed Riemann surfaces without boundaries. This is why we don't compute off-shell correlation functions like we do in QFT. If you write down some world-sheet with closed strings, e.g. at the massless level, that start from a finite location and not at the infinity, the Diff plus Weyl gauge symmetries of the Polyakov action will be violated getting something that is gauge dependent. It means that this must be non physical since gauge symmetries are required to act trivially in physical states. There are two ways to fix that.

The first one is to push this boundaries to infinity, and the Diff plus Weyl gauge will be restored. There will be a conformal transformation that maps this "boundaries at infinity" (locations where the world-sheet becomes noncompact) to punctures at the world-sheet. The state-operator correspondence gives a relation between asymptotic states and vertex operators such that you can "fill" this punctures by local operator insertions, obtaining a compact world-sheet with local insertions. This objects are the ingredients to compute the S-matrix of closed strings. You can learn more about that in the chapter 3, section 3.5 of Polchinki.

The second one is to maintain the boundary and try to tune the state/boundary condition such that the Diff plus Weyl gauge symmetry is restored. For the $X^{\mu}$ CFT there will be just two allowed boundary conditions coming from the constraint $T(z)=\tilde{T}(\bar{z})$, that is basically the condition that the energy-momentum tensor cannot flow out of the surface. This two are the Dirichlet boundary condition and the Neumann boundary condition:

$$ T(z)=\tilde{T}(\bar{z})\implies (\partial X^{\mu}-\bar\partial X^{\mu})(\partial X^{\mu}+\bar\partial X^{\mu})=0 $$

This boundaries define a state of closed strings $|B\rangle$, since boundary conditions define states in path integral, and you can look at this paper or this one to learn more about this.

This closed string at state $|B\rangle$ are coming from a object on the target space called D-brane. This is so because each Dirichlet boundary condition fix a value for one component of $X^{\mu}(\sigma)$ along the boundary, i.e.

$$ X^{\mu}(\sigma)= f(\sigma)|\,\sigma \in B $$

where $B$ is the boundary. This defines a hypersurface on the target space with same dimension as the number of Dirichlet boundary conditions. Note that if all the boundaries have Neumann conditions, this implies that this hypersurface, the D-brane, is filling all the target space. An interesting computation in supertrings is to check if the amplitude related to the disk with a puncture in the middle and Neumann condition to the boundary vanishes. This amplitude is related to the tadpole diagrams of QFT for the closed string state described by the puncture.

You can also combine both things. Have a piece of the boundary obeying $T(z)=\tilde{T}(\bar{z})$ and other pieces at the infinity. One trivial example is a infinity strip attached to the world-sheet. Again, there will be a conformal transformation that map the piece at infinity to a puncture, but now the puncture will be located at the boundary, the boundary where $T(z)=\tilde{T}(\bar{z})$. This puncture will describe a open string coming from infinity. The state-operator correspondence will give a vertex operator to the states of this open string, and it will provide the ingredient for computing the S-matrix for closed and open strings.

Summarizing:

Boundaries on the world-sheet should obey some boundary condition as $T(z)=\tilde{T}(\bar{z})$, which in that case gives two possibilities: Dirichlet or Neumann boundary condition. This boundary condition include a hypersurface on the target space, known as D-brane (that could be space filling if all the conditions are Neumann). The presence of boundaries in the world-sheet could be interpreted as endpoints of open strings or closed strings created/destroyed by D-brane in the $|B\rangle$ state.

Closed strings coming from infinity are described by non-compact regions in the interior of the world-sheet and can be mapped to punctures in the interior of the world-sheet using conformal transformations.

Open strings coming from infinity are described by non-compact intervals of the boundary and can be mapped to punctures of the boundary.

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