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Summary
A coloured illustration on page 8 of Sword Art Online Volume 10. It shows the map of the Norlangarth North Empire, with Central Centoria, in the Human Empire in Underworld.

Latest image source: Ruranobe

Sword Art Online Volume 11, Yen Press release
Dark Territory

The End Mountains

Rulid Village

Walde Farm

Forest

Zakkaria

Forest

Castle

River

Lake

Town

Road

Forest

Lake

Town

Field

Centoria

Everlasting Wall (a massive wall made of marble) - 750 km

The North Centoria Imperial Swordcraft Academy and the Integrity Knights
Within the Underworld, Centoria is the capital city, located directly in the center of the human realm. Over 20,000 citizens reside within its perfectly circular walls, encompassing a diameter of ten kilors. Within that circle, barriers known as Everlasting Walls split the city into four parts in an unusual X-shaped configuration. The four quadrants are called North Centoria, East Centoria, South Centoria, and West Centoria, and they serve as the capitals of the four empires that preside over the vast human realm.

At the very center of the city is the Axiom Church's massive, white Central Cathedral tower. Its pinnacle is so high that it can barely be seen, and looming walls hide the square grounds of the church from sight. It is from this cathedral that that the Everlasting Walls splitting the city spread outward. The Axiom Church is the overriding organization presiding over all of humanity. Its military officials known as Integrity Knights are tasked with maintaining order and serve as inspiration to the training swordfighters who look up to them.

In order to enroll at the North Centoria Imperial Swordcraft Academy and continue along the path to becoming an Integrity Knight, students must first pass the entrance exam and become primary trainees. Trainees work hard from dusk to dawn for an entire year, then take an advancement test to reach the next level. All students strive to qualify for the Imperial Battle Tournament, but Kirito and Eugeo want to be glorious Integrity Knights, which necessitates winning the Four-Empire Unification Tournament, the very highest competition of swordsmanship in the human empire.