Integrable Harmonic Higgs Bundles With Vanishing $\mathcal{U}$ And Eigenvalues of $\mathcal{Q}$

TL;DR

This paper investigates integrable harmonic Higgs bundles with vanishing endomorphism, revealing conditions under which the Higgs field and connection simplify, and analyzing eigenvalues of associated operators in complex geometric structures.

Contribution

It establishes new results on the implications of vanishing endomorphism in harmonic Higgs bundles, including conditions leading to trivial Higgs fields and properties of eigenvalues in CV-structures.

Findings

Vanishing fU implies vanishing Higgs field ff under IS condition.

Vanishing fU with holomorphic Chern connection implies vanishing Higgs field.

In odd-rank CV-structures, fQf has zero as an eigenvalue.

Abstract

We study the tt*-geometry with vanishing endormorphism $\mathcal{U}$. Given an integrable harmonic Higgs bundle $(E, h, \Phi, \mathcal{U},\mathcal{Q})$ on a complex manifold $M$, Firstly we prove that, under the \emph{IS} condition, vanishing $\mathcal{U}$ implies vanishing Higgs field $\Phi$ and the Chern connection of the Hermitian Einstein metric $h$ is a holomorphic connection, so the metric $h$ and $\mathcal{Q}$ are invariant. Secondly, without the \emph{IS} condition, we show that vanishing $\mathcal{U}$ will imply vanishing Higgs field $\Phi$ if we assume that the Chern connection of $h$ is a holomorphic connection. Finally, we add real structure $\kappa$. Given any \emph{CV}-structure, we prove that super-symmetric operator $\mathcal{Q}$ must have $0$ as an eigenvalue when the underlying bundle has odd rank.