  We analyze the coupling of elastic lattice deformations to the magnon degrees
of freedom of magnon Dirac materials. For a Honeycomb ferromagnet we find that,
as it happens in the case of graphene, elastic gauge fields appear coupled to
the magnon pseudospinors. For deformations that induce constant pseudomagnetic
fields, the spectrum around the Dirac nodes splits into pseudoLandau levels. We
show that when a Dzyaloshinskii-Moriya interaction is considered, a topological
gap opens in the system and a Chern-Simons effective action for the elastic
degrees of freedom is generated. Such a term encodes a phonon Hall viscosity
response, entirely generated by fluctuations of magnons living in the vicinity
of the Dirac points. The magnon Hall viscosity vanishes at zero temperature,
and grows as temperature is raised and the states around the Dirac points are
increasingly populated.
