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

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.