String on a Double Torus

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We have already seen how winding a string around an extra dimension gives rise to particles that can be observed from a lower dimensional space. Here, we have a g = 2 Riemann surface that looks like two donuts stuck together. A string can wind around the two different holes, and around the cross-section of each hole, giving four different winding modes. In general, a string has twice as many winding modes as there are holes in the Riemann surface to which the string is confined. Each mode corresponds to a class of particles, so the geometry of the extra dimensions has a clear physical consequence in the lower dimensional space. (Unit: 4)