Harmonic maps to buildings and singular perturbation theory
Résumé
The notion of a universal building associated with a point in the Hitchin base is introduced. This is a building equipped with a harmonic map from a
Riemann surface that is initial among harmonic maps which induce the given cameral cover of the Riemann surface. In the rank one case,
the universal building is the leaf space of the quadratic differential defining the point in the Hitchin base.
The main conjectures of this paper are: (1) the universal building always exists; (2) the harmonic map to the universal building controls the
asymptotics of the Riemann-Hilbert correspondence and the non-abelian Hodge correspondence; (3) the singularities of the universal building
give rise to Spectral Networks; and (4) the universal building encodes the data of a 3d Calabi-Yau category whose space of stability conditions
has a connected component that contains the Hitchin base.
The main theorem establishes the existence of the universal building, conjecture (3), as well as the Riemann-Hilbert part of conjecture (2), in the
case of the rank two example introduced in the seminal work of Berk-Nevins-Roberts on higher order Stokes phenomena. It is also shown that
the asymptotics of the Riemann-Hilbert correspondence is always controlled by a harmonic map to a certain building, which is constructed as
the asymptotic cone of a symmetric space.