  We study two instanton correction problems of Hitchin's moduli spaces along
with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli
space can be put into an instanton-corrected form according to physicists
Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set
of special coordinates which can be constructed as Fock-Goncharov coordinates
associated with foliations of quadratic differentials on a Riemann surface. A
wall crossing formula of Kontsevich and Soibelman arises both as a crucial
consistency condition and an effective computational tool. On the other hand
Gross and Siebert have succeeded in determining instanton corrections of
complex structures of Calabi-Yau varieties in the context of mirror symmetry
from a singular affine structure with additional data. We will show that the
two instanton correction problems are equivalent in an appropriate sense via
the identification of the wall crossing formulas in the metric problem with
consistency conditions in the complex structure problem. This result provides
examples of Calabi-Yau varieties where the instanton correction (in the sense
of mirror symmetry) of metrics and complex structures can be determined.
