Asymptotics for general connections at infinity

TL;DR

This paper studies the asymptotic behavior of monodromy matrices for connections on Riemann surfaces, showing their Laplace transforms have analytic continuations with polygonal growth rate regions, linked to spectral curves.

Contribution

It provides a detailed analysis of the asymptotics of monodromy at infinity, revealing the geometric structure of growth rates via spectral curves and Laplace transform properties.

Findings

Laplace transform of monodromy matrices has analytic continuation with finite branching.

The exponential growth rate region of monodromy forms a polygon with vertices related to spectral data.

No information obtained about singularity sizes of the Laplace transform, limiting asymptotic expansion results.

Abstract

For a standard path of connections going to a generic point at infinity in the moduli space $M_{DR}$ of connections on a compact Riemann surface, we show that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular the convex subset representing the exponential growth rate of the monodromy is a polygon, whose vertices are in a subset of points described explicitly in terms of the spectral curve. Unfortunately we don't get any information about the size of the singularities of the Laplace transform, which is why we can't get asymptotic expansions for the monodromy.