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I went to a lecture a few weeks ago and was told the following:

The world sheet of a closed string is a normal, standing cylinder. The world sheet of an open string is a cylinder on its side. This shows clear duality, therefore unifying gravity with gauge theories, as closed strings are there for gravity and open strings for gauge theories. (Bear in mind this is incredibly simplified as it would be unnecessary for me to go into great detail)

What puzzled me is why the open string produces a world sheet that is a cylinder.

Surely it would produce a 2-dimensional square/rectangle, as this is the shape that's produced when drawing out a world sheet of any horizontally straight line.

The world sheet being a cylinder would mean (according to my intuition) that the string would have to sort of split in 2 (the two strings produced being the same length as the previous lone string) and then rejoin again, producing a cylindrical shape.

This explanation seems extremely non-intuitive, which forces me to question it.

So why is the world sheet of an open string a cylinder as opposed to a 2-dimensional square/rectangle?

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It's a 1-loop open string diagram equal to a tree-level closed string one, this is covered in GSW and Polchinsky. –  Ron Maimon Nov 6 '12 at 18:01

5 Answers 5

OP is quite right: The un-compactified worldsheet of an open string has topology of a disk.

But in the open-closed duality example, the temporal coordinate of the open string worldsheet, and the spatial coordinate of the closed string worldsheet, are both assumed to be periodic, and hence in both cases they produce a cylinder $I\times S^1$. Here $I$ denotes an interval.

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Sweep a closed circular string and notice that the trajectory it leaves is a cylinder. Deform the string and you deform the cylinder. Topologically it remains a cylinder all the time, even if the string is deformed to a square one.

The same happens in space-time.

In constrast, open strings sweep a connected piece of a plane, and not a cylinder.

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He specifically asks about an open string. –  Muphrid Nov 6 '12 at 17:37
    
Tree level open string diagrams are connected pieces of plane. But the OP is asking about a 1-loop diagram, so this answer isn't really helpful. –  user1504 Nov 6 '12 at 18:24
    
@user1504: Why is a question about the world sheet of a single open string in fact a question about a 1-loop diagram (which involves a creation and destruction process of strings)? –  Arnold Neumaier Nov 6 '12 at 19:13
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@ArnoldNeumaier: The OP is asking people to explain to him in what circumstance an open string produces a cylindrical worldsheet. –  user1504 Nov 6 '12 at 19:22
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It's not at all different. Can you not clearly see that I say: "Why is the world sheet of an open string a cylinder?", which leads one to answer stating why an open string produces a cylindrical worldsheet. That's clear. –  Olly Price Nov 11 '12 at 14:46

I guess you are identifying the ends of the open string at infinity. The change in the topology is due to gluing those ends. Therefore you get a cylinder of very large radius out of the worldsheet, assuming perhaps some compacity of "space-time".

You can visualize this with a lower dimensional analogue. Take $\mathbb{R}^2$ and add the point at infinity $\mathbb{S}^2=\mathbb{R}^2\cup\{ \infty\}$. If you consider a line in $\mathbb{R}^2$, in the sphere it becomes a circle $\mathbb{S}^1$ after stereographical projection.

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You're right that the tree-level worldsheet of a single open string is a square.

The 'cylinder on its side' is the worldsheet picture of two open strings, which are pair-created from the vacuum, live their brief lives, and then arc back towards each other and annihilate. The cylinder worldsheet is obtained by taking two open string world sheets -- both squares -- and identifying the top with the top and the bottom with the bottom.

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I think what the lecturer could have been referred to is the following: Consider an open string stretched between two D-branes, a vacuum loop that string can then equivalently be seen as the exchange of a closed string between the two branes. This is explained in more detail for example on page 18 of Polchinski's lectures: http://arxiv.org/pdf/hep-th/9611050v2.pdf

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