The quiver at the bottom of the twisted nilpotent cone on $\mathbb P^1$

TL;DR

This paper characterizes the quiver structure at the bottom of the twisted nilpotent cone for Higgs bundles on the projective line, revealing its complexity and confirming topological connectedness under certain conditions.

Contribution

It determines the quiver description of the bottom of the nilpotent cone for twisted Higgs bundles on 1, extending known results and verifying conjectures about Betti numbers.

Findings

The quiver at the bottom of the cone is more complex than the A1 case.

The moduli space is topologically connected when rank and degree are coprime.

Confirmed conjectural Betti numbers from physics.

Abstract

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are coprime, thereby verifying conjectural lowest Betti numbers coming from high-energy physics.