New nonabelian Hodge graphs from twisted irregular connections

Abstract: It is known that any meromorphic connection on the Riemann sphere determines a finite diagram encoding its global Cartan matrix, and that it is invariant under the Fourier-Laplace transform. If the connection is tame at finite distance and untwisted at infinity, the diagram is actually a graph, corresponding to a symmetric generalised Cartan matrix, and it was proved by Boalch/Hiroe-Yamakawa that the corresponding nonabelian Hodge moduli space contains the Nakajima quiver variety of the graph as an open subset. In this note, we show that there exist new nonabelian Hodge diagrams that are graphs, beyond the setting of this quiver modularity theorem. The proof relies on observing that edge multiplicities in nonabelian Hodge diagrams satisfy ultrametric inequalities, which in particular gives a precise characterisation of nonabelian Hodge graphs coming from the untwisted setting.