  The phase of the quantum-mechanical wave function can encode a topological
structure with wide-ranging physical consequences, such as anomalous transport
effects and the existence of edge states robust against perturbations. While
this has been exhaustively demonstrated for electrons, properties associated
with the elementary quasiparticles in magnetic materials are still
underexplored. Here, we show theoretically and via inelastic neutron scattering
experiments that the bulk ferromagnet Mn$_5$Ge$_3$ hosts gapped topological
Dirac magnons. Although inversion symmetry prohibits a net
Dzyaloshinskii-Moriya interaction in the unit cell, it is locally allowed and
is responsible for the gap opening in the magnon spectrum. This gap is
predicted and experimentally verified to close by rotating the magnetization
away from the $c$-axis with an applied magnetic field. Hence, Mn$_5$Ge$_3$
realizes a gapped Dirac magnon material in three dimensions. Its tunability by
chemical doping or by thin film nanostructuring defines an exciting new
platform to explore and design topological magnons. More generally, our
experimental route to verify and control the topological character of the
magnons is applicable to bulk centrosymmetric hexagonal materials, which calls
for systematic investigation.
