Nilpotent Higgs bundles and families of flat connections

TL;DR

This paper studies special families of flat connections with nilpotent Higgs fields, showing they share monodromy properties with regular Higgs bundles and analyzing the asymptotic behavior of their holonomies.

Contribution

It demonstrates that nilpotent Higgs bundle families have the same monodromy as regular ones and characterizes the asymptotic exponential behavior of holonomy traces.

Findings

Families have identical monodromy to regular Higgs bundle families.

Holonomy traces grow exponentially in rational powers of the parameter.

Includes examples from real twistor lines and conformal limits.

Abstract

We investigate $\mathbb{C}^\times$-families of flat connections whose leading term is a nilpotent Higgs field. Examples of such families include real twistor lines and families arising from the conformal limit. We show that these families have the same monodromy as families whose leading term is a regular Higgs bundle and use this to deduce that traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.