The Halls of the Aesir
These Halls are the seat of power in the eyes of Humanity. Here they can walk amidst the humans, so their presence is felt and provide assurance. It is here that the Aesir formulate battle plans and regale each other with the tall tales.
The building itself is massive, rises many hundreds of stories above ground level and reaching deep into the earth as well. Amongst its many tiers lie armories, vehicle bays, loading docks, research labs, staging areas, barracks, training facilities, arboretums, hospitals and even prisons.
The building is self-sufficient enough to maintain the war effort, but it is ultimately reliant upon the Orbital String for stability. Each of the gods has his own personal Hall within this building, to be used as a sort of office, in addition to their true homes that reside atop the Orbital String tower in Asgard.