Rifts are represented in the game as floating black orbs. They are boss battles from an alternate time or another world. Oliver is requested to close several rifts in a post-completion questline by The Conductor. These rifts exist in all places where bosses were fought in the game though these versions of the bosses previously fought is more difficult than the original.
When Oliver approaches a Rift, he will get the action exclamation point over his head. Interact with the Rift. Like all Bounty Hunts, you have the choice to say no to the battle. Saving before each Rift is sealed is recommended.