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I'm reading Zwiebach's First Course in String Theory. At present I'm learning about string coupling. Zwiebach says it's possible to prove that $g_o^2=g_c$ where $g_o,\ g_c$ are open and closed string coupling constants respectively. He claims this is due to certain topological properties of world sheets.

Why, more precisely, is this true? Could someone explain and/or point me towards a proof?

Many thanks!

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The relationship holds because the Feynman diagrams in string theory are weighted by a function of the Euler characteristic $\chi$ (pronounce: chi) $${\mathcal A}_\Sigma = g_c^{-\chi_\Sigma} \cdot {\mathcal A}_\text{without couplings} $$ Now, a history of splitting and joining strings may be represented by a Riemann surface (after switching to the Euclidean world sheet signature and after some conformal transformations are being done) and the general Riemann surface may be written as a genus $h$ surface (i.e. a sphere with $h$ extra handles added, for example $h=1$ gives the torus) with $b$ extra boundaries (a disk that is cut from the surface somewhere) and $c$ extra crosscaps (boundaries with the identification of the opposite points so that the surface has no new boundaries but becomes unorientable). It may be shown that $$\chi = 2-2h - b - c,$$ see for example the article under the "Euler characteristic" link above. So the addition of a handle, $h\to h+1$, which is equivalent to adding two closed-string interaction vertices (a tube i.e. a virtual closed string gets emitted by the surface and it gets absorbed elsewhere), changes the Euler characteristic to $\chi\to \chi-2$ and adds $g_c^2$ to the dependence of the diagram on the string couplings.

That's the same change we may also obtain by adding two boundaries, $b\to b+2$. But one extra boundary adds an open-string loop i.e. two open-interaction vertices (a line interval i.e. a virtual open string gets emitted somewhere at the boundary of the world sheet and it gets reabsorbed, creating a new boundary inside), so two boundaries are four open-string interaction vertices and the diagram therefore gets an extra $g_o^4$. We therefore have $g_c^2\sim g_o^4$ or $g_c\sim g_o^2$ (up to the overall purely numerical coefficient that has to be calculated correctly, and up to a sign).

For some basics about the topology of stringy diagrams and their dependence on the string coupling, see e.g. the beginning of Chapter 3 of Joe Polchinski's String Theory, volume I. Your proportionality appears as equation (3.2.7) over there.

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Thank you very much! –  Edward Hughes Sep 3 '12 at 9:13
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Nice illustrative explanation, I like this too :-)! –  Dilaton Sep 3 '12 at 9:58

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