  LISA will open the mHz band of gravitational waves (GWs) to the astronomy
community. The strong gravity which powers the variety of GW sources in this
band is also crucial in a number of important astrophysical processes at the
current frontiers of astronomy. These range from the beginning of structure
formation in the early universe, through the origin and cosmic evolution of
massive black holes in concert with their galactic environments, to the
evolution of stellar remnant binaries in the Milky Way and in nearby galaxies.
These processes and their associated populations also drive current and future
observations across the electromagnetic (EM) spectrum. We review opportunities
for science breakthroughs, involving either direct coincident EM+GW
observations, or indirect multimessenger studies. We argue that for the US
community to fully capitalize on the opportunities from the LISA mission, the
US efforts should be accompanied by a coordinated and sustained program of
multi-disciplinary science investment, following the GW data through to its
impact on broad areas of astrophysics. Support for LISA-related multimessenger
observers and theorists should be sized appropriately for a flagship
observatory and may be coordinated through a dedicated mHz GW research center.
