The central of the three kingdoms created by the 843 Treaty of Verdun, awarded to Emperor Lothair I: an elongated realm running from Frisia through the Rhineland, Burgundy and Provence to Rome, holding both the Carolingian capital Aachen and the imperial title.
Split among Lothair I's three sons at his death in 855 into Lotharingia, the (Carolingian) Kingdom of Italy, and the Kingdom of Provence (Lower Burgundy).
No dedicated monograph on Middle Francia specifically was found on Open Library; this is the standard academic synthesis covering the 843/855 Carolingian partitions, incl. Lothair I's kingdom, in detail.