After some intense battles in the Augusta Towers, you may be relieved to note that this section in Academia keeps a somewhat more leisurely pace. There are no monsters here, although there are plenty of side objectives that you can complete if you’re so inclined. First things first, though, you should head to the Academy Headquarters located on the northern side of the map. Ride a moving walkway and head over to Chocolina to examine the terminal next to her. You will receive a Primer on the Thirteenth Ark and view a cutscene.
After watching the cutscene, proceed onwards to the north. The two automated walkways on the left and right will get you there in no time flat. From the end of the walkway, the Academy Headquarters is just a short walk away. Proceed inside and find the NPC named Dr. M. Speaking with Dr. M will initiate the Academic Rank: Monster Professor side quest, which is a sprawling and massive quest that you cannot possibly complete just yet. This quest basically tasks you with killing at least one of every single enemy in the game. If you choose to undertake this quest, you should wait until the endgame, as many monsters aren’t even available for you to battle yet.
You can also access a terminal in this area to play the Brain Blast quiz game. These terminals are scattered throughout Academia, and will test your knowledge of some rather arbitrary pieces of Final Fantasy XIII lore. Completing a terminal successfully will reward you with a Fragment, some experience, and an item from an item box.
Once you’ve exhausted yourself with the various minigames and conversations, go through the door in the back of the area to find Hope. Speak with him, and then speak with Alyssa two times. They will tell you about the Graviton Cores, special Fragments that you must have in your inventory in order to continue the main quest. Speak to Hope several more times to trigger a Live Trigger Event. A chest will appear containing a reward for your performance on the walkway to the right.
Now it’s time to begin your search for the five Graviton Cores. Go into the entrance hallway and speak with the Receptionist to unlock entries in the Fragments menu. She will also give you a few clues to the locations of the Cores. Return to the Historia Crux.