Harmonic Maps to Buildings and Singular Perturbation Theory

Abstract

The notion of a (uni)versal building associated with a point in the Hitchin base is introduced. The universal building is a building equipped with a harmonic map from the universal cover of the given Riemann surface that is initial among harmonic maps which induce the given cameral cover of the Riemann surface. The notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object. 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) a versal building always exists; (2) the asymptotics of the Riemann–Hilbert correspondence and the non-abelian Hodge correspondence are controlled by the harmonic map associated to any versal building; (3) spectral networks arise as inverse images of singularities of the versal building; and (4) versal buildings encode 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 are always controlled by a harmonic map to a certain building, which is constructed as the asymptotic cone of a symmetric space.