Persona Fragments are items in the Persona series.
Persona 4 Arena UltimaxEdit
During the P-1 Grand Prix in Persona 4 Arena, the Malevolent Entity realized that if he had succeeded in taking the Shadow Selves of the Investigation Team or the Shadow Operatives their ego would have overpowered him. As part of his plan in Ultimax, he set up a new tournament where the battles between the fake Shadow Selves and the Persona users would lead to fragments being carved off the Personas who battle in the red fog, which also leaves the winner of the battle fatigued. By using the Persona Fragments that gathered at the top of the tower, he would be able to suppress the shadow's ego in order to gain control of the body that would born when the shadows merge to summon a god.
The power of the Sagiri, which remained in Tohru Adachi after being possessed by Ameno-sagiri, is similar. Minazuki sought to use the Sagiri fragment to suppress the Malevolent Entity. The full explanation is given in Adachi's story, where Sho mentions that by taking Adachi's Persona Magatsu-Izanagi, forcibly turning it back into a shadow, and then making it the center of the fusion the power would conflict with the Malevolent Entity and would weaken it.
This would allow Sho to become capable of taming the vessel and having his wish of being the only person left in the world fulfilled. This plan ultimately failed due to Yu's untimely arrival at the top and defeating Minazuki. Adachi also plotted on destroying the collected fragments at the top, but his plan failed and he was beaten by Sho. Realizing the power is in his persona, Adachi fights ME to make use of the remaining Sagiri power as it would weaken Kagutsuchi, but it was still not enough to defeat him.