The 15th Hierarchical City of Torifune is best known as the Military Academy for the NOL. There, future librarians train and hone their skills for whatever challenges may await them outside. The story centers around Mai Natsume, a girl who had just arrived late due to a week absence, but there seems to be several phenomenons centering around her that could be more than just coincidence.
(Source: Manga Updates)

Following the plot of the game, it switches between the viewpoint of different characters as they face off against each other in the tournament. Commonly following the story from the viewpoint of Yu Narukami, Aigis, Yosuke Hanamura, Mitsuru Kirijo, and Naoto Shirogane as they face off against the other characters. Both Yosuke and Aigis face off against Shadow Labrys.

(Source: Megami Tensei Wikia)

Ragna the Bloodedge is a wanted criminal with an extraordinarily large bounty on his head (well in the trillions) for attempting to destroy the Novus Orbis Librarium, and is Jin and Saya's older brother.

In December AD 2199, several years after the Second War of Magic, a branch of the NOL was utterly destroyed by an SS-class rebel named "Ragna" also known as the "Grim Reaper", in an attempt to destroy the Librarium. The NOL, hoping to stop him, immediately announced the largest bounty ever, available to anyone who could capture him. Interestingly, Ragna possesses a powerful form of Ars Magus known as the "Azure Grimoire." This led the NOL, as well as the other fighters, to hunt Ragna not just for his bounty, but also his Azure Grimoire.

After the "Ikaruga Civil War," Mai Natsume and her friends are continuing their studies in the NOL Military Academy. They are all called to help in the preparations of Jin Kisaragi's promotion celebration. However, Mai gets caught up in a plot of the NOL's infamous Zero Squadron.