Visualizing the microscopic origins of topology in twisted molybdenum ditelluride

In moir\'e materials with flat electronic bands and suitable quantum geometry, strong correlations can give rise to novel topological states of matter. The nontrivial band topology of twisted molybdenum ditelluride (tMoTe$_2$) -- responsible for its fractional quantum anomalous Hall (FQAH) states -- is predicted to arise from a layer-pseudospin skyrmion lattice. Tracing the layer polarization of wavefunctions within the moir\'e unit cell can thus offer crucial insights into the band topology. Here, we use scanning tunneling microscopy and spectroscopy (STM/S) to probe the layer-pseudospin skyrmion textures of tMoTe$_2$. We do this by simultaneously visualizing the moir\'e lattice structure and the spatial localization of its electronic states. We find that the wavefunctions associated with the topological flat bands exhibit a spatially-dependent layer polarization within the moir\'e unit cell. This is in excellent agreement with our theoretical modeling, thereby revealing a direct microscopic connection between the structural properties of tMoTe$_2$ and its band topology. Our work enables new pathways for engineering FQAH states with strain, as well as future STM studies of the intertwined correlated and topological states arising in gate-tunable devices.