Mage Robes are a piece of armor dropped from the Necromancer Boss. Lich Robes are similar, dropped from the Lich Boss and Pig Mage Boss.
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Mage Robes and Lich Robes are worn in the chestplate slot. They provide very little protection. They do something special however: while equipped, they provide the player with passive hunger regen.
The purpose of this is to make up for hunger lost through use of the various magical staffs. The staffs have infinite durability, but use up the player's hunger bar as a form of "mana". Mage Robes and Lich Robes regenerate the hunger bar as a form of "mana regen".
Both have a durability of 850 and can be enchanted just like normal chestplates.
Unconfirmed: the Lich Robes provide faster hunger regen than the Mage Robes.