Computing K3 and CY n-fold Metrics

The derivative expnsion in the context of IIB string scattering compactified on non-trivial K3 and other Calabi-Yau manifolds is formulated. The scattering data in terms of automorphic functions can be inverted to find the these metrics. The solutions are parameterized by the moduli information, and the metrics may be found to any desired accuracy in derivatives. Metric information to low orders in derivatives allows for a counting of curves inside the manifold; in addition, the coefficients of these exponential terms via D-brane wrappings are polynomials that may admit an invariant interpretation in cohomology. An interesting case pertaining to M-theory compactifications is the collection of seven-dimensional $G_2$ manifolds; they can also be obtained when the moduli space degenerates into cases, such as a toroidal one or other limit in which modular functions on the space are known. Note: this work was written two years ago; the recipe without the explicit form of the scattering and metrics is given.