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By E. Palti

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A topology change of spacetime requires the fabric of space-time to be somehow ’ripped’ and ’glued’ back together. It seems that string theory allows such a process without breaking down. A possible explanation for this might come from the fact that string theory can only probe scales down to the self-dual scale. What happens below this scale is as yet not understood, but with the hope that eventually string theory will be a full theory of gravity and so strings will form space-time, it is possible that its quantum nature on small scales allows this kind of behaviour.

The first is a Weyl rescaling of the ten-dimensional metric by a dilaton factor to take us to the Einstein frame 1 ˆ gˆM N → gˆM N e 2 φ . 17) and finally there is a rescaling of the K¨ ahler moduli by a dilaton factor 1 ˆ vi → vi e 2 φ . 18) We also define a new field which is the four-dimensional dilaton φ that differs from the ten-dimensional dilaton φˆ by a factor of the CY volume 1 φ ≡ φˆ − lnV . 20) 1 1 + (Im M)AB F A ∧ ⋆F B + (Re M)AB F A ∧ F B , 2 2 where we define the four-dimensional Planck constant 2 2 K(4) = K(10) κ−1 .

8) XA X0 = (1, z a ). Then these co-ordinates are a good co-ordinate basis on the manifold so that the bottom half of the holomorphic section composed of the periods FA is actually a function of the top-half periods FA (X) and can be generated from a single holomorphic function of homogeneous degree two called the prepotential FA (X) = ∂ F(X) . 10) FAB = ∂A ∂B F . 11) where The bundle H clearly plays an important role in the geometry of special K¨ ahler man- ifolds and in particular its section Ωsk .

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Aspects of Moduli Stabilization in String and M-Theory [thesis] by E. Palti

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