Instanton correction, wall crossing and mirror symmetry of Hitchin's moduli spaces

Abstract

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 is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which till now has been mostly studied in the framework of geometric Langlands duality. 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. This equivalence also relates certain enumerative problems in foliations to some gluing constructions of affine varieties.