Band geometry from position-momentum duality at topological band crossings

We show that the position-momentum duality offers a transparent interpretation of band geometry at topological crossings. Under this duality, with Berry connection is dual to free-electron motion under gauge field. This identifies trace quantum metric as energy in momentum space. The crossings defects thus induce quantization metric. For $\mathbb{Z}$ nodal-point and nodal-surface semimetals three dimensions, Landau level occurs owing charges. Meanwhile, two-dimensional (2D) Dirac points exhibit vortices, leading quantized axial rotational energies. Such naturally generalizes three-dimensional (3D) nodal-loop semimetals, where nodal loops host vortex lines. ${\mathbb{Z}}_{2}$ monopoles bring about additional energies, which originate from links Nontrivial generically induces finite spread Wannier functions. While manifest lower bounds charges, logarithmic divergences occur vortices. may be probed experimentally by periodic-drive measurement.