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

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