Resolution of Topology and Geometry from Momentum-Resolved Spectroscopies

Extracting the complete quantum geometric and topological character of Bloch wavefunctions from experiments remains a challenge in condensed matter physics. Here, we resolve this by introducing the "wavefunction form factor" (WFF) matrix, a quantity directly constructible from intensities in momentum- and energy-resolved spectroscopies like ARPES and INS. We demonstrate that band topology is encoded in "spectral nodes" -- momentum-space points where the WFF determinant vanishes, providing a direct readout of topological invariants via a topological selection rule. Furthermore, when the number of independent probes exceeds the number of the target bands, our framework yields an effective band projector. This enables the determination of Wilson loop spectra and the extraction of an effective quantum geometric tensor, providing a model-independent measurement of the non-Abelian Berry curvature and quantum metric as resolved by the experimental probes.